Mathematician Cheryl Praeger has served the University of Western Australia as Lecturer, Senior Lecturer, and Professor since 1976. She leads a flourishing research group in pure mathematics and is in the top one per cent of highly cited mathematicians in the world.

Attendance is free.

Graham Farr (Monash)

will speak on

Algebraic properties of chromatic polynomials

at 1pm on Tuesday 7th of August in Maths Lecture Room 2

**Note this is the new regular seminar time for this semester**

Abstract: We give a survey of some recent work on algebraic properties of chromatic polynomials, including their roots (as algebraic numbers), factors and Galois groups. Collaborators: Adam Bohn (Queen Mary), Peter Cameron (Queen Mary), Daniel Delbourgo (Monash), Bill Jackson (Queen Mary), Kerri Morgan (Monash).

Come and find out about the courses on offer, career options, scholarship opportunities, our valuable research, community programs and facilities.

There's also residential college tours, hands-on activities, live music and entertainment, and plenty of fun activities for the whole family.

When: Tuesday 14th August 2012, 12pm to 1.30pm

Where: Science Library – 3rd Floor Seminar Room

'What Matters to me and why' is a series of lunch time talks and conversations with UWA Academics. The talks explore personal stories of family, place, formative influences and how these things continue to shape people's lives and academic work.

The next conversation is with Cheryl Praeger, who is the Director of the Centre for the Mathematics of Symmetry and Computation at UWA.

Cheryl will share some of her story and then there will be the opportunity for questions/conversation. BYO lunch. Tea/Coffee is available in the meeting room (at the request of the Science Library, please do not carry coffee through the library).

The Science Library is towards the southern end of the campus just past the Chemistry and Psychology buildings.

Sylvia Morris (UWA)

will speak on

Spreads of symplectic spaces of small order

at 1pm on Tuesday 14th of August in MLR2

Abstract: Spreads of symplectic spaces are used to construct translation planes, Kerdock codes and mutually unbiased bases. Several families of infinite symplectic spreads are known but these are far from covering all symplectic spreads. In particular, there is little known about symplectic spreads which create a non-semifield translation plane. For q=2 there is a unique spread of W(5,q) and for q=3 the symplectic spreads have been classified by Dempwolff. For q=4 there is a connection between symplectic spreads and the unique ovoid of Q^+(7,4). I have been using linear programming methods to find spreads in W(5,4) and W(5,5) which have non-trivial stabiliser. I will present my methods and results thus far, focussing on some interesting new examples of non-semifield symplectic spreads and their stabilisers.

Pablo Spiga (University of Milano-Bicocca)

will speak on

Compositions of n and an application to the covering number of the symmetric groups

at 1pm in MLR2 on Tuesday 21st of August

Abstract: Given a positive integer n, a k-composition of n is an ordered sequence of k positive integers summing up to n. In this short talk, we are interested on the number of k-compositions satisfying some "coprimeness" condition. As an application we give a Classification-free proof of some results on the covering number of the symmetric group.

All welcome

BIll Smyth (McMaster University/Kings College London/UWA)

will speak on

Are Three Squares Impossible?

at 1pm Tuesday 28th of August in MLR2.

Abstract: This talk describes work done over the last 30 years or so both to understand and to compute repetitions in strings -- especially since 1999. We will discover that, although much has been learned, much combinatorial insight gained, there remains much more that is unknown about the occurrence of repetitions in strings and the restrictions they are subject to. I present combinatorial results discovered only recently, and I suggest that possibly extensions of these results can be used to compute repetitions in an entirely new way. I hope that members of the audience will be motivated to work on some of the many open problems that remain, thus to extend combinatorial knowledge even further.

All welcome

Irene Pivotto (UWA)

will speak on

Packing Steiner trees

at 1pm Tuesday 2nd of October in MLR2

Abstract: A classic theorem of Nash-Wiliams and Tutte gives necessary and sufficient conditions for a graph to have k pairwise edge-disjoint spanning trees. We will discuss the natural generalization of this problem to trees spanning a distinguished set of vertices (which we refer to as Steiner trees). Finding edge-disjoint spanning trees is a considerably easier problem that finding edge-disjoint Steiner trees. This is due to the fact that spanning trees are bases of the natural matroid associated with a graph, while Steiner trees are not bases of any matroid. We will present a result that provides sufficient conditions for the existence of k edge-disjoint Steiner trees, reducing this problem to finding disjoint bases of a particular matroid. No prior knowledge of matroid theory is required to attend the talk.

Michael Giudici (UWA)

will speak on

Commuting graphs of groups

at 1pm Tuesday 9th of October in MLR2

Abstract: The commuting graph of a group G is the graph whose vertices are the noncentral elements of G and two vertices are adjacent if and only if they commute. Iranmanesh and Jafarzadeh conjectured that the commuting graph of a finite group is either disconnected or has diameter bounded above by some constant. I will discuss recent joint work with Chris Parker on this conjecture.

The Poincare conjecture was one of the most celebrated questions in mathematics. It was amongst the seven millennium problems of the Clay Institute, for which a prize of $1million was offered.

The Poincare conjecture asked whether a 3-dimensional space with `no holes’ is equivalent to the 3-dimensional sphere.

In 2003 Grigori Perelman posted three papers on the internet ArXiv outlining a marvellous solution to the Poincare conjecture, as part of the completion of Thurston’s geometrisation program for all 3-dimensional spaces. Perelman introduced powerful new techniques into Richard Hamilton’s Ricci flow, which `improves’ the shape of a space. Starting with any shape of a space with no holes, Perelman was able to flow the space until it became round and therefore verified it was a sphere.

A brief history of the Poincare conjecture and Thurston’s revolutionary ideas will be given. Hamilton’s Ricci flow will be illustrated.

Famously, Perelman turned down both the Clay prize and a Field’s medal for his work.

Cost: Free. RSVP to ias@uwa.edu.au

Cai Heng Li (UWA)

will speak on

Finite meta-primitive permutation groups

at 1pm Tuesday 16th of October in MLR2

Abstract: A transitive permutation group is called meta-primitive if its any imprimitive quotient action is primitive, namely, each of the block systems is maximal. I will discuss the structural properties of meta-primitive groups.

All welcome.

Arun Ram (University of Melbourne)

will speak on

Clifford theory and Hecke algebras

at 1pm on Tuesday the 23rd of October in MLR2

Abstract: The usual Clifford theory describes the irreducible representations of group G in terms of those of a normal subgroup. Generalizing, Clifford theory constructs the irreducible representations of semidirect product rings and invariant rings. In this work with Z. Daugherty we use Clifford theory to index the irreducible representations of two pole Hecke algebras and relate this indexing to a labeling coming from statistical mechanics (following work of de Gier and Nichols) and to a geometric labeling (coming from K-theory of Steinberg varieties following Kazhdan-Lusztig). Despite the maths-physics and geometric motivations for the project, in the talk I shall assume only that the audience is familiar with the notions of groups, rings, and modules.

All welcome

Wei Jin (UWA)

will speak on

Finite s-Geodesic Transitive Graphs

at 1pm on Tuesday 20th of November in Maths Lecture Room 2

Abstract: A geodesic from a vertex u to a vertex v in a graph is one of the shortest paths from u to v, and this geodesic is called an s-geodesic if the distance between u and v is s.

A graph is said to be s-geodesic transitive if, for each i less than or equal to s, all i-geodesics are equivalent under the group of graph automorphisms. In this talk, I will show the relationship of 2-geodesic transitive graphs with a certain family of partial linear spaces. I will also compare s-geodesic transitivity of graphs with two other well-known transitivity properties, namely s-arc transitivity and s-distance transitivity.

This is a joint work with my supervisors.

All welcome

Bojan Kuzma (University of Primorska, Slovenia)

will speak on

Graphs and general preservers of zero products

at 1pm on Tuesday 27th of November in MLR2

Abstract: We survey some results in preserver problems where graphs were used as the main tool. In particular, the classification of maps which preserve Jordan orthogonality (AB+BA=0) reduces to the fact that a certain graph is a core and has chromatic number 4. We also give a classification of certain matrices (rank-ones, semisimple, non-derogatory) in terms of a commuting graph.

All welcome.

Neil Gillespie (UWA)

will speak on

Completely regular codes with large minimum distance

and

Daniel Hawtin (UWA) will speak on Elusive Codes in Hamming Graphs

at 1pm Tuesday 4th of December in MLR2

Abstracts:

Completely regular codes with large minimum distance: In 1973 Delsarte introduced completely regular codes as a generalisation of perfect codes. Not only are completely regular codes of interest to coding theorists due to their nice regularity properties, but they also characterise certain families of distance regular graphs. Although no complete classification of these codes is known, there have been several attempts to classify various subfamilies. For example, Borges, Rifa and Zinoviev classified all binary non-antipodal completely regular codes. Similarly, in joint work with Praeger, we characterised particular families of completely regular codes by their length and minimum distance, and additionally with Giudici, we also classified a family of completely transitive codes, which are necessarily completely regular. In this work with Praeger, and also with Giudici, the classification given by Borges, Rifa and Zinoviev was critical to the final result. However, recently Rifa and Zinoviev constructed an infinite family of non-antipodal completely regular codes that does not appear in their classification. This, in particular, led to a degree of uncertainty about the results with Praeger and with Giudici. In this talk I demonstrate how I overcame this uncertainty by classifying all binary completely regular codes of length m and minimum distance $ elta$ such that $ elta>m/2$.

Elusive Codes in Hamming Graphs:

We consider a code to be a subset of the vertex set of a Hamming graph. We examine elusive pairs, code-group pairs where the code is not determined by knowledge of its set of neighbours. We provide an infinite family of elusive pairs, where the group in question acts transitively on the set of neighbours of the code. In our examples, we find that the alphabet size always divides the length of the code, and prove that there is no elusive pair for the smallest set of parameters for which this is not the case.

Padraig Ó Catháin (The University of Queensland)

will speak on

Algebraic aspects of Hadamard matrices

at 3pm Friday 22nd of February in MLR2.

Abstract:

Hadamard matrices have applications in the design of experiments, signal processing, coding theory and many other areas. They have been extensively studied for many years, and are known to be closely related to symmetric designs with certain parameters. Many constructions for Hadamard matrices are known. Some are combinatorial in nature, others make use of finite fields and tools from abstract algebra.

In this talk I will give an introduction to Hadamard matrices, their automorphism groups, and their relations to other combinatorial objects. As a corollary of the classification of finite doubly transitive permutation groups, a classification of 'highly symmetric' Hadamard matrices is obtained. I will also look at the problem of constructing Hadamard matrices with primitive automorphism groups.

In 1961 G.E. Wall conjectured that the number of maximal subgroups of a finite group is less than the order of the group. The conjecture holds for all finite solvable groups (proved by Wall himself in his original paper) and holds for almost all finite simple groups, possibly all of them (proved by Liebeck, Pyber and Shalev in 2007). It is now known to be false in general, at least as originally stated, with infinitely many negative composite group examples found through a combination of computational and theoretical techniques. (I cite in particular computer calculations of Frank Luebeck, as partly inspired and later confirmed by calculations of my undergraduate student, Tim Sprowl, with theoretical input from myself and Bob Guralnick.) In this talk I will try to discuss the ingredients in this quite remarkable story, and I will mention as much of the legacy of positive consequences as time permits.

will speak on

Control of fusions in fusion systems and applications

at 1pm on Tuesday 5th of March, in MLR2

Abstract:

Fusion systems were introduced by L. Puig in early 1990's mainly for the purpose of block theory. Fusion systems are also of interest in homotopy theory. In this talk we will define a new control of fusion in fusion systems and apply it to the study of maximal Sylow intersections.

will speak on

Generalised n-gons and the Feit-Higman theorem

at 3pm on Friday 8th of March.

Abstract:

Jacques Tits' theory of buildings played a vital role in the proof of the classification theorem on finite simple groups. The class of rank 2 buildings are also known as generalised n-gons.

In my talk, generalised n-gons will be defined as a certain class of bipartite graphs, so as to skip the (rather abstruse) building-theoretic definition. I will also state and outline a proof of the Feit-Higman theorem, which states that the majority of generalised n-gons can only exist for certain n. The proof, due to Kilmoyer and Solomon (1973), weaves together representation theory and graph theory.

To finish off, I will talk a little about what I've been doing here at UWA.

Erdös-Ko-Rado sets (EKR sets) are a family of k-sets of { 1, ..., n } that pairwise intersect in at least one element and were first studied by Erdös, Ko, and Rado in 1961. There are several generalizations of EKR sets. The speaker's main interest is study of EKR sets in polar spaces. These are sets of generators (maximal totally isotropic subspaces) that pairwise intersect in at least a point and were recently studied by Valentina Pepe, Leo Storme, and Frédéric Vanhove. After introducing EKR sets for sets, projective spaces, and polar spaces, some specific results using algebraic as well as geometric techniques will be presented.

Let G be a group, let H be a subgroup of G and let V be an irreducible KG-module over a field K. We say that (G,H,V) is an irreducible triple if V is an irreducible KH-module. Classifying the irreducible triples of a group is a fundamental problem in representation theory, with a long history and several applications.

The case where G is a simple algebraic group over an algebraically closed field can be traced back to work of Dynkin in the 1950s (H connected, char(K) = 0). Through work of Seitz and Testerman in the 1980s, and more recent work of Ghandour, the problem of determining the irreducible triples (G,H,V) for simple algebraic groups has essentially been reduced to the case where G is a classical group and H is disconnected.

In this talk I will report on recent work that determines all the irreducible triples (G,H,V) when G is classical and H is a disconnected, infinite, maximal subgroup. This is an important step towards a complete classification of the irreducible triples for simple algebraic groups. I will briefly recall some of the basic results on algebraic groups and representation theory that we will need, and I will describe some of the main ideas that are used in the proofs.

This is joint work with Soumaia Ghandour, Claude Marion and Donna Testerman.

will speak on

Bacterial genome evolution with algebra

at 2pm Thursday 28th of March in Blakers Lecture Theatre

NOTE CHANGE OF DAY, TIME AND VENUE

Abstract:

The genome of a bacterial organism consists of a single circular chromosome that can undergo changes at several different levels. There is the very local level of errors that are introduced through the replication process, giving rise to changes in the nucleotide sequence (A,C,G,T); there are larger scale sequence changes occurring during the lifetime of the cell that are able to insert whole segments of foreign DNA, delete segments, or invert segments (among other things); and there are even topological changes that give rise to knotting in DNA.

Algebra might be defined as the study of ``sets with structure", and has been used over the past century to describe the symmetries of nature, most especially in areas like physics and crystallography, but it also plays a role in technological problems such a cryptography. In this talk I will describe how algebraic ideas can be used to model some bacterial evolutionary processes. In particular I will give an example in which modelling the inversion process gives rise to new algebraic questions, and show how algebraic results about the affine symmetric group can be used to calculate the ``inversion distance" between bacterial genomes. This has applications to phylogeny reconstruction.

All welcome.

L. Redei has studied in a detailed way so-called "lacunary" polynomials over finite fields. One of the applications described is to investigate the number of values the difference quotient of a polynomial over a finite field can have. This result has a direct implication in the theory of blocking sets of finite Desarguesian projective planes, and this connection is the start of the use of "Redei-polynomials" in finite geometry. We will discuss some cases to explain the principle of using Redei-polynomials finite projective spaces and some particular generalized quadrangle. Then we discuss a problem on maximal partial ovoids, that has been partially solved using Redei-polynomials, but that can be expressed in terms of transitive subsets of the group SL(2,q).

For any class of graphs, the growth function h(n) of the class is defined to be the maximum number of edges in a graph in the class on n vertices. The Erdos-Stone Theorem remarkably states that, for any class of graphs that is closed under taking subgraphs, the asymptotic behaviour of h(n) can (almost) be precisely determined just by the minimum chromatic number of a graph not in the class. I will present a surprising version of this theorem for finite geometries, obtained in joint work with Jim Geelen. This result is a corollary of the famous Density Hales-Jewett Theorem of Furstenberg and Katznelson.

Matroids are combinatorial structures that generalize graphs as well as configurations of points in projective space. They consist of a finite ground set E and a set of subsets of E called B, such that B satisfies certain axioms. We consider the problem of bounding the number m_n of matroids on a fixed ground set of size n. In 1973, Piff showed that log log m_n < n- log n + O(log log n) In 1974, Knuth gave a lower bound of log log m_n > n- (3/2) log n + (1/2) log(2/pi) - o(1) In the talk, I will present a recent result with Nikhil Bansal and Jorn van der Pol, that log log m_n < n- (3/2) log n + (1/2) log(2/pi) + 1 + o(1)

An arc in a graph is an unordered pair of adjacent vertices. A graph is called arc-transitive if its automorphism group acts transitively on its set of arcs. We consider the problem of bounding the size of the automorphism group of an arc-transitive graph in terms of its order. We consider the impact of the local action on this problem. (The local action is the permutation group induced by the action of the stabiliser of a vertex on its neighbourhood.) This is joint work with Primož Potocnik and Pablo Spiga.

The normal quotient method has made the study of certain families of finite graphs (for instance, s-arc transitive and locally s-arc transitive graphs) more approachable by dividing the problem into two parts: (I) Study the "basic graphs," those graphs in the family that are not covers of anything but "trivial" graphs; (II) Study the regular covers of the basic graphs. While (I) has been studied extensively, far less work has been done toward (II). In this talk, I will discuss how voltage graphs can be used to find regular covers of graphs where certain symmetries lift, and specifically look at the problem of determining the locally s-arc transitive regular covers of complete bipartite graphs. No previous knowledge of any of these topics will be assumed.

This talk is a survey of one of the driving topics in finite geometry, and the connections that ovoids and spreads have to other areas of finite geometry and permutation groups. Apart from a presentation of the history of the field and the main open problems, the speaker will give an overview of his most recent collaboration with Ferdinand Ihringer and Jan De Beule on ovoids of Hermitian polar spaces.

One of the earliest triumphs in applying the finite simple group classification in algebraic graph theory was the characterization of finite distance transitive graphs. Recent work by Devillers, Giudici, Li and myself focuses on a generalisation of this class of graphs: locally $s$-distance transitive graphs where ordered vertex-pairs, with given first vertex and at a given distance at most $s$, are equivalent under automorphisms. One basic type of example is closely linked with the existence of very symmetrical point-line incidence structures which we call pairwise transitive designs. I will trace these developments and their links.

The speaker will give the second half of his talk from a fortnight ago, but with a different context. We will first go through some background on the basics of algebraic graph theory, eigenvalue techniques, and strongly regular graphs, before giving some short non-existence proofs of ovoids and spreads in particular polar spaces.

Given an integer k and a graph L, a (k,L)-complex is a polygonal complex consisting of vertices, edges and faces such that each face is a regular k-gon and the graph induced by the edges and faces at each vertex is isomorphic to L. Many questions about the existence and uniquenes of (k,L)-complexes and about their automorphism groups can be converted to questions about automorphisms of the graph L. I will discuss some of these ideas and report on work initiated at last year's retreat.

Merino and Welsh conjectured that for any loopless bridgeless graph, the number of spanning trees is dominated either by the number of acyclic orientations of the graph, or by the number of totally cyclic orientations of the graph.

In this talk, I will discuss this conjecture and the partial results that have been previously obtained, along with a proof (which is joint work with Steve Noble) that the conjecture is true for series-parallel graphs.

The fundamental goal of Computational Group Theory is recognition: given a group somehow represented in a computer (usually by generators), determine any information about the group that you can, as quickly as possible. This is `easy' if we have enough time and memory to write the group in its entirety, but when the group is input as a generating set of matrices, this is impractical. We are explorers in an unknown land, attempting to determine the global structure of our setting from limited, local information.

The Matrix Group Recognition Project provides a recursive framework for full recognition of an arbitrary input group, by searching for a normal subgroup $N$ of $G$ and dealing with $N$ and $G/N$ separately. We present a recognition algorithm for some of the `base cases' of this recursion process, namely certain irreducible representations of a classical group, for which general methods are least effective and specialised algorithms are needed for efficiency.

--Talk 1--

Mark Ioppolo

will speak on

Symmetry in coding theory: Constructing error control codes with group theory

Abstract:

When data is transmitted over a noisy communication channel there is a possibility that the received message will be different to what the sender intended. A frequently made assumption in coding theory is that the probability of an error occurring does not depend on the position of the error in the codeword, or on the value of the error. The group theoretic analogue of this assumption is known as neighbour-transitivity. This talk will introduce the study of neighbour-transitive codes, focusing on the case where the automorphism group of the code in question is contained in a group of symplectic matrices.

--Talk 2--

David Raithel

will speak on

Structures of Symmetries

Abstract:

Permutation groups are the tools with which we understand and study symmetry. For over a century, mathematicians have been endeavouring to classify classes of permutation groups, which in turn classifies classes of symmetries. Transitive groups lie at the heart of permutation groups, and one of the large overarching themes of permutation group theory has been to characterise transitive groups. In this talk I shall outline four major structure theorems from as early as 1911 to as recently as 2004. These theorems have proven to be powerful tools which have allowed group theorists to sledgehammer their way through some otherwise insurmountable problems.

--Talk 3--

Daniel Hawtin

will speak on

Affine Elusive Codes

Abstract:

An Elusive pair $(C,X)$ is a code-group pair where $X$ fixes the neighbour set setwise, and contains an automorphism which does not fix $C$ setwise. This implies that there are multiple codes, each with the same neighbour set. The concept was introduced by Gillespie and Praeger in order to discern the correct definition for neighbour transitive codes. We discuss a family of exmples which are as large as possible, in some sense, and display properties which previous examples have not.

A module M is called pseudo-injective if for every submodule X of M, any monomorphism f : X -------> M can be extended to a homomorphism g : M -------> M. Let K be a field and G a group. It is well known that a group algebra K[G] is self-injective if and only if the group G is finite. We show that if a group algebra K[G] is pseudo-injective then G is locally finite. It is also shown that if a group algebra K[G] has no non-trivial idempotent then K[G] is pseudo-injective if and only if it is self-injective. Furthermore, if K[G] is pseudo-injective then K[H] is pseudo-injective for every subgroup H of G.

A maximal n-arc is a set of q(n-1) + n points in a projective plane such that any line of the plane meets 0 or n of them. The most common maximal arcs, called the Mathon arcs, are constructed by taking the union of regular hyperovals. In particular, there are no known maximal 4-arcs other than the Mathon arcs and their duals. This talk will cover the history of maximal arcs, including the construction of the Mathon arcs, as well as new results. The most important result is that every maximal 4-arc in PG(2,q), that is a union of regular hyperovals, is a Mathon arc. This is joint work with Nicola Durante from Universita' degli Studi di Napoli Federico II.

A graph is called X-semisymmetric if the group X acts transitively on edges, but not on vertices. Such graphs are necessarily bipartite and bi-regular, of valencies k and l say. There is a natural relationship between semisymmetric graphs and amalgams of groups. This leads us to consider the "universal" example: the bi-regular tree T of valencies k and l. The conjecture of Goldschmidt says that when k and l are primes there are (up to isomorphism) finitely many locally finite groups X such that T is X-semisymmetric (X is locally finite if it acts with finite vertex stabilisers). For k=l=3 it was shown by Goldschmidt that there are 15 such groups. In the talk I will give an overview of some results which bear relevance to the conjecture, and report on some recent progress with respect to certain small primes.

There's also residential college tours, hands-on activities, live music, entertainment, and plenty of fun activities for the whole family as we celebrate our 100th birthday.

One motivation for this talk comes from representation theory: decomposing a tensor product of irreducible (or indecomposable) representations as a sum of smaller degree irreducible (or indecomposable) representations. Other motivations come from quantum mechanics and Frobenius algebras.

Consider an $r imes r$ matrix $K_r$ over a field $F$ with 1s on the main diagonal and first upper diagonal (positions $(i,i)$ and $(i,i+1)$) and zeros elsewhere. The tensor product $K_r times K_s$ is a unipotent matrix whose Jordan canonical form is determined by some partition of $rs$. We will show that this partition enjoys surprising symmetries: duality, periodicity, regularity. Our original motivation was to study this partition when the characteristic $p$ of $F$ is small (i.e. $p<r+s-1$). The large characteristic case ($p e r+s-1$) was solved recently by Iima and Iwamatsu.

This is joint work with Cheryl E. Praeger and Binzhou Xia.

Symmetry breaking involves colouring the elements of a combinatorial structure so that the resulting structure has no nontrivial symmetries. In this talk I'll give an introduction to symmetry breaking, with a particular focus on infinite graphs. I'll also discuss a number of research directions that are opening up. Along the way I'll highlight some of the interesting open questions and conjectures that are being worked upon.

Many of these problems relate to very deep problems in group theory, but I'll try to make the group theory as accessible as possible.

In this talk, I will give an introduction to decomposition theory of 3-connected matroids. In order to make it more accessible I will also introduce some related definitions and examples.

A string or word is usually thought of as a sequence of letters drawn from some alphabet. Applications to bioinformatics and other areas suggest the utility of defining strings on subsets of the alphabet instead -- so-called "indeterminate" strings. I describe recent work that connects such strings to ideas from graph theory, and wonder if graph theoretical concepts and knowledge might be still further applied to their analysis and use.

In this talk, we will consider the non-commuting graph of a non-abelian finite group G; its vertex set is the set of non-central elements of G, and two distinct vertices x and y are joined by an edge if they do not commute together. Actually, we study some properties of the non-commuting graph such as connectivity, regularity, etc., and we show that, for many groups G, if H is a group which has the same non-commuting graph of G, then they have the same order. We determine the structure of any finite non-abelian group G (up to isomorphism) for which its non-commuting graph is a complete multipartite graph. We also show that a non-commuting graph is a strongly regular graph if and only if it is a complete multipartite graph.

This is just a survey talk of various ways to construct all of the known finite generalised quadrangles starting with a group and a configuration of subgroups of that group. In particular, the speaker will give a summary of where one of the "retreat" problems is at.

will speak on

The Cohen-Lenstra heuristics: from arithmetic to topology and back again.

at 11am in the Science Library Access Grid room.

I will discuss some models of what a "random abelian group" is, and some conjectures (the Cohen-Lenstra heuristics of the title) about how they show up in number theory. I'll then discuss the function field setting and a proof of these heuristics, with Ellenberg and Westerland. The proof is an example of a link between analytic number theory and certain classes of results in algebraic topology ("homological stability").

Straight-line programs offer a method for encoding group computations in a "black box" sense, namely without using specifics of the group's representation or how the group operations are performed. We advocate that straight-line programs designed for group computations should be accompanied by comprehensive complexity analyses that take into account not only the number of group operations needed, but also memory requirements arising during evaluation. We introduce an approach for formalising this idea and discuss a fundamental example for which our methods can drastically improve upon existing implementations. This is joint work (in progress!) with Alice Niemeyer and Cheryl Praeger.

We study the problem of covering Euclidean space R^d by possibly overlapping translates of a convex body P, such that almost every point is covered exactly k times, for a fixed integer k. Such a covering of Euclidean space by translations is called a k-tiling. We will first give a historical survey that includes the investigations of classical tilings by translations (which we call 1-tilings in this context). They began with the work of the famous crystallographer Fedorov and with the work of Minkowski, who founded the Geometry of Numbers. Some 50 years later Venkov and McMullen gave a complete characterization of all convex objects that 1-tile Euclidean space.

Today we know that k-tilings can be tackled by methods from Fourier analysis, though some of their aspects can be studied using purely combinatorial means. For many of our results, there is both a combinatorial proof and a Harmonic analysis proof. For k larger than 1, the collection of convex objects that k-tile is much wider than the collection of objects that 1-tile, and there is currently no complete knowledge of the polytopes that k-tile, even in 2 dimensions. We will cover both ``ancient'', as well as very recent, results concerning 1-tilings and more generally k-tilings. These results are joint work with Nick Gravin, Mihalis Kolountzakis, and Dmitry Shiryaev.

Given a finite group G and a faithful irreducible FG-module V where F is a field of prime order, we can ask whether G has a regular orbit on the vectors of V. This problem is related to determining which primitive permutation groups of affine type have a base of size 2, as well as the famous k(GV)-problem and a conjecture of Brauer concerning defect groups of blocks. We will consider the regular orbit problem for the symmetric and alternating groups.

Let H be a finite linear group acting completely reducibly on a finite vector space V. Gabriel Navarro asked: if the H-orbits containing vectors a and b have coprime lengths m and n, is there an H-orbit of length mn? We answered, by showing that the H-orbit containing a + b has length mn, and by showing, moreover, that in this situation H cannot be irreducible. That is to say, a stabiliser in an affine primitive permutation group does not have a pair of orbits of coprime lengths. I will make some comments, if time permits, about coprime orbit lengths for stabilisers in arbitrary primitive permutation groups. This is joint work with Silvio Dolfi, Bob Guralnick and Pablo Spiga.

Codes arising from algebraic geometry, first introduced by Goppa, gained attention when Tsfasman–Vladut–Zink used them to improve the Gilbert-Varshamow bound. We will give a gentle introduction to some of the beautiful ideas from algebraic geometry used to build these codes. We will then show how to construct them, and then discuss the Tsfasman–Vladut–Zink bound. There will be an emphasis on examples.

I will discuss a few things, all related to semiregular graph automorphisms : the polycirculant conjecture, the abelian normal quotient method, an interesting class of graphs...

More recently biologists have used them as models of DNA folding, and to model experiments in which biological molecules are pulled from a surface. I will describe the rather short list of rigorous results, the longer list of what we "know" to be true but can't prove, and describe some numerical results that are of interest in applications. No prior knowledge is assumed.

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Analysis and Implementation: the Two 'Editions' of a Matrix Group Algorithm

at 3pm Friday January the 10th in Blakers Lecture Theatre.

Abstract :

The quality of an algorithm in computational mathematics is represented by two separate, yet equally important measures: the theoretical analyses which measure the runtime's growth for large input, and the implementations whose runtimes we can measure in seconds. This leads to different aspects of algorithm design being prioritised in different settings, and often two very different algorithms are produced: one is described in a journal article analysing the worst-case complexity, and a very different procedure is implemented in practice.

In this talk I present the motivation, overall structure, and details of a reduction algorithm for specific irreducible modules of a classical group G, and discuss issues specific to the implementation of the algorithm in the Magma computer algebra system.

This is joint work with Cheryl Praeger and Akos Seress, with special thanks to Eamonn O'Brien.

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Graphs and transitivity on 2-geodesics

at 3pm Friday January the 17th in Blakers Lecture Theatre.

Abstract :

Joint work with Wei Jin, Cai Heng Li, Cheryl Praeger, Akos Seress.

An s-geodesic in a graph is a shortest path connecting two vertices at distance s. We say that a graph is locally transitive on s-geodsics if the stabiliser of any vertex is transitive on the s-geodesics starting at that vertex. Being locally transitive on s-geodesic is not a monotone property: if an automorphism group G of a graph is locally transitive on s-geodesics, it does not follow that G is locally transitive on shorter geodesics. For instance, (local) transitivity on 2-geodesics does not imply local transitivity on arcs (1-geodesics).

In this talk, I will first show a nice characterisation of all graphs that are locally transitive on 2-geodesics, but not locally transitive on 1-geodesics.

Then I will describe graphs that are (locally) transitive on 2-geodesics and on arcs, in terms of their local structure.

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An introduction to IPE

at 3pm Friday January the 24th in Blakers Lecture Theatre.

Abstract:

IPE is a drawing editor for creating figures in PDF or EPS format, which can then be inserted into LaTeX documents. I will show the basic features of IPE, as well as some of the more advanced ones.

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Groups and first-order logic

at 3pm Friday January the 31st in Blakers Lecture Theatre.

Abstract :

We study the expressive power of first-order logic for groups. A finitely generated group is called quasi-finitely axiomatizable if a single sentence characterizes it within the class of finitely generated groups. I showed in 2005 that, for instance, the Heisenberg group is quasi-finitely axiomatizable. Recent work of Lasserre provides new examples, such as the Thompson groups.

A group is homogeneous if the orbit of every tuple under the action of automorphisms is described by its first-order properties. I proved (J. Algebra, 2003) that the free group F_2 has this property. Recent work of Perrin and Sklinos (Duke Math. J. 2013) extends this to F_n for larger n.

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Row echelon matrices, flags and Grassmannians

at 3pm Friday February the 7th in Blakers Lecture Theatre.

Abstract:

There is a well trodden path from finite dimensional vector spaces to algebraic geometry. How much progress along this path is possible if fields are replaced by rings?

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Metrically homogeneous graphs

at 3pm Friday February the 21st in Blakers Lecture Theatre.

Abstract:

A `homogeneous structure' is a countably infinite relational structure M (e.g. graph, k-uniform hypergraph, digraph,...) with the property that every isomorphism between finite induced substructures of M extends to an automorphism of M. These are constructed by an amalgamation method developed by Fraisse (and independently Jonsson) in the 1950s. There are classification results, often very hard, in restricted contexts (e.g. graphs, partial orders, digraphs, totally ordered graphs) but in general classification seems out of reach.

I will discuss an attempt to classify `metrically homogeneous graphs', that is, countably infinite graphs M which become homogeneous when enriched by binary `distance relations' corresponding to graph-distance in M. This notion generalises distance transitivity for countably infinite graphs. Cherlin has produced a `catalogue' of metrically homogeneous graph and conjectures that it is complete. In joint work in progress with Amato and Cherlin, we verify the conjecture for metrically homogeneous graphs of diameter at most three.

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Subgroup covering numbers of symmetric groups

at 3pm Friday February the 28th in Weatherburn Lecture Theatre.

Abstract:

Let G be a group. The subgroup covering number of G is defined to be the least integer m such that G is equal to the set theoretic union of m proper subgroups of G. In 2005, Maroti determined the subgroup covering number for the symmetric group S_n, when n > 9 is odd, and he provided bounds for sufficiently large even values of n. I will discuss these previous results, joint work with Luise-Charlotte Kappe and Daniela Nikolova towards filling in the gap for small values of n, and ongoing work to determine the exact value for large even values of n.

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Factorizations of almost simple groups with a soluble factor

at 3pm Friday March the 7th in Weatherburn Lecture Theatre.

Abstract:

Factorizations of almost simple groups arise in many contexts. I will talk about the factorizations with one factor soluble, including construction of examples in classical groups and the classification result.

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The brilliant career of Frédéric Vanhove

at 3pm Friday March the 14th in Weatherburn Lecture Theatre.

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Have we ever tried to count Cayley graphs?

at 3pm Friday March the 21st in Weatherburn Lecture Theatre.

Abstract:

In this talk we give some elementary upper bounds on the number of finite Cayley graphs. The asymptotic number of Cayley graphs is much harder to pin down and we give a brief outline of the main technique and the main ingredients needed for this counting. On the way we leave some problems on oriented Cayley graphs and tournaments.

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Primitive groups, diophantine equations, and functional equations

at 3pm Friday March the 28th in Weatherburn Lecture Theatre.

Abstract:

I will explain how results about primitive groups play a crucial role in proving results about diophantine equations and functional equations. A sample application is that, for any polynomial f(X) with rational coefficients, the function Q-->Q defined by c --> f(c) is (<=6)-to-1 over all but finitely many values.

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Edge transitive dessins d'enfant

at 3pm Friday April the 4th in Weatherburn Lecture Theatre.

Abstract:

A 2-cell embedding of a bipartite graph in an orientable surface is called a dessin d'enfant. Thus a dessin d'enfant is an orientable bipartite map. I will present an explicit representation of an edge transitive dessin in terms of a group with two distinguished generators, and apply it to study the dessin.

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Graphs are to matroids what ribbon graphs are to ...?

at 3pm Friday April the 11th in Weatherburn Lecture Theatre.

Abstract:

Much of the combinatorial structure of an abstract connected graph is encoded in its set of spanning trees. These form one of the canonical examples of the bases of a matroid. Ribbon graphs contain extra topological information on the embedding of the underlying abstract graph in a surface. The structures playing the role of spanning trees are the subgraphs having one boundary component. The combinatorial structure which they form is a delta-matroid, which roughly speaking is like a matroid except that the bases do not need to have the same size. We will discuss three classes of delta-matroids, some natural operations on ribbon graphs which correspond to natural operations on delta-matroids, give an excluded minor theorem for one of the classes of delta-matroids and, time-permitting describe the Bollobas-Riordan polynomial which forms one way of generalizing the Tutte polynomial to ribbon graphs.

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Some recent results on elusive groups

at 3pm Friday May the 2nd in Weatherburn Lecture Theatre.

Abstract:

Every transitive permutation group has a derangement of prime power order but not necessarily a derangement of prime order. A transitive permutation group is called elusive if it has no derangements of prime order. I will talk about some recent results on this topic including elusive groups of automorphisms of graphs of small valency and a new construction of elusive groups.

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Locally-transitive graphs and their vertex stabilisers

at 3pm Friday May the 9th in Weatherburn Lecture Theatre.

Abstract:

Let D be a graph with a group of automorphisms G and suppose that D is G-locally-arc-transitive (for each vertex x of D the vertex stabiliser G(x) acts transitively on the neighbourhood of x). Fixing the valency of D one can ask if there is a bound on the order of G(x)? For valency three there are fundamental results due to Tutte and Goldschmidt. We'll instead fix the `local actions’, that is, the possible permutation groups induced on neighbourhoods in D (there can be at most two of these) and ask again: is there a bound on the order of G(x)? I will revisit some of the interesting results on this question. Then I will talk about some recent results like: when there can be a bound; when there is no bound; given a bound (and a little more) an instance when G(x) can be completely determined. Spread throughout this is joint work with Giudici, Giudici-Ivanov-Praeger and Spiga-Verret.

Gledden Visiting Fellow, Institute for Advanced Studies, UWA

and

School of Physics, Trinity College, Dublin, Ireland

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Classic problems of packing in 2d, 3d and on a cylinder

at 3pm Friday May the 16th in Weatherburn Lecture Theatre.

Abstract:

Some classic problems of optimal packing are reviewed: the Kepler Problem, the Kelvin Problem, and cylindrical packings of spheres and disks. The latter are tentatively associated with the celebrated phenomenon of spiral Phyllotaxis, much in evidence in the splendid gardens of UWA. If the speaker omits or forgets anything, it is to be found in the book Pursuit of Perfect Packing, T. Aste and D. Weaire, 2nd Edition.

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Chromatic roots of graphs and matroids

at 3pm Friday May the 23rd in Weatherburn Lecture Theatre.

Abstract:

The location of the real and/or complex roots of the chromatic polynomial of a graph has been studied for many years, both by combinatorial mathematicians and statistical physicists, yet despite this many fundamental questions remain unsolved. And even though the chromatic polynomial is most generally a matroidal concept, very little indeed is known about the location of chromatic roots of matroids that are not graphic or cographic. In this talk, I will present a necessarily-personal survey of the major results and my favourite open problems in this area.

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Taking products of permutation groups

at 3pm Friday May the 30th in Weatherburn Lecture Theatre.

Abstract:

At the AustMS conference, I introduced a new product for permutation groups. At the time, I was in a rush to announce the result because it could be used to solve an open question of P. E. Caprace and N. Monod. Since that talk, much has changed: I have an entirely new construction method that expands the scope of the construction, and I have extended many of the results describing its properties.

In this talk I'll introduce the product again, but this time in more detail. I'll sketch a proof of one of its most striking properties, and I'll talk about three areas of research in which the product has proved to be important (one of which is my DECRA project!).

The content of the talk will be largely combinatorial, and there will be no topology (although I may use the word "closed" accidentally, for which I hope I will be forgiven).

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Computing Kazhdan-Lusztig Polynomials and some Applications

at 3pm Thursday June the 5th in Maths Lecture Room 2.

Abstract:

My interest in Kazhdan-Lusztig polynomials comes from a certain character formula for reductive algebraic groups, which was first conjectured by Lusztig.

More generally, parabolic Kazhdan-Lusztig polynomials can be defined for arbitrary Coxeter groups. These polynomials are notoriously difficult to compute. I will report on an implementation of an algorithm to compute them, which goes much further than previous programs.

It turned out that some of the coefficients of Kazhdan-Lusztig polynomials that I was able to compute have interesting (and for me unexpected) interpretations in the context of conjectures by Guralnick (on a bound of the dimension of first cohomology groups for finite groups) and Wall (on the number of maximal subgroups of any finite group). I will also explain these conjectures and sketch the connection between these topics.

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Quasi-isometry and commensurability for right-angled Coxeter groups

at 3pm Friday 1 August in Weatherburn Lecture Theatre

Abstract: Let Gamma be a finite simple graph with vertex set S. The associated right-angled Coxeter group W_Gamma is the group with generating set S, and relations s^2 = 1 for all s in S and st = ts if and only if s and t are adjacent vertices. We investigate the classification of such W_Gamma up to quasi-isometry, which is a "coarse" equivalence relation on finitely generated groups formulated by Gromov, and also up to commensurability, where two groups G and H are commensurable if they have isomorphic finite index subgroups. Our methods are geometric and topological. This is joint work with Pallavi Dani (Louisiana State University) and Emily Stark (Tufts University).

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Codes from quadrics in symplectic space

at 3pm 8 August in Weatherburn LT

Abstract: Whether combinatorial or group theoretic, symmetry plays an important role in the construction and analysis of error correcting codes in graphs. A code in a graph is called neighbour-transitive if it admits a group of automorphisms which stabilises the codewords set-wise, while acting transitively on the codewords and code-neighbours.

I will discuss the work of Liebler and Praeger towards a classification of error correcting codes in Johnson graphs and present recent progress relating to families of codes associated with embeddings of hyperbolic and elliptic quadrics in binary symplectic space.

This year there will be campus tram tours, hands-on activities, live music and entertainment, as well as plenty of fun activities for the whole family to enjoy.

Join us for Open Day 2014 from 10.00am to 4.00pm on Sunday 10 August.

Joanna Fawcett (UWA)

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Locally triangular graphs and rectagraphs with symmetry

at 3pm on Friday 15 of August in Weatherburn LT

Abstract: Locally triangular graphs are known to be halved graphs of bipartite rectagraphs, which are connected triangle-free graphs in which every 2-arc lies in a unique quadrangle. A graph is locally rank 3 if it has a group of automorphisms G such that for each vertex u, the permutation group induced by the vertex stabiliser of u in G on the neighbourhood of u is transitive of rank 3. One natural place to seek locally rank 3 graphs is among locally triangular graphs. In this talk, we will discuss our classification of the connected locally triangular graphs that are also locally rank 3, which we obtain by classifying the locally 4-homogeneous rectagraphs (with some additional structure). This is joint work with J. Bamberg, A. Devillers and C. Praeger.

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Designs and Elusive Codes in Hamming Graphs

at 3pm Friday August the 22nd in Weatherburn Lecture Theatre.

Abstract:

We consider a code to be a subset of the vertices of a Hamming graph. The set of s-neighbours of a code are those vertices which are distance s from some codeword, but not distance r from any codeword, for 0 <= r < s. The automorphism group of a code necessarily fixes the set of s-neighbours of the code. An s-elusive code is a code such that the automorphism group of the set of s-neighbours is strictly larger than the automorphism group of the code itself. We provide examples for s=1,2,3, including a family of Reed-Muller codes and the Preparata codes. We also discuss some restrictions on the parameters of elusive codes and show that an elusive code gives rise to a q-ary t-design. This fact is used to provide a new construction for the Nordstrom-Robinson code.

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Point-primitive generalised polygons

at 3pm Friday 29 August in Weatherburn Lecture Theatre.

Abstract:

This talk will be about some on-going joint work with John Bamberg, Stephen Glasby, Cheryl Praeger and Csaba Schneider on classifying point-primitive generalised hexagons and octagons. I will go through our proof that if a (thick) generalised hexagon or octagon admits an automorphism group G that acts primitively on points, then G must be an almost simple group. If there is time, I will say a bit about how we can go further.

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Counting surfaces: ribbon graphs, branched covers, and more

at 3pm Friday 5 September in Weatherburn Lecture Theatre.

Abstract:

The enumeration of ribbon graphs (graphs embedded in surfaces) satisfies a so-called Tutte recursion, while the enumeration of branched covers (maps from a surface to a sphere) satisfies a so-called cut-and-join recursion. In fact, both are instances of the topological recursion, which emerged several years ago from the theory of matrix models. The topological recursion is now known to govern myriad problems from enumerative geometry, knot theory, theoretical physics, and more. We will give a very gentle introduction to these ideas, focusing on the combinatorial and algebraic aspects.

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Generalized quadrangles and AS-configurations of finite p-groups

at 3pm Friday 12 September in Weatherburn Lecture Theatre.

Abstract:

This is joint work with John Bamberg and Eric Swartz at UWA. One method for constructing generalized quadrangles involves finding a certain family of subgroups of a group. Given a group G of order q^3 we seek a family of q+2 subgroups U_0, U_1,..., U_{q+1} of G, each of order q, such that U_0 is normal in G and U_iU_j ap U_k={1} for all distinct i,j,k in {0,1,...,q+1}. The family is called an AS-configuration for G. We determine all the AS-configurations for q odd as well as q=2,4,8. To determine q=8 we showed, using extensive theory and computation, that only one of the 10,494,213 groups of order 2^9 admits an AS-configuration!

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Finite 2-set-homogeneous graphs

at 3pm Friday 19 September in Weatherburn Lecture Theatre.

Abstract:

Let k be a positive integer. A finite graph G is said to be k-set-homogeneous if for any two isomorphic subgraphs, A and B, of G, each of order at most k, there is an automorphism of G taking A to B. Clearly, every k-set-homogeneous graph is also 2-set-homogeneous for k >= 2. We study the 2-set-homogeneous graphs. It is shown that a 2-set-homogeneous graph G either belongs to some known families of graphs or Aut(G) is an almost simple primitive rank 4 permutation group with exactly one orbital self-paired.

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A geometric proof of Wedderburn's Theorem

at 3pm Friday 26 September in Weatherburn Lecture Theatre.

Abstract:

J. H. Maclagan-Wedderburn (1905) exploited the interplay between a finite division ring and its group of units to prove that a finite division ring is commutative (and hence a field). One of the most well-known applications of this result is that a finite projective plane is Desarguesian if and only if it is Pappian. It is perhaps not as well-known that Wedderburn's Theorem also implies the Dandelin-Galluci Theorem for finite projective 3-spaces. In joint work with Tim Penttila, the Dandelin-Galluci Theorem plays the central role in a proof of Wedderburn's Theorem, thus completing Beniamino Segre's dream of developing a proof of Wedderburn's Theorem that is truly geometric.

Cheryl Praeger (UWA)

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Simple group factorisations and applications

at 3pm Friday 3 October in Weatherburn Lecture Theatre.

Abstract:

Factorisations of the finite simple groups yield important information about their subgroup structure. Moreover, a group factorisation of the form G=AB with A, B proper subgroups, has at least two useful interpretations when studying symmetric structures, such as graphs or designs. For example, if G is a given group of automorphisms acting transitively with stabiliser B, then the factorisation G=AB reveals that A is a proper transitive subgroup. If in addition A and B have trivial intersection then the points of the graph or design may be identified with elements of A. For the second interpretation, A may be a known transitive automorphism group of a graph or design, and then factorisations G=AB (or their absence) can help to determine the full automorphism group. The lecture will survey results about simple group factorisations and how they can be used. A complete classification of simple group factorisations is as yet beyond our reach.

Scientists have now mapped the human genome - the next frontier is understanding human epigenomes; the ‘instructions’ which tell the DNA whether to make skin cells or blood cells or other body parts. Apart from a few exceptions, the DNA sequence of an organism is the same whatever cell is considered. So why are the blood, nerve, skin and muscle cells so different and what mechanism is employed to create this difference? The answer lies in epigenetics. If we compare the genome sequence to text, the epigenome is the punctuation and shows how the DNA should be read. Advances in DNA sequencing in the last five years have allowed large amounts of DNA sequence data to be compiled. For every single reference human genome, there will be literally hundreds of reference epigenomes, and their analysis could occupy biologists, bioinformaticians and biostatisticians for some time to come.

This lecture is presented by the Australian Mathematical Sciences Institute, The Statistical Society of Australia and the UWA Institute of Advanced Studies.

Professor Speed will be touring the Australia as the 2014 AMSI-SSAI Lecturer.

Cost: Free, but RSVP required via https://www.ias.uwa.edu.au/lectures/speed

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Vertex-transitive graphs with large automorphism groups: a potpourri

at 3pm Friday 10 October in Weatherburn Lecture Theatre.

Abstract:

I would like to understand connected vertex-transitive graphs which have "large" automorphism groups (with respect to their order and valency). I'll discuss a few approaches that have been used successfully to approach this problem as well as some new potential directions.

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Magic words

at 3pm Friday 17 October in Weatherburn Lecture Theatre.

Abstract:

A word map on a group G is a map sending (g_1,...,g_k) to w(g_1,...,g_k), where w is a fixed word in k variables. For example, the commutator map and power maps are word maps. I shall discuss various questions about word maps, such as surjectivity and distribution of values, with particular emphasis on simple groups.

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Self-complementary vertex-transitive graphs

at 3pm Friday 24 October in Weatherburn Lecture Theatre.

Abstract:

A graph is self-complementary if the graph and its complement are isomorphic. In this talk I will present the main results from my PhD project.

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Commuting graphs

at 3pm Friday 7 November in Mathematics Lecture Room 1.

Abstract:

Given a group G, the commuting graph of G is the graph with vertices the noncentral elements of G, and two vertices are adjacent if and only if they commute. Commuting graphs of other algebraic structures can be defined similarly. Commuting graphs of groups have received a lot of attention in recent years. I will discuss some recent results about the structure of such graphs, and in particular their diameter and which graphs can be the commuting graphs of groups and other algebraic structures.

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Antiflag-transitive generalized quadrangles

at 3pm Friday 14 November in Mathematics Lecture Room 1.

Abstract:

A generalized quadrangle is a point-line incidence geometry Q such that (1) any two points lie on at most one line, and (2) given a line l and a point P not incident with l, P is collinear with a unique point of l. An antiflag of a generalized quadrangle is a non-incident point-line pair (P, l), and we say that the generalized quadrangle Q is antiflag-transitive if the group of collineations (automorphisms that send points to points and lines to lines) is transitive on the set of all antiflags. We prove that if a finite, thick generalized quadrangle Q is antiflag-transitive, then Q is one of the following: the unique generalized quadrangle of order (3,5), a classical generalized quadrangle, or a dual of one of these. This is joint work with John Bamberg and Cai-Heng Li.

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Tits' buildings as combinatorial objects

at 3pm Friday 28 November in Mathematics Lecture Room 1.

Abstract:

Buildings were invented in the 1960s by the Belgian-French mathematician Jacques Tits. They are a useful tool to "visualise" algebraic groups (such as classical and exceptional Lie groups). The simplest examples are the projective spaces. Other examples include polar spaces, generalised polygons, infinite trees. I will give a general introduction to the concept of buildings, from several points of view, with special emphasis on the chamber system point of view, which is very combinatorial. I will then illustrate how that combinatorial structure can be used to show results that apply to many different types of buildings at once.

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Symmetric p-groups

at 11am in Weatherburn LT

(The time will be finalised on Wed evening. There may be the need for a change of time to 1:30pm.)

Abstract: In 1978 R.M. Bryant and L.G. Kov acs showed that for every subgroup H of GL(d,p) for a prime p there is a finite p-group P such that the automorphism group of P induces a subgroup on the Frattini quotient P/ hi(P) which is isomorphic to H. Their proof demonstrates the existence of such a p-group by considering sufficiently large quotients to terms of the exponent-p lower central series of the free group F on d generators.

In joint work with J. Bamberg, S. Glasby and L. Morgan we consider maximal subgroups of GL(d,p) for odd primes p and d at least 4 and show that in many cases we only need to consider the exponent-p lower central series of F of class 3 to find such a p-group.

Speaker: Irene Pivotto (UWA)

Title: Growth rates of binary matroids.

Abstract: The growth rate of a class of matroids is a function of the rank which expresses how large a matroid in the class can be. For example, a planar graph on n vertices has at most 3n-6 edges. So the corresponding class of matroids has growth rate 3r-3 (since the rank of the matroid of a graph on n vertices is r=n-1). The Growth Rate Theorem says that any minor-closed class of matroids has growth rate that is either linear, quadratic or exponential (or undefined). This result is amazingly general, and thus it is of a qualitative nature. This talk focuses instead on the problem of determining the exact growth rate of some classes. In particular, I will discuss quadratic and linear growth rates in binary matroids. No prior knowledge of matroids is necessary.

Speaker: Colva Roney-Dougal (University of St Andrews)

Title: Generation of finite groups.

Abstract: There is a beautiful result, due to Liebeck, Shalev, and others, that the probability that two random elements of a finite simple group G generate the whole of G tends to 1 as the order of G tends to infinity. Many variations on this result are known -- for example, we could insist that one of the two elements lay in a specified conjugacy class, or had a specified order -- and I shall discuss a few of them, together with some open problems. Far less is known about random generation of arbitrary non-finite groups, but I'll briefly discuss this too. If time permits, I'll finish with a discussion of the Product Replacement Algorithm, which is used to make random elements of a group, and some open problems relating to this.

Speaker: Colva Roney-Dougal, University of St. Andrews.

Title: Groups, diagrams and geometries.

Abstract: The study of finitely-presented groups has been ongoing since the work of Hamilton in the 1850s - almost as long as group theory itself! This talk will be a gentle introduction to finitely-presented groups, with an emphasis on algorithms. I'll describe some finite diagrams, and some potentially infinite geometries, that are naturally associated with any finitely-presented group, and show how results about the diagrams and geometries prove structural results about the group, and vice versa.

Speaker: Seyed Hassan Alavi (Buali Sina University)

Title: Recent studies on symmetric designs.

Abstract: A symmetric design (v,k,l) is an incidence structure consisting of a set of v points and a set of v blocks with an incidence relation such that every block is incident with exactly k points, and every pair of points is incident with exactly l blocks. An automorphism group of a symmetric design is a group of permutations on points of the design which maps blocks to blocks and preserves incidence and non-incidence. The main part of this talk is devoted to giving a survey on recent study of symmetric designs which admit a group of automorphisms acting primitively (resp. transitively) on the set of points (resp. blocks).

Speaker: Gordon Royle (UWA)

Title: Homomorphisms and Endomorphisms of Graphs.

Abstract: A homomorphism from a graph X to a graph Y is a function f from V(X) to V(Y) that preserves edges (but not necessarily non-edges); if Y = X, then it is called an endomorphism. Just as all the automorphisms of a graph form its automorphism group, all the endomorphisms of a graph form its endomorphism monoid. After covering the background definitions, I will discuss a number of problems that have in common that any solution will rely on understanding - either theoretically or computationally - various aspects of the endomorphisms of the graphs involved.

Speaker: Prof. Nozer Singpurwallah, City University of Hong Kong.

Title: The Bayesian Paradigm for Statistical Inference and Decision Making.

Abstract: In this expository talk, open to a general audience, I outline the essence of the Bayesian paradigm for inference and decision making. After an overview of the subjective nature of probability, I discuss the notion of a likelihood, and the genesis of a probability model. The material here is standard, but the perspective is not. It will be illustrated by some simple examples.

Speaker: Cai-Heng Li (UWA)

Title: The regular one-face dessins and surface coverings.

Abstract: I will present a classification of edge-transitive dessins (orientable bipartite maps) with a single face, and applying it to determine ramification coverings between surfaces.

Speaker: John Bamberg (UWA)

Title: Multimarkdown and typesetting mathematics.

Abstract: Markdown has been around for around a decade or so, and is a simpler alternative to xml for producing simple easy-to-read text with powerful mark-up capability. If you don't already know what Markdown is, then you really ought to know! The genius of Markdown is that it has an almost flat learning curve, and it is ideal for people who have no programming experience. It can be easily converted to RTF or HTML and is essentially platform independent. For more advanced use, there is "Multimarkdown", and yet it is still amazingly simple and intuitive. Multimarkdown can be easily converted to HTML and LaTeX, and so it has the potential to be very useful to mathematicians. This seminar is a basic introduction to this fantastic mark-up language.

Speaker: Luke Morgan (UWA)

Title: A non-classification theorem for 2-arc-transitive graphs.

Abstract: It would be great to have a classification of 2-arc-transitive graphs! Sadly, this is probably out of reach. However, there are many results on classifications of 2-arc-transitive graphs of certain orders, valencies, girth, etc... In this talk, I'll describe a recent result of Eric Swartz, Gabriel Verret and myself where we do something slightly different. Rather than classify all the possible graphs, we show that for fixed n and k, the family of 2-arc-transitive graphs of order kp^n is finite, with p a prime, once the valency is big enough (relative to n). So, our "non-classification" shows that the most interesting 2-arc-transitive graphs are those of small valency (relative to n).

Time and date: 4pm, Thursday 19th March

Venue: Blakers Lecture Theatre

Speaker: Prof. Michael Small, The University of Western Australia.

Title: Complex Systems: From nonlinear dynamics to graphs, via time series.

Abstract: Given a deterministic dynamical system - possibly contaminated by noise - what can I say about that system by measuring the time evolution of a single state? There are standard methods to answer this question, and I will review these. I will also show that by transforming the reconstructed system into a large graph, it is possible to learn even more.

Speaker: Frederico A. M. Ribeiro (Federal University of Minas Gerais)

Title: UCS groups and associated algebras.

Abstract: Groups having many characteristic subgroups tend to have simpler automorphism subgroups, so it might be a good idea to look into the automorphism subgroups of groups having few characteristic subgroups. While characteristically simple groups are well understood, this is not the case when "few" is greater than zero. UCS groups are groups having a unique non-obvious characteristic subgroup. When investigating one of the two classes of UCS p-groups, I, together with Csaba Schneider, showed that they could be associated bijectively with a class of algebras we called IAC (irreducible and anticommutative). We can then, by looking into properties of these algebras, gather information about the groups.

Speaker: Gabriel Verret (University of Western Australia)

Title: Vertex-primitive digraphs having vertices with almost equal neighbourhoods.

Abstract: Apart from some very trivial examples, a vertex-primitive digraph cannot have two vertices with equal neighbourhoods. (This is an easy exercise.) The first non-trivial case is thus when a vertex-primitive digraph has two vertices with neighbourhoods differing by one. I will give a proof of the classification of such digraphs, and describe a more general result and some applications.

Speaker: Padraig O Cathain (Monash University)

Title: Combinatorial designs and compressed sensing.

Abstract: Traditionally signal sampling and signal processing have been regarded as two separate tasks. Shannon's theorem relates the number of samples to the quality of the reconstruction: more samples are required for higher quality data. Compressed sensing is a new paradigm in signal processing in which sampling and compressing are combined into a single step. Under certain weak conditions, this reduces the number of samples required below the Shannon limit, without any loss in quality.

Tao's breakthrough papers on this topic showed that random matrices make good compressed sensing matrices. But such arrays are difficult to compute and to store, so are of limited practical use. In this talk we will outline the properties required of a good compressed sensing matrix, and describe a construction for such arrays using Hadamard matrices and pairwise balanced designs.

Speaker: Jason Semeraro (University of Bristol)

Title: Conway Groupoids.

Abstract: To a supersimple 2-(n,4,lambda) design (a simple design where lines intersect in at most two points), one may associate a "Conway groupoid." This is a generalization of the Mathieu groupoid associated to Conway's M_13 using a `game' played on P_3, the finite projective plane of order 3. We will discuss what is known so far about Conway groupoids. Highlights include a full classification for lambda<3, a structural characterization of the Mathieu groupoid, and a surprising connection with 3-transposition groups. We also speculate on the possibility of a full classification subject to the condition that the analogue of the set M_13 actually forms a group. If time permits, we will describe a new infinite family of completely transitive codes which were discovered along the way. The talk is intended to be accessible, with plenty of time given to motivation and context. This is joint work with Nick Gill and Neil Gillespie.

Speaker: Laura Mancinska (National University of Singapore)

Title: Graph homomorphisms for quantum players.

Abstract: A homomorphism from a graph X to a graph Y is an adjacency preserving mapping f:V(X)->V(Y). We consider a nonlocal game in which Alice and Bob are trying to convince a verifier with certainty that a graph X admits a homomorphism to Y. Classical players can succeed if and only if X admits a homomorphism to Y. In contrast, entangled quantum players can sometimes succeed even when the corresponding homomorphism does not exist. This motivates the introduction of quantum homomorphisms which turn out to be natural graph-theoretic objects and can also be defined in purely combinatorial terms.

Via systematic study of quantum homomorphisms we prove new results for the previously studied quantum chromatic number. Most importantly, we show that the Lovasz theta number of the complement lower bounds the quantum chromatic number, which itself is not known to be computable. We also show that other quantum graph parameters, such as quantum independence number, can differ from their classical counterparts. Finally, we show that quantum homomorphisms closely relate to zero-error channel capacity. In particular, we use quantum homomorphisms to construct graphs for which entanglement-assistance increases their one-shot zero-error capacity. This talk is based on https://arxiv.org/abs/1212.1724 which is a joint work with David E. Roberson.

Speaker: Alastair Litterick (University of Auckland)

Title: The Many Faces of Triangle Groups.

Abstract: Do the finite groups in a given class have generating sets of a prescribed form? Versions of this problem are as old as group theory itself, and still arise naturally today in enumerative geometry, combinatorics, the inverse Galois problem, and computational algebra.

This talk will consider `triangle-generated finite groups', i.e. 2-generated groups where the generators and their product have specified orders. We will discuss in particular the `rigidity conjecture' of Claude Marion, which relates triangle generation in a family of finite groups of Lie type to a property of the corresponding algebraic group. Recent progress on this conjecture has required a surprising variety of approaches, and we will discuss each of these and their limitations..

Speaker: Bart De Bruyn (Ghent University)

Title: Regular near hexagons and Q-polynomial distance-regular graphs.

Abstract: A near 2d-gon is a point-line geometry with diameter d having the property that for every point x and every line L, there exists a unique point on L nearest to x. A near polygon is called thick if every line is incident with at least three points and regular if its collinearity graph is a so-called distance-regular graph. In my talk, I will discuss thick regular near 2d-gons with a so-called Q-polynomial collinearity graph. For d > 3, we show that apart from Hamming near polygons and dual polar spaces there are no thick Q-polynomial regular near polygons. A thick regular near hexagon is Q-polynomial if and only if t = s^3 + t_2 (s^2 - s + 1), where t + 1 is the number of lines through each point, s + 1 is the number of points on each line and t_2 + 1 is the constant number of common neighbors that two points at distance 2 have. We also show that there cannot exist (necessarily Q-polynomial) regular near hexagons whose parameters (s,t_2,t) are equal to either (3,1,34), (8,4,740), (92,64,1314560), (95,19,1027064) or (105,147,2763012). All these nonexistence results imply the nonexistence of distance-regular graphs with certain parameters. We also mention some applications of these non-existence results.

(Joint work with Frederic Vanhove)

Speaker: David Roberson (Nanyang Technological University)

Title: Cores of cubelike graphs.

Abstract: A graph is said to be cubelike if it is a Cayley graph for some power of the group of order two. The core of a graph is its smallest subgraph to which it admits a homomorphism. In this talk we consider the following open question: Is the core of a cubelike graph cubelike? We will discuss the previously known results on this problem before proving the following:

Theorem: If X is a core of a cubelike graph and has degree k, then X has at most 2^(k-1) vertices or is K_2. Furthermore, if k is at least 2 and the above bound is met, then k is odd and X is the folded cube of order k.

This result is as expected if the answer to the above question is "yes", and thus it provides some evidence that this is in fact the case. We will also discuss some properties that would be required of a minimal counterexample to the conjecture if time permits.

Speaker: Shujiao Song (University of Western Australia)

Title: On the stabilisers of locally 2-transitive graphs.

Abstract: For a connected locally (G,2)-transitive graph and an edge {v,w}, both the induced actions of G_v on the neighbor set of v and G_w on the neighbor set of w are 2-transitive permutation groups. However, not every pair of 2-transitive permutation groups can be taken as such a pair. In this talk we give a description of such pairs of 2-transitive permutation groups. One interesting corollary tell us that a locally 3-arc-transitive amalgam with trivial vertex kernel is one of the following triples: (A_7,A_7,A_6),(S_7,S_7,S_6),(A_7,2^4:A_6,A_6),(S_7,2^4:S_6,S_6),(A_8,2^4:A_7,A_7),(A_9,2^4:A_8,A_8).

Speaker: Stephen Glasby (University of Western Australia)

Title: p-groups, Weyl modules, and maximal subgroups of linear groups.

Abstract: Given a subgroup H of GL(d,p), there exists a d-generated p-group G whose automorphism group induces the linear group H on the Frattini quotient G/Phi(G). The proof of this fact is non-constructive. In some sense G is a "non-linear representation" of H.

We consider how to go from a *maximal* subgroup H of GL(d,p) to a nonabelian p-group G with minimal class/exponent/order. This involves understanding the H-submodule structure of Weyl modules and some intriguing H-homomorphisms. This work was motivated by applications to geometry and algorithms for p-groups, and is joint with John Bamberg, Luke Morgan and Alice Niemeyer.

Speaker: Dr. Murray Elder, University of Newcastle

Title. Solving equations in free groups.

Abstract. An equation in a free group is an expression U=V where U,V are words over elements of the group and variables X,Y,Z, etc. A solution is an assignment of group elements to the variables which make the equation true.

In the 1970s, Makanin constructed a (really complicated) algorithm which decides if an equation has a solution or not. Later, Razborov extended Makanin's result to find all solutions. The complexity of these algorithms was subsequently shown to be pretty bad.

In this talk I will present a new approach, describing a finite graph that encodes all solutions in reduced words, which has exponential size and can be constructed in nondeterministic quasilinear space (in the length of the equation). I will try to motivate and explain the problem, how it relates to some questions in logic, and give some of the ingredients of the proof.

Speaker: Murray Elder (University of Newcastle)

Title: Solving equations in free groups and free monoids with constraints.

Abstract: My talk will give some details of the proof of our result - that the set of all solutions to an equation over a free group or free monoid has a simple description encoded by a finite graph, that can be constructed in space O(n log n).

- reduction from an equation over a free group to an equation over a free monoid with involution with constraints

- extended equations and partial commutation

- construction of the finite graph encoding all solutions

- two propositions: any path in the graph from an initial to final vertex is a solution; and any solution is realised by a finite path in the graph from an initial to final vertex

- algorithm to prove the second proposition - block and pair compression. This is joint work with Laura Ciobanu, Neuchatel and Volker Diekert, Stuttgart.

Speaker: Heiko Dietrich (Monash University)

Title: Computing with real Lie algebras.

Abstract: The structure theory of complex semisimple Lie algebras uses many combinatorial objects such as root systems, Weyl groups, and Dynkin diagrams, which makes the theory accessible for computer investigations. Computer algebra systems, like GAP, Magma, and LiE, contain software for computing with complex Lie algebras. Also the real semisimple Lie algebras are classified, and there exists a detailed structure theory. However, with the exception of the ATLAS project (USA), there has not been much effort to develop software for computing with real semisimple Lie algebras. We report on the functionality and the underlying theory of our GAP package 'CoReLG' (Computing with Real Lie Groups); it provides functions to construct real semisimple Lie algebras, to check for isomorphisms, and to compute Cartan decompositions, Cartan subalgebras, and nilpotent orbits.

Speaker: Istvan Kovacs (University of Primorska)

Title: On regular Cayley maps for dihedral groups.

Abstract: The classification regular Cayley maps for cyclic groups was recently completed by Conder and Tucker. In the talk I will present a few results on regular Cayley maps for dihedral groups. This is based on joint works with Y. S. Kwon, D. Marusic and M. Muzychuk.

Speaker: Simon Smith (City University of New York)

Title: The suborbits of infinite primitive permutation groups

Abstract: In this talk I'll discuss the subdegrees (cardinalities of suborbits) of infinite primitive permutation groups. In particular I'll be looking at the extent to which a group is determined by its subdegrees. This is related to an old remark made by Samson Adeleke and Peter Neumann which I've recently started thinking about again. It will contain a number of conjectures and open problems.

Speaker: Cheryl Praeger (University of Western Australia)

Title: Permutation groups with "LPS"

Abstract: My collaborations with Jan and Martin were centred on group actions, that is, on permutation groups. My talk is in tribute to our wonderful mathematical journey together.

Speaker: Qing Xiang (University of Delaware)

Title: Cameron-Liebler line classes and two-intersection sets

Abstract: Cameron-Liebler line classes are sets of lines in PG(3,q) having many interesting combinatorial properties. These line classes were first introduced by Cameron and Liebler in their study of collineation groups of PG(3,q) having the same number of orbits on points and lines of PG(3,q). In the last few years, Cameron-Liebler line classes have received considerable attention from researchers in both finite geometry and algebraic combinatorics. In [1], the authors gave several equivalent conditions for a set of lines of PG(3,q) to be a Cameron-Liebler line class; Penttila gave a few more of such characterizations. We will use one of these characterizations as the definition of Cameron-Liebler line class. Let L be a set of lines of PG(3,q) with |L| = x(q^2+q+1), x a nonnegative integer. We say that L is a Cameron-Liebler line class with parameter x if every spread of PG(3,q) contains x lines of L. It turned out that Cameron-Liebler line classes are closely related to certain projective two-weight codes (equivalently, certain two-intersection sets in PG(5,q)).

We will talk about a recent construction of a new infinite family of Cameron-Liebler line classes with parameter x=(q^2-1)/2 for q = 5 or 9 (mod 12). This family of Cameron-Liebler line classes generalizes the examples found by Rodgers in through a computer search, and represents the second infinite family of Cameron-Liebler line classes. Furthermore, in the case where q is an even power of 3, we construct the first infinite family of affine two-intersection sets.

We should remark that De Beule, Demeyer, Metsch and Rodgers also independently obtained the same result as ours on Cameron-Liebler line classes with parameter x=(q^2-1)/2 at almost the same time.

[1] P. J. Cameron, R.A. Liebler, Tactical decompositions and orbits of projective groups, Linear Algebra Appl., 46 (1982), 91-102.

Speaker: Jingbo Wang (University of Western Australia)

Title: Quantum walk on graphs, unitary operation, Hamiltonian simulation, and efficient quantum circuits

Abstract: Quantum walk represents a generalised version of the well-known classical random walk. Regardless of their apparent connection, the dynamics of a quantum walk is often non-intuitive and far deviates from its classical counterpart. A multi-particle quantum walk presents an even richer dynamical system due to intrinsic quantum correlation and interaction. Current research is suggesting potential applications across a wide range of different fields. In this talk, I will give a brief introduction to quantum walks, and then focus on efficient quantum circuit implementation that holds promise of solving practical problems otherwise intractable.

Speaker: Daniel Hawtin (University of Western Australia)

Title: 2-Neighbour Transitive Codes in Hamming Graphs

Abstract: We consider a code C to be a subset of the vertex set of a Hamming graph, with alphabet Q and entries M. A 1-neighbour is a vertex which is not contained in C, but is distance 1 from some codeword in C. A 2-neighbour is a vertex of the Hamming graph which is distance 2 from some codeword, but not distance 1 or 0 from any codeword. Let X be a subgroup of the automorphism group of the Hamming graph. We say a code C is (X,2)-neighbour transitive if X is transitive on C, the set of 1-neighbours, and the set of 2-neighbours.

Let X_1 be the subgroup of X which fixes the first entry. Gillespie, Giudici and Praeger showed that the action X_1^Q, of X_1 on the alphabet in the first entry, is 2-transitive, and thus affine or almost-simple. In this talk we first address the case where X acts faithfully on M, and then when X_1^Q is almost-simple.

Time and date: 4pm, Thursday 13th August

Venue: Blakers Lecture Theatre

Speaker: Francis Woodhouse (The University of Western Australia)

Title: Mimicking magnets with lattices of bacterial vortices.

Abstract:When alone in an unbounded fluid, a rod-shaped motile bacterium like E. coli will swim in straight lines punctuated by random turns. Pack many of them together in the same fluid, however, and they adopt collective swirling patterns akin to macroscopic turbulence. Confining the bacteria within a small circular cavity tames this turbulence and leads instead to a steadily spinning bacterial vortex. When many such vortices are linked together in a square lattice of cavities, the rotation sense of a vortex becomes dependent on those of its neighbours. By declaring the senses to be 'up' and 'down' spins, the result is a bacterial analogue of an Ising ferromagnet. After explaining the background to these so-called 'active matter' systems, I will explore the challenges involved in mapping classical statistical physics models to this decidedly non-classical system - but only after revealing an entirely unexpected twist in the experiments.

Speaker: Joanna Fawcett (University of Western Australia)

Title: Affine rank 3 partial linear spaces

Abstract: A partial linear space consists of a finite non-empty set P of points and a collection L of subsets of P called lines such that every pair of points lies on at most one line, and every line contains at least two points. A partial linear space is proper if it is not a graph (i.e., every line contains exactly two points) or a linear space (i.e., every pair of points lies on exactly one line). Proper partial linear spaces with the most symmetry are those for which the automorphism group acts transitively on pairs of collinear points and pairs of non-collinear points. Such a geometry admits a transitive rank 3 group of automorphisms, and when this group is primitive, it is of almost simple, grid or affine type. Devillers classified the proper partial linear spaces admitting a primitive rank 3 group of automorphisms of almost simple or grid type. In this talk, we will consider some recent progress on the affine case. This is joint work with Bamberg, Devillers and Praeger.

Speaker: Mark Ioppolo (University of Western Australia)

Title: Triality via Geometric Algebra

Abstract: The automorphisms of the orthogonal group SO(+1,2n,q) generally correspond to semi-similarities of the underlying vector space. However, in eight dimensions the group SO(+1,8,q) has unexpected "triality" automorphisms. My main goal is to offer an informal explanation of the analogous result for SO(8) within the framework of "Clifford algebras". The theory of Clifford algebras is extremely beautiful in its own right, so I may also allow myself to be sidetracked by spinor representations of SO(n), Octonions and other topics.

Speaker: Tomasz Popiel (University of Western Australia)

Title: Point-primitive, line-transitive generalised quadrangles of 'holomorph type'.

Abstract: In a 2012 paper, Bamberg, Giudici, Morris, Royle and Spiga showed that a group acting primitively on both the points and the lines of a (finite and thick) generalised quadrangle must be an almost simple group of Lie type. In on-going work, we seek to weaken their assumption to primitivity on points and transitivity on lines, and classify the resulting examples. This has proved extremely difficult. In joint work with Bamberg, Glasby and Praeger, we have classified the examples for which the primitive action on points is of affine type. In joint work with Bamberg and Praeger, we have now also shown that no examples arise when the primitive action on points is of holomorph simple or holomorph compound type. In this highly anticipated talk, I will give some ideas of the proof of the latter result.

Speaker: Gordon Royle (University of Western Australia)

Title: Almost synchronising groups.

Abstract: A permutation group G on a set X is said to be synchronising if for every transformation f of X that is not a permutation, the semigroup generated by G and f contains a constant transformation. A synchronising group is necessarily primitive, but there are examples of non-synchronising primitive groups. For every non-synchronising primitive group, there is a uniform transformation witnessing the fact that the group is non-synchronising, and non-synchronising primitive groups for which every witness is uniform are called almost synchronising. Araujo, Bentz and Cameron conjectured that all primitive groups are either synchronising or almost synchronising, but then shortly afterwards found counterexamples to their own conjecture. In this talk, I will discuss subsequent work and remaining open questions on the topic of almost synchronising groups.

Speaker: Melissa Lee (University of Western Australia)

Title: The 3rd Heidelberg Laureate Forum and related mathematics.

Abstract: The Heidelberg Laureate Forum is a prestigious conference held in Heidelberg every August/September. It brings together 200 young researchers in mathematics and computer science from around the world, and laureates of the Abel, Fields and Turing awards for a week of networking, lectures and discussions. In this fairly general talk, I will outline my experiences at this year's forum. I will also speak about some topics in mathematics and computer science covered by the laureates, including password algorithms, proof formalisation and a graph theory problem.

Time and date: 4pm, Thursday 22nd October

Venue: Blakers Lecture Theatre

Speaker: Paul Baird (Universite de Bretagne Occidentale)

Title: Encoding geometric information into combinatorial structure

Abstract: There may be several reasons why we might wish to encode geometric information in this way: to transmit 3D-images; to provide algorithms by which an autonomous robot may navigate in the world; to optimise complex networks. My approach will be to consider how we ourselves reconstruct 3D-images, or frameworks, from suitably drawn lines on a piece of paper, the so-called Gestalt effect. One crucial aspect that I wish to capture is invariance with respect to similarity transformations, so that the geometry of the framework doesn't depend upon the way in which it is embedded in space, or in other words, the perspective from which it is viewed. To do this I will consider a set of equations that one can associate to a combinatorial graph which define its "geometric spectrum". An element of the spectrum allows one at least locally to realise the graph as an invariant framework in Euclidean space. Comparison with smooth counterparts provides insight into the interpretation of the equations.

Very little background is required to follow this talk: a basic knowledge of complex numbers, linear algebra and the geometry of curves and surfaces would be useful.

Speaker: Adrian Petersen (University of Western Australia)

Title: Pairwise transitive 1-designs

Abstract: Design theory has extensive interactions with other areas of mathematics, especially group theory and graph theory. Indeed, the problem investigated here arises from the study of a particular family of graphs, called locally s-distance transitive graphs. In this dissertation we investigate the relationship between a 1-design and its automorphism group, specifically we aim to classify pairwise transitive 1-designs. We say that a design is G-pairwise transitive if a group G of automorphisms of the design is transitive on the following ordered sets: collinear point pairs, non-collinear point pairs, intersecting block pairs, non-intersecting block pairs, flags, and anti-flags. Some of the transitivity conditions have been studied previously for 1-designs, but never have they been imposed all at once. In a way, a stronger symmetry property could not be imposed on a 1-design, as we require transitivity on every possible ordered pair of a design.

We draw inspiration from work by Devillers and Praeger, who classified the less general class of pairwise transitive 2-designs. This classification relied on the 2-transitive permutation groups, which are all known. In our case an original classification scheme is developed, which says that rank 3 permutation groups are needed to construct pairwise transitive 1-designs. Rank 3 permutation groups can be either primitive or imprimitive. The former case is very well understood; all primitive rank 3 actions are classified, and we take advantage of this classification. The latter case is not well understood. The primitive rank 3 groups can be further subdivided into the almost simple, affine, and grid type groups. This dissertation presents a partial classification of G-pairwise transitive strict 1-designs for primitive almost simple G.

Speaker: Irene Pivotto (University of Western Australia)

Title: Decomposition theorems for graphs and matroids

Abstract: Wagner proved that every graph that does not contain a clique on 5 vertices as a minor may be constructed by glueing together (in specified ways) planar graphs and copies of a specific graph. This is a prototypical example of a decomposition theorem: a result that explains how to construct all graphs in a class by glueing together graphs in well-understood classes and graphs from some finite set. Besides being useful tools in proving properties of decomposable graphs, these types of theorems have algorithmic consequences.

In this talk I will present some interesting decomposition theorems for graphs and matroids and discuss some progress toward proving a new result of this type.

Title: On automorphisms and opposition in spherical buildings.

Abstract: The opposition relation is fundamental in Tits' theory of spherical and twin buildings. When studying automorphisms of such buildings it is of course natural to consider fixed element structures, however it is also fruitful and instructive to investigate the structure of the elements mapped to opposite elements by the automorphism. In this talk we discuss recent developments in this theory, drawing from joint work with A. Devillers, B. Temmermans, and H. Van Maldeghem.

Speaker: Yian Xu (University of Western Australia)

Title: Generalized Thrackle and Graph Surface Embeddings

Abstract: A thrackle on a surface X is a graph of size e and order n drawn on X such that every two distinct edges of G meet exactly once either at their common endpoint, or at a proper crossing. An unsolved conjecture of Conway (1969) asserts that e is at most n for every thrackle on a sphere. Until now, the best known bound is that e is at most 1.428n. By using discharging rules we show that e is at most 1.4n. Furthermore we show that the following are equivalent: G has a drawing on X where every two edges meet an odd number of times (a generalized thrackle); G has a drawing on X where every two edges meet exactly once (a one-thrackle); G has a special embedding on a surface whose genus differs from the genus of X by at most one.

Speaker: David Raithel (University of Western Australia)

Title: Innately transitive groups

Abstract: An innately transitive permutation group is a permutation group with a transitive minimal normal subgroup. Innately transitive groups as a family contain all quasiprimitive (and hence primitive) groups, as well as some interesting groups in their own right (such as the symmetries of the icosahedron). In his PhD, Bamberg under the supervision of Praeger, was able to generalise the O’Nan-Scott Theorem to innately transitive groups. The first half of this talk will discuss Bamberg’s result, as well as the theory of innately transitive groups which developed from it. The rest of the talk will be discussing rank 3 innately transitive groups, and some of the work I have done under the supervision of Bamberg, Devillers and Praeger, including how to construct purely innately transitive rank 3 groups.

Speaker: Reza Naserasr (Université Paris-Sud)

Title: Bounding K_4-minor-free graphs in the homomorphism order

Abstract: Given a graph B of odd-girth 2k+1 we are interested in the following question: Does every K_4-minor-free graph of odd-girth at least 2k+1 admit a homomorphism to B? We present a necessary and sufficient condition for B to minimally be a yes instance. Using this, we present a polynomial time algorithm (in the order of B) to answer the question for a given B. We then turn our focus on finding smallest order of YES-instances. We provide a few families of them, relating the question to a conjecture in extension of the 4CT and to fractional and circular chromatic number of K_4-minor-free graphs of odd-girth 2k+1. Finally we present an application to edge-coloring of regular K_4-minor-free plane graphs.

Joint work with: L. Beaudou, F. Foucaud and Q. Sen

Time and date: 4pm, Thursday 19th November

Venue: Blakers Lecture Theatre

Speaker: Professor Andrew Bassom (The University of Western Australia)

Title: Grrrrr...... linear stability should be simple -- the saga of the Stokes' layer

Abstract: The linear stability of boundary layers is a subject which was thought to have been essentially solved long ago. During my interview at UWA over 12 years ago I talked about some calculations directed towards understanding the stability properties of a Stokes layer, which is the fluid flow set up when an oscillatory viscous flow moves over a rigid boundary. Those computations gave results very different from experimental observations and it is only relatively recently that we believe we have found a plausible explanation for the discrepancy. Here I shall review a number of the various frustrations experienced in the research into this ostensibly straightforward problem (and conclude that I should have given up long ago).

Speaker: Stephen Glasby (University of Western Australia)

Title: How big is the Sylow p-subgroup of an irreducible solvable subgroup of GL(d,p^f)?

Abstract: Let G be a subgroup of GL(d,q) with q = p^f, where p is a prime. The order |G|_p of a Sylow p-subgroup of G is at most p^{d(d-1)f/2}. We prove that if G is solvable and completely reducible, then there is a much smaller bound. Our first bound |G|_p e p^{(d-1)f} has recently been improved to |G|_p e 2^{(d-1)f} if p e 3. With thought, the hypothesis of solvability can be eliminated!

This is joint work with Michael Giudici, Cai Heng Li and Gabriel Verret.

Speaker: Michael Giudici (University of Western Australia)

Title: Classical groups, derangements and primes

Abstract: An elementary result of Jordan shows that every finite transitive group of degree at least two contains a derangement, that is an element that fixes no points. It was later shown by Fein, Kantor and Schacher that there is actually a derangement of prime power order. There is not always one of prime order but in the primitive group case all primitive groups without a derangement of prime order are known. When Tim Burness came to visit in 2008, we started a program of trying to determine for a given primitive group, for which primes p there is a derangement of order p. A large part of these investigations will be appearing in our very soon to be released booked of the same title. In this talk I will give a brief outline of some of the contents. It will be a sequel to my talk in this series on the 8th of September 2009. No knowledge of that talk will be required.

Speaker: Luke Morgan (University of Western Australia)

Title: Semiprimitive Groups: Are they wild or just misunderstood?

Abstract: The class of semiprimitive groups contains the classes of primitive, quasiprimitive and innately transitive groups. Every regular group creeps in too. Also every Frobenius group. There are just lots of semiprimitive groups! A goal of Michael Giudici's and mine is to bring some order to this chaos. I will talk about the first scratchings of our theory, and discuss the following big question: Can semiprimitive groups be classified, or are they just wild?

Speaker: Marcel Fernandez (Universitat Politecnica de Catalunya)

Title: Codes with traceability properties

Abstract: The talk will present codes with traceability properties. These codes are used in digital content distribution schemes to prevent dishonest users from redistributing copyrighted material. After showing that separation is a necessary property, we will move on to separating codes. In the crypto literature, separating codes have been rediscovered under the name of secure frameproof codes. We will gain some insight about both properties by discussing weaker versions, namely almost separating and almost secure frameproof codes. Silverberg, Staddon and Walker asked if for Reed-Solomon codes the separating property was equivalent to the stronger TA property. It will be shown that it is true for particular code parameters. The general answer remains an open problem.

Speaker: Sara Zemljic (University of Iceland)

Title: Wiener dimension and Sierpinski graphs

Abstract: Wiener dimension is a relatively new graph invariant closely connected to the Wiener index of graphs. It is defined as the number of different (total) distances of its vertices. The (total) distance of a vertex of a graph is the sum of all the distances to that vertex. Despite a clear definition of the Wiener dimension, it is not always straightforward how to determine this dimension for some particular graphs. In this talk I will go through some known results about the Wiener dimension and mostly focus on the family of Sierpinski graphs, for which the Wiener dimension is still unknown.

Speaker: Dillon Mayhew (Victoria University of Wellington)

Title: Finite vs. infinite fields in matroid representation

Abstract: We can think of a matroid as an abstraction of a discrete configuration of points in space. We say that a matroid is representable over a field if those points can be assigned coordinates from the field in a consistent way. There is now a substantial body of work suggesting that matroids representable over finite fields are in some ways well-behaved. Over the last (nearly) two decades, Geelen, Gerards, and Whittle have developed matroid analogues of the Robertson-Seymour theory for graphs. Amongst other results, they have proved Rota's conjecture, and showed that there is no infinite antichain under the minor order for matroids representable over a finite field. On the other hand, the situation for matroids representable over infinite fields seems to be as bad as it possibly can be. This talk will survey all these results, covering joint work with Mike Newman and Geoff Whittle. No knowledge of matroid theory will be assumed.

Speaker: Alexander Bors (University of Salzburg)

Title: Finite groups with an automorphism inverting, squaring or cubing a non-negligible fraction of elements

Abstract: There are various results in the literature stating that a finite group for which a certain parameter (e.g., the number of conjugacy classes or of prime order subgroups) is large enough must be in some sense close to being abelian. In this talk, I will present some recent results of that type on finite groups in which the maximum fraction of elements inverted resp. squared resp. cubed by a single automorphism is bounded away from 0. In spite of the elementarity of the problem, we will require some nontrivial tools for its solution, including the classification of finite simple groups and a bit of character theory.

Speaker: Joy Morris (University of Lethbridge)

Title: Automorphisms of circulant graphs

Abstract: Determining the full automorphism group of a graph is a hard problem with a long history. I will discuss some of the major results that involve finding graphs with a given automorphism group. I will then focus on circulant graphs, and describe some structural results and algorithms that help us determine the full automorphism group of the graph. I will also give some asymptotic results about how many circulant graphs fall into different categories.

Speaker: Binzhou Xia (Peking University)

Title: 2-arc-transitive Cayley graphs of soluble groups

Abstract: Classifying certain classes of 2-arc-transitive graphs has received considerable attention. There have been results on 2-arc-transitive Cayley graphs of abelian groups and dihedral groups. Generalisation of these results oriented, a project was started by Cai Heng Li and me, aiming to classify 2-arc-transitive Cayley graphs of soluble groups. In this seminar, I will talk about the recent progress on this project.

Speaker: Gabriel Verret (UWA)

Title: Asymptotic enumeration of vertex-transitive and Cayley graphs

Abstract: I am interested in determining the number f(n) of graphs of order n from various "symmetric" families. (Ex: vertex-transitive graphs, Cayley (di)graphs, GRRs... of arbitrary or fixed valency...) While obtaining exact counts for small n is an interesting problem, I will focus in this talk on the opposite end of the spectrum: determining the asymptotic behaviour of f(n) for large n.

Speaker: Steven Noble (Brunel University London)

Title: Counting labelled delta-matroids

Abstract: Delta-matroids were introduced and studied by Bouchet in the 1980s, but have largely been ignored until the last few years. They generalize matroids and include classes coming from embedded graphs and from symmetric matrices. This talk is based on recent joint work with Darryl Funk and Dillon Mayhew, in which we give bounds on the number of labelled delta-matroids with n elements. As expected there are many more delta-matroids than matroids, but the number of even delta-matroids, a class that is generally thought to behave more like matroids, is not so much larger than the number of matroids.

Abstract: Scuba divers are trained to follow procedures that keep them safe while diving and afterward. Surprisingly, these safety measures are based on just a handful of simple mathematical and statistical principles, which will be explained in this colloquium. Better public understanding of these basic principles would dispel many common misconceptions about scuba safety. Modern diving practices also pose some challenging mathematical problems, with interesting solutions, which are often treated as an industrial secret by the manufacturers of safety devices.

About the speaker: Adrian Baddeley PhD DSc FAA is Professor of Computational Statistics at Curtin University in Perth, Australia. He is a former UWA professor and head of department, and now adjunct professor. Adrian is a keen diver with over 1100 scuba dives logged. He has recently published 3 journal papers on scuba decompression theory and scuba accident statistics.

Speaker: Kyle Rosa (UWA)

Title: Invariable generation of permutation groups

Abstract: An invariable generating set for a group is a collection of conjugacy classes from which any choice of representatives form a generating set. In this talk I will discuss some general results about invariable generation, and in particular the content of my Honours thesis on this topic. We culminate with a refinement of a theorem of Detomi and Lucchini which states that every permutation group except Sym(3) admits an invariable generating set of order half its degree.

Speaker: Ty Ghaswala (University of Waterloo)

Title: The superelliptic covers and the lifting mapping class group

Abstract: Given a finite sheeted (possibly branched) covering space over a surface, one can ask the following question: Which homeomorphisms of the base space lift to homeomorphisms of the total space? If we take the quotient of this question by isotopy, it becomes a much more interesting one: What can we say about the subgroup of the mapping class group of the base space that consists of isotopy classes of homeomorphisms that lift to the total space? This subgroup is the lifting mapping class group.

This question was completely answered by Birman and Hilden when the deck group is the two element group generated by a fixed hyperelliptic involution. In this case, everything lifts. Interestingly, this does not happen in general.

In this talk, I will give an introduction to the mapping class group of a surface and a history of this lifting problem. I will then focus on the algebraic structure of the lifting mapping class group in the case of the superelliptic covers, which are $n$-sheeted generalisations of the 2-sheeted covering spaces studied by Birman and Hilden. Time permitting, I will outline some interesting questions that arise from this work.

This is joint work with Rebecca Winarski.

Speaker: Cai-Heng Li (UWA)

Title: Relative elementary groups and their applications

Abstract: Motivated by some problems about Cayley graphs and permutation groups, we introduce and study a class of finite groups G, called REA-groups, which behave similarly to elementary abelian p-groups with p prime, that is, there exists a subgroup N such that all elements of G-N are conjugate or inverse-conjugate under Aut(G).

In this talk, I will report on some interesting properties of REA-groups and explain the relations between REA-groups with imprimitive permutation groups of rank 3 and normal Cayley graph representations of multi-partite complete graphs.

Speaker: Jesse Lansdown (UWA)

Title: Bruck nets and partial Sherk planes

Abstract: In Bachmann’s Aufbau der Geometrie aus dem Spiegelungsbegriff, it was shown that a finite metric plane is a Desarguesian affine plane of odd order equipped with a perpendicularity relation on lines, and conversely. Sherk generalised this result to characterise the finite affine planes of odd order by removing the ‘three reflections axioms’ from a metric plane. We show that one can obtain a larger class of natural finite geometries, the Bruck nets of even degree, by weakening Sherk’s axioms to allow non-collinear points.

Speaker: Norman Do (Monash University)

Title: Counting curves on surfaces

Abstract: Choose an even number of points on the boundary of a surface. How many ways are there to pair up these points with disjoint arcs on the surface? The most basic instance of this problem produces the Catalan numbers while the problem in general exhibits a surprisingly rich structure. For example, we will show that this enumeration obeys an effective recursion and exhibits polynomial behaviour. Moreover, there are unexpected connections to algebraic geometry and mathematical physics.

Speaker: John Bamberg (UWA)

Title: Hemisystem-like structures in finite geometries

Abstract: This is a longer version of the talk I will be giving at “New Directions in Combinatorics” (Singapore). Beniamino Segre showed in his 1965 manuscript 'Forme e geometrie hermitiane, con particolare riguardo al caso finito' that there is no way to partition the points of the Hermitian surface H(3,q^2) into lines, when q is odd. Moreover, Segre showed that if there is an m-cover of H(3,q^2), a set of lines covering each point m times, then m=(q+1)/2; half the number of lines on a point. Such a configuration of lines is known as a hemisystem and they give rise to interesting combinatorial objects such as partial quadrangles, strongly regular graphs, and imprimitive cometric Q-antipodal association schemes. This talk will be on developments in the field of hemisystems of polar spaces and regular near polygons and their connections to other interesting combinatorial objects. No background in finite geometry will be assumed.

Speaker: Luke Morgan (UWA)

Title: The Weiss Conjecture and inspirations thereof

Abstract: In 1979 Richard Weiss made a beautiful conjecture: There exists a function f on the natural numbers such that, for each G-vertex-transitive locally-primitive graph X, the order of a vertex-stabiliser is bounded by f(d), with d the valency X. Although the conjecture remains open in full generality, many mathematicians have been attracted to work on this problem, and in several interesting cases it has been shown that the conjecture is true. As well as proofs of (cases of) the statement, Richard’s conjecture also spurred people to understand vertex-transitive graphs with local actions of certain flavours, such as locally-quasiprimitive graphs. This lead to the Praeger Conjecture of 2000, and for locally-semiprimitive graphs, the Potocnik-Spiga-Verret Conjecture of 2012. At this point the graphs faded to the background and general questions about semiprimitive permutation groups were raised. These turned out to be quite difficult, but one can say something. I’ll give an overview of some of these recent results and describe what we know on the various conjectures - that is - an overview of the wonderful mathematics that was inspired by Richard’s Conjecture.

Speaker: Phill Schultz (UWA)

Title: What do you do when Krull-Schmidt fails?

Abstract: I present some old and some new results on partitions of positive integers and direct decompositions of torsion-free abelian groups.

Abstract: When Galois invented groups they were very different from the structures taught and learned and loved in undergraduate courses at UWA and other modern universities. My purpose in this lecture will be to explain the differences and calibrate the similarities. As a by-product I hope to show that topics in the History of Mathematics can be just as exciting, subtle and difficult as mathematics itself.

About our speaker: Peter was awarded a DPhil in 1966, written under the supervision of Professor Graham Higman FRS, and a DSc by Oxford in 1976. In Oxford, Peter 38 students completed doctorates under his supervision, including one Cheryl E. Praeger.

Peter has been honoured with the Lester R. Ford Award by the Mathematical Association of America in 1987, the Senior Whitehead Prize by the London Mathematical Society in 2003, and the David Crighton Medal jointly by the Institute of Mathematics and its Applications and the London Mathematical Society in 2012. Peter was elected to an Emeritus Fellowship of Queen’s in 2008.

Peter’s research has contributed to a range of areas of algebra and its history. Some include: finite permutation groups; infinite permutation groups; soluble groups; design of group-theoretic algorithms; history of group theory.

Speaker: Peter Neumann (Oxford)

Title: TPP subgroup triples in finite groups

Abstract: About a dozen years ago, Henry Cohn and Christopher Umans proposed a group-theoretic attack on a famous problem in the theory of computational complexity. The problem is to pin down the exponent of matrix multiplication. Their attack involves a mixture of combinatorics and character theory of finite groups. The combinatorial side is based on what they call TPP triples of subsets of finite groups. If those subsets are subgroups $S$, $T$, $U$ then the TPP condition may be expressed simply as the condition that $S ap T$ is trivial and $ST ap U$ is trivial. I propose to give a little background, and then say a little about such triples of subgroups, focussing on finite nilpotent groups of class $2$.

This free event will see researchers presenting about their experiences as high performance computing (HPC) users and how their projects have been impacted by Pawsey services and expertise. Staff from Pawsey will also be presenting information about the organisation and answering questions from researchers who would like to take advantage of HPC and related services.

If you are a researcher or student who is interested in high performance computing, or if you want to see if your work can be taken to the next level by using powerful computing resources, we welcome you to come along and meet with Pawsey users and staff.

If you wish to attend, please RSVP at https://www.pawsey.org.au/pawsey-roadshow-at-uwa-rsvp-form/

Speaker: Gordon Royle (UWA)

Title: The characteristic polynomial of a graph

Abstract: The characteristic polynomial of a graph G is the characteristic polynomial of its adjacency matrix. While there are many different graph polynomials (chromatic, Tutte, matching etc), the characteristic polynomial is perhaps the most heavily studied of all, primarily because the roots of the characteristic polynomial (i.e. the eigenvalues of its adjacency matrix) carry so much information about structure of the graph and its subgraphs. Indeed, a large proportion of the entire field of algebraic graph theory can be viewed as exploring exactly which properties of graphs are, or are not, reflected in its spectrum. In this talk, I will outline some of the main properties of the characteristic polynomial of a graph, but also introduce some of the interesting open questions that remain. As an example, it is not currently known whether or not almost all graphs are determined up to isomorphism by their characteristic polynomials.

Speaker: John Bamberg (UWA)

Title: Hemisystem-like structures in finite geometries

Abstract: This is a longer version of the talk I will be giving at “New Directions in Combinatorics” (Singapore). Beniamino Segre showed in his 1965 manuscript 'Forme e geometrie hermitiane, con particolare riguardo al caso finito' that there is no way to partition the points of the Hermitian surface H(3,q^2) into lines, when q is odd. Moreover, Segre showed that if there is an m-cover of H(3,q^2), a set of lines covering each point m times, then m=(q+1)/2; half the number of lines on a point. Such a configuration of lines is known as a hemisystem and they give rise to interesting combinatorial objects such as partial quadrangles, strongly regular graphs, and imprimitive cometric Q-antipodal association schemes. This talk will be on developments in the field of hemisystems of polar spaces and regular near polygons and their connections to other interesting combinatorial objects. No background in finite geometry will be assumed.

Please note: this talk has been rescheduled from Friday 13 May.

Speaker: Shu-Jiao Song (UWA)

Title: On constructions of arc-transitive maps

Abstract: A map is a 2-cell embedding of a graph into a surface. An automorphism of a map is an automorphism of the underlying graph which preserves the supporting surface. A map is called arc-transitive if its automorphism group is transitive on the arc set. Arc-transitive maps include chiral maps, regular maps and non-rotary arc-regular maps. In this talk, I will talk about the constructions of these three types.

Speaker: Don Taylor (The University of Sydney)

Title: Reflection subgroups of unitary reflection groups

Abstract: The usual notion of a reflection in Euclidean space has a natural analogue in vector spaces over the complex numbers. In this talk I will give a brief introduction to finite groups generated by complex (i.e., unitary) reflections and report on some results characterising reflection subgroups and their normalisers. An imprimitive unitary reflection group is a normal subgroup of the group of m-th roots of unity by a symmetric group. For these groups there are uniform methods of proof. However, for 15 exceptional (primitive) groups I resort to calculations using the Magma computation algebra system.

Talk title: The Discovery of Janko's Sporadic Simple Groups. Talk abstract: In 1966, Zvonimir Janko published a paper which revolutionised finite group theory. The previous year, working as a Research Fellow at the Institute of Advanced Study within the Australian National University, Janko constructed a new simple group which was neither an alternating group nor a group of Lie type. The 1966 paper contains the complete details.

Before 1965 only five sporadic simple groups were known. They had been discovered almost exactly one hundred years prior (1861 and 1873) by Émile Mathieu but it was not until 1900 that G. A. Miller proved their simplicity.

By 1976 the number of new sporadic simple groups had risen to 21 and Janko had found four of them, including the first and the last. This talk recounts some of the history of those exciting times.

About our speaker: In 1968 Don Taylor graduated from Monash University with an MSc supervised by Professor Zvonimir Janko. He then travelled to the University of Oxford where he completed a DPhil with Professor Graham Higman FRS.

In 1972 Don took up a lectureship at La Trobe University and in 1975 he moved to Sydney where he has been ever since. He has written several books on group theory, the most recent (with Gus Lehrer) on complex reflection groups.

Since 2007 Don has been an Honorary Associate Professor at the University of Sydney, working on reflection groups and algorithms for groups of Lie type.

Speaker: Melissa Lee (UWA)

Title: Generalising hemisystems and relative hemisystems

Abstract: Segre determined in 1965 that the only nontrivial m-covers of H(3,q^2), q odd, have m=(q+1)/2. He named these covers hemisystems because they constitute half of the lines on every point. Since then, there have been a variety of generalisations of hemisystems, including to nonclassical generalised quadrangles. Vanhove showed that the dual of a hemisystem, namely a (q+1)/2-ovoid of a dual Hermitian space DH(2d-1,q^2), gives rise to a distance regular graph, and such distance regular graphs would be new for d > 2. Relative hemisystems, defined by Penttila and Williford in 2011, are the analogues of hemisystems for q even. Their definition was motivated by the desire to generate rare primitive Q-polynomial 3-class association schemes. There has been no exploration of relative hemisystems on nonclassical generalised quadrangles in the literature, nor has there been any discussion about relative hemisystems as the unique nontrivial relative m-covers. In this talk, I will explore such generalisations of relative hemisystems, as well as presenting some results on m-ovoids of dual polar spaces, in our search for new distance regular graphs. This talk is a summary of my recently accepted Masters dissertation.

Speaker: Cheryl Praeger (UWA)

Title: Infinite permutation groups with all infinite normal subgroups transitive

Abstract: The "O'Nan-Scott approach" to describing structure for finite primitive permutation groups is now a standard way to understand and analyse these groups. I will discuss an exploration made with Peter Neumann and Simon Smith of the extent to which a similar approach might be successful in the realm of infinite groups.

Speaker: Joanna Fawcett (UWA)

Title: Partial linear spaces with a primitive affine automorphism group of rank 3

Abstract: A partial linear space consists of a finite non-empty set P of points and a collection L of subsets of P called lines such that each pair of points lies on at most one line, and each line contains at least two points. A partial linear space is proper if it is not a linear space or a graph. In this talk, we will consider some recent progress on classifying the proper partial linear spaces with a primitive affine automorphism group of rank 3.

Talk title: The random graph and its friends

Talk abstract: There is a countably infinite graph R (first explicitly constructed by Richard Rado) with the following remarkable property: If we choose a countable random graph by selecting edges independently with probability 1/2, then with probability 1 it is isomorphic to R. (This fact was implicit in a paper of Erdos and Renyi at about the same time as Rado's construction.) The graph has many other surprising properties, and occurs in a number of guises.

It turns out that the graph is produced by a construction by Fraisse more than ten years earlier, which builds homogeneous relational structures with prescribed finite substructures, and shows its uniqueness. But even Fraisse had been anticipated by Urysohn, who showed (in a posthumous paper a quarter of a century earlier) that there is a unique homogeneous Polish space containing all finite metric spaces.

It is natural to ask what happens if we dualise Fraisse's construction by turning the arrows around. There is indeed a dual construction; among other things it gives a new way to build a remarkable topological space, the pseudo-arc.

Speaker: Rosemary Bailey (St. Andrews)

Title: Circular designs balanced for neighbours at distances one and two

Abstract: We consider experiments where the experimental units are arranged in a circle or in a single line in space or time. If neighbouring treatments may affect the response on an experimental unit, then we need a model which includes the effects of direct treatments, left neighbours and right neighbours. It is desirable that each ordered pair of treatments occurs just once as neighbours and just once with a single unit in between. A circular design with this property is equivalent to a special type of quasigroup. In one variant of this, self-neighbours are forbidden. In a further variant, it is assumed that the left-neighbour effect is the same as the right-neighbour effect, so all that is needed is that each unordered pair of treatments occurs just once as neighbours and just once with a single unit in between. I shall report progress on finding methods of constructing the three types of design.

Speaker: Peter Cameron (St. Andrews)

Title: Idempotent generation and road closures

Abstract: This is part of a big project linking the properties of a transformation semigroup to those of the permutation group forming its group of units. The particular question I am discussing is this: For which permutation groups $G$ is it the case that, for any transformation $a$ of rank 2, the semigroup $ angle G,a angle etminus G$ is generated by its idempotents? Such a group must be primitive, but not all primitive groups have this property. It turns out to be equivalent to the property that, for any orbital graph for $G$ (regarded as a road network), if we close the edges in a block of imprimitivity for the action of $G$, the network remains connected. We have a conjecture about the primitive groups which satisfy this condition. This is joint work with João Arajúo (Lisbon).

Talk title: Design of dose-escalation trials: Research spurred by a trial that went wrong

Talk abstract: In March 2006 the topic of designed experiments briefly hit the British newspaper headlines when a clinical trial near London went badly wrong. When a working party of the Royal Statistical Society looked into this, my experience from designing experiments in other areas, such as agricultural field trials and microarray experiments, proved useful in improving the design of dose-escalation trials to obtain better information without compromising safety or using more volunteers. I shall say something about the recommendations of the RSS working party, something about the ethical constraints on Phase I clinical trials, and something about the new designs.

Speaker: Stephen Glasby (UWA)

Title: Linear codes from matrices: twisted centralizer codes

Abstract: A linear code is a subspace of a finite vector space. These are the codes most commonly used in error-correcting applications to digital communication and storage. I will report on new classes of linear codes we construct using matrices. I will focus on the dimension and minimal distance of the special case of "twisted centralizer codes". Prior knowledge of codes is not a prerequisite.

Speaker: Barbara Baumeister (Bielefeld University)

Title: Covering a group by conjugates of a coset

Abstract: For every doubly transitive permutation group G the conjugates of a non-trivial coset of a point stabilizer H cover the group. This implies that every non-trivial conjugacy class of elements of G that contains an element of H does contain a transversal of H in G and that every other non-trivial conjugacy class contains a transversal for the set of cosets of H in G different from H. We study the finite groups satisfying this property; more precisely the class of primitive permutation groups, called CCI groups. In fact they properly contain the class of 2-transitive permutation groups.

Speaker: Tomasz Popiel (UWA)

Title: Generalised quadrangles and simple groups

Abstract: Generalised quadrangles (GQs) with point-regular automorphism groups have been studied since at least 1992, when Dina Ghinelli conjectured (but fell slightly short of proving) that if a GQ of order (s,t) admits a group of automorphisms acting regularly on points, then s=t. The abelian case is well understood, in the sense that a GQ with an abelian point-regular group of automorphisms necessarily arises from a so-called pseudo-hyperoval in a certain projective space. In particular, there are many examples of such GQs. A natural question in the non-abelian case is the following. Let T be a non-abelian finite simple group and r a positive integer, and consider the group M=T^r acting on itself by both right and left multiplication. Can there be a generalised quadrangle Q with point set M that admits such a group? This scenario arises, in particular, if we assume that Q admits a point-primitive automorphism group of O’Nan-Scott type HS (r=1) or HC (r>1). The latter situation has previously been shown to be impossible under certain additional assumptions: first line-primitivity (Bamberg, Giuidici, Morris, Royle & Spiga, 2012), and then line-transitivity (Bamberg, Popiel & Praeger, to appear). As part of a long-term project to classify the point-primitive GQs, we have almost managed to show that, in fact, no symmetry assumption on lines is necessary. As of this writing, it is known that r must equal 1, and that T must be one of 17 groups of Lie type (each of Lie rank at most 7). I will sketch the proof of this near-result and, time permitting, summarise our progress on the overarching classification project. This is joint work with John Bamberg and Cheryl Praeger.

Speaker: Michael Giudici (UWA)

Title: Simple automorphic loops

Abstract: A loop is a set Q with binary operation * such that the Cayley table for * is a Latin square and there is a neutral element 1 such that 1*x=x*1=x for all x in Q. Groups are examples of what are called automorphic loops. In this talk I will discuss a project in progress with Colva Roney-Dougal that is attempting to classify all simple automorphic loops.

Talk title: Conformal geometry and taming infinity

Abstract:

The world around us appears to involve lengths and angles. From these emerge the classical notions of shape and symmetry configured in 3 dimensions. Our mathematical ancestors realised that these notions should be important, not only for construction and surveying, but also for understanding "life, the universe and everything". In the process of simplifying complicated structures it turns out that an important role is played by conformal geometries - these are spaces where there is a notion of angle but not length.

We will discuss some elegant tools for working with these less than rigid geometries and how they help treat other problems such as writing massive particle equations and taming non-compact spaces.

Title: "Higher-Spin Gauge Symmetries and Space-Time"

Abstract: Higher-spin gauge theory is a theory exhibiting an infinite extension of usual space-time and internal symmetries of modern models of fundamental interactions. I will discuss general aspects of the higher-spin gauge theory with the emphasis on the higher-spin symmetry and its implications for fundamental concepts of space-time.

Cheese and wine will be served in the Physics Common Room from 5.00 pm.

Speaker: Yian Xu (UWA)

Title: Normal and non-normal Cayley graphs of non-prime power order

Abstract: Let Q be a known generalised quadrangle of order (q - 1, q + 1), where q is a prime power p^k with p eq 5. The point graph of Q has order q^3 and is called a BG-graph. In Bamberg and Giudici (2011), the authors proved that a BG-graph is a normal and non-normal Cayley graph for two isomorphic groups. In this paper, we show that the Cayley graphs obtained from a finite number of BG-graphs by Cartesian product, direct product, and strong product respectively also possess this property. In particular, we construct normal and non-normal Cayley graphs of non-prime power order.

Title: Queues on Interacting Networks

Abstract: We have all had the unpleasant experience of waiting for too long at some queue. We seem to lose a significant amount of time waiting for some operator to reply to our call or for the doctor to be able to see us. Queues are the object of study of queuing theory, i.e., the branch of applied mathematics that studies models involving a number of servers providing service to at least one queue of customers. Queues are an example of a stochastic process and a group of connected queues is an example of a network.

In this talk, we will give a brief overview of the area of stochastic processes, ranging from classroom examples to their impact on industry and technology. We then introduce networks with interacting architectures and look at different architectures through examples. The aim is to give an idea of the mathematical challenges that these interactions create and the importance of incorporating this level of detail in mathematical analysis.

Time and place: 16:00 Friday 03/03/2017 in Weatherburn LT

Title: Colour-preserving automorphisms of Cayley graphs

Abstract: A Cayley digraph Cay(G,S) comes equipped with a natural colouring of its arcs: the arc (g,sg) has colour s. It is an easy exercise to show that the group of colour-preserving automorphisms of the graph is exactly G. If the digraph is actually a graph, one can forget about the orientation of the arcs and colour the edges instead: the edge {g,sg} has colour {s,s^(-1)}. Determining the group of colour-preserving automorphisms in this case is much harder. This is the subject of this talk, with a specific focus on the case when the colour-preserving group does not normalise G.

Time and place: 16:00 Friday 10/03/2017 in Weatherburn LT

Title: Orbital graphs

Abstract: In joint work with Markus Pfeiffer and Chris Jefferson (both in St Andrews) I got interested in algorithms for permutation groups, in particular search algorithms that use partition backtracking. Typical examples would be algorithms that calculate the intersection of two permutation groups, a set stabiliser in a given group or the normaliser of a subgroup. We noticed that some algorithms could be improved by exploiting some group action, and this is where graphs enter the scene. In my talk I will introduce orbital graphs in this specific context and explain how they improve our algorithms. Moreover I will discuss in detail why, sometimes, these graphs do not help at all!

Talk title: Snowflakes, viruses and algorithms

Talk abstract: In this talk we will look at different objects that have one thing in common: they exhibit symmetry. Why is this relevant? How can we grasp the concept of symmetry mathematically and how can we understand it? Inspired by examples, we will look at symmetry and discuss how it can be used to simplify complex calculations, to understand and classify objects from real life, and to improve algorithms.

Time and place: 16:00 Friday 17/03/2017 in Weatherburn LT

Title: On top locally-s-distance-transitive graphs

Abstract: A graph with some graph symmetry property is called "top", if it cannot be viewed as a nontrivial normal quotient of some other graph with the same graph symmetry property. Therefore, a graph being top implies that it has no nontrivial normal multicovers, including normal covers. John Conway proved that every s-arc-transitive graph has a nontrivial s-arc-transitive normal cover, so there is no top s-arc-transitive graph. However, there exist top locally-s-distance-transitive graphs, and complete multipartite graphs are examples of this. In this talk, I will give a generic condition (E graphs) for locally-s-distance-transitive graphs to be top, and examples and characterizations of locally-s-distance transitive graphs.

Time and place: 16:00 Friday 24/03/2017 in Weatherburn LT

Title: Symmetric designs and their automorphism groups

Abstract: A symmetric design (v,k,λ) is an incidence structure consisting of a set of v points and a set of v blocks with an incidence relation such that every block is incident with exactly k points, and every pair of points is incident with exactly λ blocks. An automorphism group of a symmetric design is a group of permutations on points of the design which maps blocks to blocks and preserves the incidence and non-incidence relations. An automorphism group of a design naturally acts on the elements of the design, namely, points, blocks and flags. The main part of this talk is devoted to giving a survey on recent study of symmetric designs which admit a group of automorphism acting primitively (resp. transitively) on the set of points (resp. blocks). We in addition give a list of possible parameters of designs admitting a flag-transitive and point-primitive almost simple automorphism group with socle a finite exceptional simple group.

Time and place: 16:00 Friday 31/03/2017 in ENCM:G04 Engineering LT2

Title: Expander Mixing Lemma in Finite Geometry

Abstract: https://www.maths.uwa.edu.au/~glasby/GandCSeminars.html

Time and place: 16:00 Friday 07/04/2017 in Weatherburn LT

Title: Regular maps and simple groups

Abstract: A regular map is a highly symmetric embedding of a finite graph into a closed surface. I will describe a programme to study such embeddings for a rather large class of graphs: namely, the class of orbital graphs of finite simple groups.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GandCSeminars.html

WHAT IS A PAWSEY CLINIC?

Pawsey Clinics are events organised for researchers who need to use Pawsey services. They can find out how to gain access to supercomputing, data or visualisation systems and how they can benefit from the expertise of Pawsey staff in transitioning their research. This is also an opportunity for current users who need one-on-one advice from one of Pawsey’s experts to take their research to the next level or get help with code issues.

WHO SHOULD ATTEND?

Researchers who do not know how supercomputing, data and visualisation services can improve their projects.

Pawsey users that need a one-on-one session with a Pawsey expert to get answers about issues including queue scripts, source code compilation and debugging, profiling, data and workflow needs, and any other matters they may have.

TIME AND VENUE:

The University of Western Australia, Conference Room in the Pawsey at UWA offices located on the ground floor of the Physics building. 9.00 am to 12.00 pm, Thursday 13 April 2017. The is a free event. Please RSVP before Friday 7 April Please feel free to share this information with your colleagues if you think it might be of interest. The information has been uploaded into our calendar of events: https://www.pawsey.org.au/events/?date=Apr%202017 and also into the Pawsey Clinics page: https://www.pawsey.org.au/pawseyclinics/

Cheryl began her career at the University of Western Australia in 1976, and has recently retired after 40 years of service. To celebrate her amazing and influential career, there will be a mini-symposium hosted by the University of Western Australia on April 13 (2017).

There will be four plenary lectures devoted to Cheryl’s legacy given by some of her closest colleagues:

- Cai Heng Li (Southern University of Science and Technology China)

- Martin Liebeck (Imperial College, University of London)

- Alice Niemeyer (RWTH Aachen University)

- Jacqui Ramagge (University of Sydney)

See

https://www.cmsc.uwa.edu.au/news/cheryl-praeger-retirement-mini-symposium

for more details.

Time and place: 16:00 Friday 21/04/2017 in Weatherburn LT

Title: Cyclotomic difference sets

Abstract: The study of cyclotomic difference sets dates back to Paley in 1930s ' and has become well-known for its connection to the long-standing conjecture that every finite flag-transitive projective plane is Desarguesian. In this talk, I will propose a new approach to this classic problem, which gives insights from character sums, $p$-adic gamma functions and systems of polynomial equations.

Time and place: 16:00 Friday 28/04/2017 in ENCM: G04 Engineering LT2 (N.B. alternative venue!)

Title: Maximally symmetric p-groups

Abstract: The set of finite p-groups of nilpotency class r, rank d and exponent p includes a universal group P such that every group in this set is a quotient of P. For certain values of r and p, Bamberg, Glasby, Morgan and Niemeyer (BGMN) constructed P as a group whose underlying set is a Cartesian product of vector spaces, and proved some results about the structure of Aut(P). The main goal of BGMN's work was to determine the maximal subgroups H of the general linear group GL(d,p) for which there exists an associated p-group G with nilpotency class at most 4, rank d and exponent p, such that Aut(G) induces H on the Frattini quotient of G. BGMN showed that such a group G exists when p > 3, and when H lies in an Aschbacher class other than C_6 or C_9.

In this talk, I will present new results about the structure of Aut(P) for certain universal groups P. I will also give new examples of matrix subgroups H, which lie in class C_6 or C_9 and are not necessarily maximal, with associated p-groups G. In particular, I will show that for each odd prime p, there exists a p-group related to the exceptional group of Lie type G_2(p).

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GandCSeminars.html

Michael Giudici (University of Western Australia)

Time and place: 16:00 Friday 05/05/2017 in Weatherburn LT

Title: Symmetry of digraphs

Abstract: The symmetry of graphs is a widely studied topic, but less has been done on the symmetry of digraphs. In this talk I will outline some of the fundamental differences between the two topics and outline some recent research with Glasby, Li, Verret and Xia.

Time and place: 16:00 Friday 12/05/2017 in Weatherburn LT

Title: m-ovoids of regular near polygons

Abstract: In this talk, an introduction to the basics of regular near polygons will be given together with a summary of the work of Bart De Bruyn and Frederic Vanhove (2011). In joint work with Jesse Lansdown and Melissa Lee, the speaker will present some recent work which improves some of the results of De Bruyn and Vanhove. No knowledge of finite geometry or regular near polygons will be required.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GandCSeminars.html

Time and place: 16:00 Friday 19/05/2017 in Weatherburn LT

Title: Edge-transitive oriented graphs of valency four: quotients and cycles

Abstract: I will report on our analysis of the structure of this family of oriented graphs in terms of their quotients – especially their cyclic normal quotients. This approach was an experiment to see how a quotient analysis would work for oriented graphs. It turns out it gives new insights, and makes possible various classifications. For example, each candidate graph with independent cyclic normal quotients is shown to be a normal cover of a graph from one of five explicit families. Graphs from most of these five families had previously been used in the literature to illustrate various local properties. This new quotient approach highlights their importance. There are several open problems which I’ll mention

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/S17.html

Title: Canonical Decompositions of Abelian Groups

Time and place: 16:00 Friday 26/05/2017 in Weatherburn LT

Abstract: A canonical decomposition theorem for a category of algebraic structures states that every object A has an essentially unique direct sum decomposition A = B_1 + ... + B_n in which each B_i has a specified structure. For example, every finitely generated abelian group A can be decomposed as A = F + T where F is free and T is finite. I describe a canonical decomposition theorem for the category of finite rank torsion-free abelian groups.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/S17.html

Title: The isomorphism problem for universal enveloping algebras of Lie algebras

Time and place: 16:00 Friday 02/06/2017 in Weatherburn LT

Abstract: Let U(L) denote the universal enveloping algebra of a Lie algebra L. George Bergman in his influential article of 1978 remarked that it was an open question whether an isomorphism between two universal enveloping algebras U(L) and U(K) implied the isomorphism between the Lie algebras L and K. Since the publication of Bergman’s paper we have learned that the answer to this problem is no. On the other hand, as far as we know, it is a rather rare phenomenon among small-dimensional Lie algebras that non-isomorphic algebras have isomorphic enveloping algebras. This is what makes this problem so interesting.

In the talk, I will review the known results on general Lie algebras and will focus on small-dimensional solvable Lie algebras. I will present some interesting examples to show that isomorphism between universal enveloping algebras is possible without isomorphism of the corresponding Lie algebras.

Future seminars (and past ones back to 2007) may be found at https://www.maths.uwa.edu.au/~glasby/S17.html

Speaker: Gordon Royle (University of Western Australia) Title: Hamilton Cycles in Cubic Graphs Time and place: 16:00 Friday 01/09/2017 in Weatherburn LT

Abstract: Starting from early attempts to prove the 4-colour theorem, the study of hamilton cycles in cubic graphs has been an integral part of graph theory for well over a century. Despite a number of beautiful results, there are still many fundamental unresolved questions about both the existence and enumeration of hamilton cycles in various classes of cubic graphs. In this talk I will describe various results and open problems from this body of work, including some recent counting results obtained with Irene Pivotto.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/S17.html

Speaker: Nick Gill (University of South Wales) Title: On Cherlin's conjecture for primitive binary groups Time and place: 16:00 Friday 08/09/2017 in Weatherburn LT

Abstract: Roughly speaking, a permutation group G on a set X is called "binary" if one can calculate the orbits of G on X^n for any n once one knows the orbits of G on X^2. This definition was conceived in the 1990's by Cherlin who was interested in ideas coming out of model theory. He also formulated a conjecture which asserts that all primitive binary permutation groups lie in one of a number of known families.

Thanks to work by Cherlin himself, and by Wiscons, this conjecture has been reduced to a statement about almost simple groups. I will describe work with Francesco Dalla Volta (Milano-Bicocca), Francis Hunt (South Wales), Martin Liebeck (Imperial) and Pablo Spiga (Milano-Bicocca) which aims at resolving the conjecture in full. This work is group theoretic in nature.

I will also try and explain the model theoretic context of the conjecture, with a particular focus on Lachlan's theory of homogeneous relational structures.

Abstract: We define a class of objects in pappian projective planes by a simple algebraic formula and parametrized by quadratic field extensions. These objects turn out to be ordinary hermitian curves if the extension is separable, and projections of certain quadrics otherwise. Endowed with the secant lines, we call the resulting point-block incidence structures "Pappian unitals". These have some remarkable properties such as the lack of O'Nan configurations, the admittance of translations and a nontrivial group of projectivities, and a characterization via a geometric construction using the André representation of the projective plane relative to the quadratic extension. We classify the embeddings of all Pappian unitals in arbitrary pappian projective planes, recovering and extending a recent result by Korchmáros, Cossidente and Szönyi for finite hermitian unitals.

Abstract: We study random constructions in incidence structures using a general theorem on set systems. Our main result applies to a wide variety of well-studied problems in finite geometry to give almost tight bounds on the sizes of various substructures. This is joint work with Jacques Verstraete (UCSD).

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/S17.html

Abstract: The Cayley index of a Cayley digraph on a finite group G is the index of (the regular representation of) G in the automorphism group of that digraph. The minimum Cayley index of one is attained by digraphs called DRRs (Digraphical Regular Representations). These are classified by Babai who shows that, apart from five groups, every finite group admits a DRR. That is, apart from five exceptions, every finite group G has a Cayley digraph such that G is the full automorphism group of that digraph. In this talk, we'll consider the question of what might make the Cayley index "large" relative to the number of vertices (= the order of the group). A result of Morris shows that any Cayley digraph on a cyclic $p$-group (with $p$ an odd prime) has Cayley index super-exponential in $p$, if there exists another distinct regular subgroup. This result was later generalised to $p=2$. So these results say that if a digraph is Cayley for two distinct $p$-groups, one of which is cyclic, the Cayley index is "large". In joint work with Morris and Verret, we considered if cyclic $p$-groups are exceptional in this respect. We found that, in contrast to the previous results, every non-cyclic abelian $p$-group ($p$ odd) of order at least $p^3$ admits a Cayley digraph of Cayley index $p$ that admits two distinct regular subgroups. I'll show how this result works, and give some further questions on this topic.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/S17.html

Speaker: John Bamberg (University of Western Australia) Title: q-analogues of designs and the 2-Fano plane Time and place: 16:00 Friday 13/10/2017 in Weatherburn LT

Abstract: A q-analogue of a t-design, called a t-(n,k, ambda)_q design, is a set of k-subspaces of F_q^n such that each t-subspace is contained in exactly ambda elements. If t = 1, the design is a q-analogue of a Steiner system, and is denoted S_q(t,k,n). Braun, Østergård, Vardy and Wassermann have shown that q-Steiner systems do exist, but existence is not known in the case of S_q[2; 3; 7], the q-analogue of a Fano plane. Kiermeir, Kurz and Wassermann have shown that if a S_2[2; 3; 7] exists, then the order of its automorphism group would be at most 2. We will present the progress of our search for S_2[2; 3; 7] (or its non-existence), in particular, we have found that no 2-Fano plane exists with automorphism group of order 2. This is join work with Ferdinand Ihringer, Jesse Lansdown, and Gordon Royle.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/S17.html

Speaker: Hongxue Liang (University of Western Australia)

Title: Flag-transitive point-primitive non-symmetric 2-(v,k,2) designs

Time and place: 16:00 Friday 27/10/2017 in Woolnough Lecture Theatre 2 GGGL:107

Abstract: A 2-(v,k,λ) design is a finite incidence structure D=(P, B) consisting of v points and b blocks such that every block is incident with k points, every point is incident with r blocks, and any two distinct points are incident with exactly λ blocks. D is called symmetric if v=b (or equivalently r=k), and non-trivial if 1< k< v. A flag of D is an incident point-block pair (α, B) where α is a point and B is a block. An automorphism of D is a permutation of the points which also permutes the blocks. The set of all automorphisms of D with the composition of maps is a group, denoted by Aut(D). A subgroup G≤Aut( D) is called point-primitive if it acts primitively on P and flag-transitive if it acts transitively on the set of flags of D. In this talk, we will focus on the flag-transitive point-primitive automorphism groups of non-symmetric 2-(v,k,2) designs.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/S17.html

Abstract: The canonical basis, which is a particular type of basis of a vector space will be introduced in this talk, and a sufficient and necessary condition is given to determine the existence of such a basis for a vector space. The structures of canonical bases are then used to study Cayley graphs of extraspecial $2$-groups of order $2^{2r+1}$ ($r eq 1$), which are further shown to be normal Cayley graphs and $2$-arc-transitive covers of $2r$-dimensional hypercubes. ​

Title: Worst-case approximability of functions on finite groups by endomorphisms and affine maps

Time and place: 16:00 Friday 10/11/2017 in Weatherburn LT

Abstract: https://www.maths.uwa.edu.au/~glasby/S17.html

Speaker: Stephen Glasby (University of Western Australia)

Title: Norman involutions and tensor products of unipotent Jordan blocks

Time and place: 16:00 Friday 17/11/2017 in Weatherburn LT

Abstract: Suppose R is an rxr unipotent matrix over some field F, i.e. its characteristic polynomial is (t-1)^r. The Jordan form of R is a sum of unipotent Jordan blocks, so we obtain some partition of r. If S is a unipotent sxs matrix over F, then so is R times S. To understand the partition of rs afforded by R times S it suffices to understand the partition afforded by J_r times J_s where J_r denotes a single rxr unipotent Jordan block. When char(F)=p, we denote this partition by lambda(r,s,p).

When p>0, the partitions lambda(r,s,p) are shrouded in mystery. Assume r<= s. We show that there is a larger set of 2^{r-1}-1 partitions (which is independent of p) and contains the mysterious partitions. These partitions correspond to involutions in the symmetric group S_r of degree r, and also to nonempty subsets of the set {1,2,...,r-1}. We also show that the group G(r,p)=<lambda(r,s,p) | s>= r>, is a wreath product, and we determine its structure.

One motivation for this research comes from representation theory: understanding the structure of the Green ring. This is joint work with Cheryl E. Praeger and Binzhou Xia.

Paul Dirac proposed the baryon symmetric universe in 1933. This proposal has become very attractive now since it seems that all pre-existing asymmetry would have been diluted if we had an inflationary stage in the early universe. However, if our universe began baryon symmetric, the tiny imbalance in numbers of baryons and anti-baryons which leads to our existence, must have been generated by some physical processes in the early universe. In my talk I will show why the small neutrino mass is a key for solving this long standing problem in understanding the universe we observe.

Bio:

Professor Tsutomu Yanagida is a world-renowned expert on theoretical high energy physics and cosmology. He is famous, in particular, for the Seesaw mechanism (proposed in 1979) and for the Leptogenesis (proposed in 1986). The Seesaw mechanism predicts very small neutrino masses; the 2015 Nobel Prize in Physics was awarded for the discovery of neutrino oscillations, which show that neutrinos have small masses. The Leptogenesis explains the baryon asymmetry observed in the Universe. Professor Tsutomu Yanagida has published more that 500 papers, which have generated 29,666 citations (as of 28 November 2017). His h-index is 80. He co-authored the book ``Physics of Neutrinos and Applications to Astrophysics’’ written jointly with M. Fukugita and published in 2003.

Professor Tsutomu Yanagida obtained his PhD in 1977 from Hiroshima University. In 1979, he joined Tohoku University in Japan, first as Assistant Professor, then Associate Professor (1987) and finally Professor (1990). In the period 1996—2010, he was Professor at Tokyo University. He is currently Professor at Kavli Institute for the Physics and Mathematics of the Universe, Tokyo where he has been since 2010.

Title: Transitivity Notions for Groups

Time and place: 16:00 Friday 02/02/2018 in Social Sciences LT (Note the unusual location!)

Abstract: https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

There will be `cake' in the Mathematics and Statistics tea room at 15:40. The seminar starts at 16:00, and after 17:05 we go to the UniClub or Student Tavern for a drink.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

Similarly, principals of symmetry are also some of the fundamental and most useful tools in modern mathematical natural science that play a major role in theory and applications. As a consequence, it is not surprising that the desire to understand the origin of life, based on the genetic code, forces us to involve symmetry as a mathematical concept.

The genetic code can be seen as a key to biological self-organisation. All living organisms have the same molecular bases - an alphabet consisting of four letters (nitrogenous bases): adenine, cytosine, guanine, and thymine. Linearly ordered sequences of these bases contain the genetic information for synthesis of proteins in all forms of life. Thus, one of the most fascinating riddles of nature is to explain why the genetic code is as it is.

Genetic coding possesses noise immunity which is the fundamental feature that allows to pass on the genetic information from parents to their descendants. Hence, since the time of the discovery of the genetic code, scientists have tried to explain the noise immunity of the genetic information. In this talk we will discuss recent results in mathematical modelling of the genetic code with respect to noise immunity, in particular error-detection and error-correction.

Cheese and wine to follow in the Maths common room (5pm-6pm)

Title: Abstract ovals

Time and place: 16:00 Friday 09/02/2018 in Woolnough LT (note unusual location)

Abstract: This talk will be about “abstract ovals” which were introduced in 1966 by Francis Buekenhout. In particular, we will present some new results, that have some interesting connections with the theory of Moufang sets and classical permutation group theory. This is joint work with Tim Harris and Tim Penttila.

Title: Uniform Domination for Simple Groups

Time and place: 16:00 Friday 23/02/2018 in Robert Street LT

Abstract: It is well known that every finite simple group can be generated by just two elements. In fact, by a theorem of Guralnick and Kantor, there is a conjugacy class C such that for each non-identity element x there exists an element y in C such that x and y generate the entire group. Motivated by this, we introduce a new invariant for finite groups: the uniform domination number. This is the minimal size of a subset S of conjugate elements such that for each non-identity element x there exists an element s in S such that x and s generate the group. This invariant arises naturally in the study of generating graphs.

In this talk, I will present recent joint work with Tim Burness, which establishes best possible results on the uniform domination number for finite simple groups, using a mix of probabilistic and computational methods, together with recent results on the base sizes of primitive permutation groups.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

There will be cake in the Mathematics and Statistics tea room at 15:40. The seminar starts at 16:00, and after 17:05 we go to the UniClub or Student Tavern for a drink.

Title: Algebraic Models of Computation: Nearrings of Group Automata

Time and place: 16:00 Friday 2/3/2018 in Weatherburn LT

Abstract: Input-Output Automata are a simple but powerful model of processes. Inputs arrive and, depending upon the current state, the automaton outputs a symbol and moves to a new state. In the case that the input and output alphabets are identical, we can compose them by connecting the input of one automaton to the output of another. By equipping the alphabet with a group operation, we can compose automata in parallel, feeding each the same input and summing their output using the group operation. We obtain a collection of synchronous state automata, equivalently a set of mappings on infinite sequences.

The algebraic tool that best allows us to model, manipulate and analyse collections of such automata are nearrings, the nonlinear analogue of rings. A (right) nearring (N,+,*) is a (not necessarily abelian) group (N,+), a semigroup (N,*) with one distributive law (a+b)*c= a*c+b*c. Standard examples are the set of all mappings of a group to itself under pointwise addition and functional composition, the closed subalgebras of this, or the class of nearfields as used in describing certain projective planes including the Hall Plane. In this talk we will outline this construction in more detail, demonstrate the complexity of the nearring of automata and introduce radicals as simplifying tools. In particular we will show that the amnesiac mapping and the set of delay automata help bound the Jacobson 2-radical, defining it in certain cases. In these cases we can write down the 2-semisimple automata nearrings explicitly. Input-Output Automata are a simple but powerful model of mechanical, electronic and other processes. In the case that the input and output alphabets are identical, we can compose them by connecting the input of one automaton to the output of another. By equipping the alphabet with a group operation, we can compose automata in parallel, feeding each the same input and summing their output using the group operation.

This work is supported by the grant P29931 and SFB Project F5004 of the Austrian Science Foundation, FWF.

Title: Structural results on tetravalent half-arc-transitive graphs

Time and place: 16:00 Friday 09/03/2018 in Weatherburn LT

Abstract: In this talk we focus on tetravalent graphs admitting a half-arc-transitive subgroup of automorphisms, that is a subgroup acting transitively on its vertices and its edges but not on its arcs. One of the most fruitful approaches for the study of structural properties of such graphs is the well known paradigm of alternating cycles and their intersections which was introduced by Marušič 20 years ago.

We introduce a new parameter for such graphs, giving a further insight into their structure. Various properties of this parameter are given. The obtained results are used to establish a link between two frameworks for a possible classification of all tetravalent graphs admitting a half-arc-transitive subgroup of automorphisms, the one proposed by Marušič and Praeger in 1999, and the much more recent one proposed by Al-bar, Al-kenai, Muthana, Praeger and Spiga which is based on the normal quotients method.

We also present results on the graph of alternating cycles of a tetravalent graph admitting a half-arc-transitive subgroup of automorphisms. A considerable step towards the complete answer to the question of whether the attachment number necessarily divides the radius in tetravalent half-arc-transitive graphs is made.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

There will be cake in the Mathematics and Statistics tea room at 15:40. The seminar starts at 16:00, and after 17:05 we go to the UniClub or Student Tavern for a drink.

Title: Finite two-distance-transitive circulants

Time and place: 16:00 Friday 16/03/2018 in Weatherburn LT

Abstract: In this talk, we give a complete classification of the family of 2-distance-transitive circulants. We show that a 2-distance-transitive circulant is a cycle, a Paley graph of prime order, a regular complete multipartite graph, or a regular complete bipartite graph of order twice an odd integer minus a 1-factor.

This is joint work with J.Y. Chen and C.H. Li.

Title: Bases for permutation groups and the Saxl graph

Time and place: 16:00 Friday 23/03/2018 in Weatherburn LT

Abstract: Let G be a permutation group on a set X. A base for G, is a subset B of X such that the pointwise stabiliser of the elements of B is trivial. There has been a large amount of recent research on the size of a base of a primitive permutation group, culminating with the recent proof of Pyber’s Conjecture. At the same time there has been a large amount of work devoted to finding the primitive groups with a base of size two. For such groups we can define the Saxl graph of G to be the graph with vertex set X and two elements are joined by an edge if they are a base. I will discuss some recent work with Tim Burness that investigates some of the properties of this graph.

Please come at 15:00 this Friday to meet our BPhil students over cake (which may go quickly).

All welcome.

Title: Vector Colorings of the Categorical Product of Graphs

Time and place: 16:00 Friday 06/04/2018 in Weatherburn LT

Abstract:In 1966 Hedetniemi conjectured the that chromatic number of the categorical product (sometimes called the direct product) of two graphs is equal to the minimum of their individual chromatic numbers. This remains one of the major open questions in the field of graph colorings to this day. We will prove this conjecture for the vector chromatic number, a vector/semidefinite relaxation of the chromatic number. Moreover, we will provide a necessary and sufficient condition for when every optimal vector coloring of the product is induced by optimal vector colorings of the factors.

Title: Random groups

Time and place: 16:00 Friday 13/04/2018 in Weatherburn LT

Abstract: What does a random group look like? To answer this question, we first need to specify which class of groups we are talking about, and to then put a probability distribution on them. I’ll talk about the answer for two different distributions on the class of all finitely-presented groups, presenting some recent joint work with Calum Ashcroft. I’ll then go on to discuss what can be said in the finite case, concentrating on subgroups of the symmetric group, and covering some work in progress with Gareth Tracey.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

There will be cake in the Mathematics and Statistics tea room at 15:40. The seminar starts at 16:00, and after 17:05 we go to the UniClub or Student Tavern for a drink.

All welcome.

Title: Finite groups with a large automorphism orbit

Time and place: Fri 27 Apr, 2018 in Weatherburn LT

Abstract and Slides: https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

Title: Derangement action digraphs and graphs

Time and place: Fri 04 May, 2018 in Weatherburn LT

Abstract: Derangement action digraphs are closely related to group action digraphs, which were introduced by Annexstein, Baumslag and Rosenberg in 1990 as models for interconnection networks underpinning parallel computer architectures. They were introduced by Moharram Iradmusa and me as a generalisation of Cayley digraphs. Some group action digraphs have loops or multiple arcs, and we wished to avoid these. We were in fact looking for a natural family of simple digraphs which properly contains all finite simple vertex-transitive graphs and digraphs, but is not `too large'. We believe that the family of derangement action digraphs satisfies these criteria.

For a non-empty set X and a non-empty subset S of derangements of X (fixed-point-free permutations of X), we define the derangement action digraph DA(X, S) to have vertex set X, and an arc from x to y if and only if y is the image of x under the action of some element of S. Thus by definition it is a simple digraph (no loops, no multiple arcs). It was easy to see that the family of derangement action digraphs contains all Cayley digraphs, but not so easy to see (though it is true) that it also contains all finite vertex-transitive graphs. I’ll report on what we discovered: in particular we found necessary and sufficient conditions on S under which DA(X, S) may be viewed as a simple undirected graph of valency |S|. We investigated various structural and symmetry properties of these digraphs, but are left with several open problems.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

Title: A cycle of maximum order in a graph of high minimum degree has a chord

Time and place: 4pm 11 May 2018, Weatherburn LT

Abstract: A long standing conjecture of Thomassen states that every cycle of maximum order in a 3-connected graph contains a chord. While many partial results towards this conjecture have been obtained, the conjecture itself remains unsolved. In this talk, we shall discuss a result for graphs of high minimum degree, which shows Thomassen's conjecture holds in that case. This result holds for all graphs with high minimum degree, rather than just 3-connected graphs, and this result is within a constant factor of best possible when no assumption is made on the connectivity.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

There will be cake in the Mathematics and Statistics tea room at 15:40. The seminar starts at 16:00, and after 17:05 we go to the UniClub for a drink.

All welcome.

Title: Quotient-complete arc-transitive latin square graphs

Time and place: 4pm 18 May 2018, Weatherburn LT

Abstract: This talk deals with an open problem in my PhD thesis, which investigated arc-transitive diameter two graphs. In that thesis, via normal quotient reduction, we identified two main families of such graphs, namely vertex-quasiprimitive graphs and the so-called quotient-complete graphs. The focus of this talk is the quotient-complete case with at most two complete normal quotients, and the aim is to find an infinite family of examples that fall under this case. In particular we consider latin square graphs corresponding to the Cayley table of finite groups.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

There will be cake in the Mathematics and Statistics tea room at 15:40. The seminar starts at 16:00, and after 17:05 we go to the UniClub for a drink.

All welcome.

Title: Finite simple quotients of mapping class groups

Time and place: 4pm Friday 25 May 2018, Weatherburn LT

Abstract: A natural approach to studying an infinite group is to try to understand its finite quotients (or non-quotients). We discuss the introduction of some new techniques to determine non-quotients in the particular case of the surface mapping class groups, however they apply much more generally. This is joint work with Dawid Kielak.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

There will be cake in the Mathematics and Statistics tea room at 15:40. The seminar starts at 16:00, and after 17:05 we go to the UniClub for a drink.

All welcome.

Title: The Distinguishing Number of Finite Permutation Groups

Time and place: 4pm Friday 15 Jun 2018, Robert Street LT (note unusual venue)

Abstract: The distinguishing number of a permutation group G acting on a set X is the smallest size of a partition of X such that only the identity of G fixes each part of the partition. The symmetric and alternating groups of degree n have distinguishing number n and n-1 respectively. On the other hand, most primitive groups one meets in the wild have distinguishing number two. In fact, apart from the symmetric and alternating groups, the distinguishing number of a primitive group is bounded by an absolute constant. At the recent CMSC retreat, Alice Devillers, Scott Harper and myself looked at a larger class of finite permutation groups, the quasiprimitive and semiprimitive groups. I'll report on what we have found happens in these cases.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

All welcome.

Title: Compatible local actions in arc-transitive digraphs

Time and place: 4pm Friday 22 Jun 2018, Blakers LT

Abstract: Given a G-arc-transitive graph, the local action is the permutation group induced by a vertex-stabiliser Gv on the neighbourhood of v. It plays an extremely important role in the study of these objects. The situation for digraphs is a little more complicated. One can consider both the in- and out-local-action, that is, the permutation group induced by a vertex-stabiliser on the corresponding in- and out-neighbourhood, respectively. Perhaps surprisingly, these two permutation groups need not be isomorphic, they may not even have the same order. We say that two permutation groups are compatible if they arise in this way, that is, as the in- and out-local action of some finite G-arc-transitive digraph. We will discuss the problem of characterising compatible permutation groups. It turns out that the question can be reformulated in a purely group theoretic manner, in a very nice way.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

There will be cake in the Mathematics and Statistics tea room at 15:40. The seminar starts at 16:00, and after 17:05 we go to the UniClub for a drink.

All welcome.

Title: Subspaces of Matrices over Finite Fields with Restricted Rank

Time and place: 4pm Friday 20 Jul 2018, Weatherburn LT

Abstract: Rank-metric codes are codes consisting of matrices over a finite field, with the distance between two matrices defined as the rank of their difference. Delsarte showed that if U is a subspace of m x n matrices in which the rank of every nonzero element is at least k, then the dimension of U is at most n(m-k+1). Linear maximum rank-distance (MRD) codes are subspaces obtaining this bound. They are the rank-metric analogue of MDS codes. Delsarte showed the existence of examples for all parameters. Interest in the topic has increased in recent years due to potential applications, for example in random network coding, and in cryptography.

Finite semifields are division algebras over a finite field, where multiplication is not assumed to be associative. Some constructions are known, but classification remains a difficult open problem. They have been studied in part due to their connections to objects in finite geometry such as projective planes, spreads, ovoids, and flocks. Semifields also correspond to n-dimensional subspaces of n x n matrices where every nonzero element is invertible; i.e. MRD codes with minimum distance n.

In this talk we will give an overview of the known constructions and classifications of semifields and MRD codes. We will present recent algebraic constructions using linearized polynomials and skew-polynomial rings, which constitutes the largest known family.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

Title: The curious incident of the Leningrad mathematicians

Time and place: 4pm Friday 27 Jul 2018, Weatherburn LT

Abstract: The solution of a famous Sherlock Holmes mystery hinges on the fact that none of the characters noticed a dog howling in the nighttime.

Something similar occurred in Leningrad in the 1980s: two issues of the Proceedings of the Steklov Institute contained incompatible theorems about direct decompositions of finite rank torsion-free abelian groups, and no-one seems to have noticed.

I will describe the theorems and try to resolve the mystery.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

Title: Online Collaborative LaTeX Authoring with Overleaf

Time and place: 4pm Friday 03 Aug 2018, Weatherburn LT

Abstract: This seminar is in the (very) occasional CMSC Technical Seminar Series covering useful computational tools for mathematicians.

All of us have extensive collaborations both locally and internationally, and face the logistical challenge of coordinating the efforts of multiple authors in different time zones, on different operating systems and with differing levels of technical interest and acumen.

For many of us, the emergence of Dropbox was the first technical development since the advent of email that changed our collaboration habits, pushing us from “revise-and-reply” to the “shared folder” paradigm.

Over the last 5 or so years, web technologies have advanced greatly, and a number of fully online LaTeX editing platforms (using a web browser as the GUI) have been developed. After a half-decade of launches, name-changes and mergers, one platform, namely Overleaf, is now clearly dominant, both in total numbers of users and range of services offered.

In this seminar, I will briefly describe various collaboration paradigms, give their pros and cons, before giving a fairly extensive introduction to / demonstration of Overleaf.

As with Dropbox, there is always some concern about entrusting one’s work to some initially-unknown remote server, and as with Dropbox, Overleaf’s free service provides something useful enough to engage your attention, but with key limitations that only a paid subscription will remove.

Title: Hemisystems in Finite Geometries

Time and place: 4pm Friday 17 Aug 2018, Weatherburn LT

Abstract: Given an incidence geometry Γ with elements P called points, and elements L called lines, an m-ovoid O is a subset of P such that every line is incident with exactly m elements of O. If m is exactly half the number of lines on a point, then O is called a hemisystem.

Fr´ed´eric Vanhove showed that the existence of hemisystems in the dual Hermitian polar space implies the existence of new distance-regular graphs with classical parameters, so the existence of hemisystems in this case is of great interest.

In this talk I will define these objects and their geometric settings in greater detail, explain some of the history, and present some recent new results.

Title: Structures invariant under maximal subgroups of symmetric groups: a more geometrical O’Nan-Scott Theorem

Time and place: 4pm Friday 24 Jul 2018, Weatherburn LT

Abstract: Maximal intransitive and imprimitive finite permutation groups are stabilisers of subsets, and partitions (sets of subsets) respectively. Similarly maximal finite primitive permutation groups of product type are stabilisers of Cartesian decompositions (sets of partitions). We study whether groups in certain families of infinite primitive groups can be described in a similar way as stabilisers of similar structures. More speculatively we ask whether primitive groups of simple diagonal type arise as stabilisers of some combinatorial structure.

This is joint work with Csaba Schneider.

Past and future seminars may be found at https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

Title: Classification of hyperovals and KM-arcs in small projective planes

Time and place: 4pm Friday 21 Sep 2018, Weatherburn LT

Abstract: Hyperovals (resp. KM-arcs) are point sets in PG(2,q) (resp. in AG(2,q)^D) such that every line contains 0 or 2 of these points. Every hyperoval can be seen as a KM-arc, but not vice versa; both only exist when q is even. Hyperovals always have size q+2, KM-arcs have size q+t for some t|q. A commonly studied problem for any projective substructure is to classify its examples in small projective planes. We give an overview of the known results, with particular focus on the most recent result: a full classification of the KM-arcs in PG(2,64).

(Actually what I write above is not true, since the computations are not finished, but I know they'll finish in a reasonable time and by now it is unlikely that any new examples will pop up.)

Title: Non-crossing partitions in Coxeter and Artin groups

Time and place: 4pm Friday 28 Sep 2018, Weatherburn LT

Abstract: The talk will start with an introduction to non-crossing partitions, Coxeter and braid or more generally Artin groups and the relations between these objects. A motivation for the study of these objects is the word and the conjugacy problem in Artin groups. We will describe the non-crossing partitions in the Artin groups and will apply it in the end to the study of Hecke algebras.

The following week Carlisle King will speak. Future seminars will be updated later https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

Title: Generation of finite simple groups

Time and place: 4pm Friday 5 Oct 2018, Weatherburn LT

Abstract: Let G be a finite simple group. It is well-known that G is generated by a pair of elements – we say G is 2-generated. It has been shown that 2 elements chosen at random from G generate G with probability tending to 1 as |G| tends to infinity. A natural refinement is to ask, given a pair of integers (a,b), whether G is 2-generated by an element of order a and an element of order b. If such a pair exists, we say that G is (a,b)-generated. We will explore some past results regarding (2,3)-generation proved using probabilistic methods, as well as a recent result on (2,p)-generation for some prime p.

An equivalent statement of the 2-generation theorem is that every finite simple is an image of F2, the free group on 2 generators. More generally, given a finitely presented group Gamma, one can ask which finite simple groups are images of Gamma. We will study a new result in this area concerning the free product Gamma=A*B of nontrivial finite groups A and B, also proved using probabilistic methods.

For past and future seminars see https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

Title: p-groups related to exceptional Chevalley groups

Time and place: 4pm Friday 19 Oct 2018, Weatherburn LT

Abstract: In 1978, Bryant and Kovács proved that if H is a subgroup of the general linear group GL(d,p), with d > 1 and p prime, then there exists a p-group P such that Aut(P) induces H on the Frattini quotient of P. However, it is not known in general when we can choose P to be small, in terms of its exponent-p class, exponent, nilpotency class and order. In this talk, we consider the representation theory of the (finite) simply connected versions of the exceptional Chevalley groups, and their overgroups in corresponding general linear groups, in order to construct small related p-groups. This follows on from the recent work of Bamberg, Glasby, Morgan and Niemeyer, who constructed small p-groups related to maximal subgroups of GL(d,p).

Title: Bounds for Semiprimitive Permutation Groups

Time and place: 4pm Friday 2 Nov 2018, Weatherburn LT

Abstract: Semiprimitive groups are permutation groups for which every normal subgroup is either transitive, semiregular, or both. These groups naturally generalise primitive and quasiprimitive groups, and inherit many of their structural properties. I will present the results from my master's thesis, in which I establish bounds on quantities such as order, element degree, and base size.

For abstracts and titles of seminars (upcoming and past), see https://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

Title: The p-part of the order of an almost simple group of Lie type

Time and place: 4pm Friday 23 Nov 2018, Robert Street LT (not Weatherburn LT)

Abstract: Primitive permutation groups are fundamental building blocks in the sense that every finite permutation group can be built from the primitive ones. Apart from the alternating group A_n and the symmetric group S_n of degree n, the primitive subgroups G of S_n are small. For example, in 1980 Praeger and Saxl showed that |G| e 4^n, which is much smaller than n!/2. Since this time, powerful results such as the O’Nan-Scott Theorem, which classifies the primitive permutation groups, and the Classification of the Finite Simple Groups, have become available. We will bound the p-part |G|_p of |G| for some prime p. This is, the largest p-power p^{ u_p(G)} that divides |G|. The bound |G| e 4^n implies nu_p(G) e n og_p(4). We prove the stronger bound nu_p(G) e frac{2sqrt{n}}{(p-1)}+1 (with five exceptions). For several cases, we even obtain a bound that is logarithmic in n. Our proof uses the O'Nan-Scott theorem to reduce to simple groups. The hardest case, and the one I will discuss, is when the simple group is of Lie type.

Title: Erdös-Ko-Rado Problem for Permutation Groups

Time and place: 4pm Friday 01 Feb 2019, Weatherburn LT

Abstract: A classical result of Erdös-Ko-Rado in extremal set theory is about intersections of subsets of a set, leading to the so-called Erdös-Ko-Rado problem in various versions. I will explain the problem for permutation group version, and then address a conjecture about the upper-bound for the numbers of intersecting sets.

Title: Conjugacy class sizes in finite groups: variations on the theme

Time and place: 4pm Tuesday 12 Feb 2019, Weatherburn LT

Abstract: The study of conjugacy class sizes goes back to the beginning of the last century with Burnside's result, Miller's studies on groups with few conjugacy classes and Ito who began the study of the structure of a group in terms of the number of conjugacy classes. Over the last 30 years the subject has become fashionable and many papers have been written on this topic. In this talk I try to summarize some results obtained by using two particular graphs: the common divisor graph and the prime graph in relationship with conjugacy class sizes and finally the variation regarding the so-called vanishing classes, that are those classes of elements g for which there exists an irreducible non linear character chi such that chi(g)=0.

Title: An update on the Polycirculant Conjecture

Time and place: 4pm Friday 15 Feb 2019, Robert Street LT

Abstract: One version of the Polycirculant Conjecture is that every finite vertex-transitive digraphs admits a non-trivial semiregular automorphism. I will give an overview of the status of this conjecture, as well as describe some recent progress with Michael Giudici.

Title: Searching for partial congruence partitions in groups of order p^8

Time and Place: 4pm Friday 08 Mar 2019, Weatherburn LT

Abstract: A partial congruence partition (or PCP) of a group is defined simply as a set of pairwise disjoint subgroups which pairwise factorise the whole group. This project has focused on finding examples of PCP that are 'large' in the sense that they are close the best known theoretical bounds on the maximum number of subgroups comprising a PCP. In particular, we focused on the non-elementary abelian groups of order 2^8 and 3^8, where no large examples were previously known. These groups, with one exception, were exhaustively searched for large PCP. In this talk I will discuss some of the theoretical bounds with a focus on deriving results that are useful computationally, before moving on to some details of the computational enumeration.

Title: Elements with large irreducible submodules contained in maximal subgroups of the general linear group

Time and place: 4pm Tuesday 12 Mar 2019, Blakers LT

Abstract: We refer to an element of the finite general linear group GL(V) as being fat if it leaves invariant, and acts irreducibly on, a subspace of dimension greater than dim(V)/2. Fat elements generalise the concept of ppd-elements, which are defined by the property of having orders divisible by certain primes called primitive prime divisors. In 1997, Guralnick, Penttila, Praeger and Saxl classified all subgroups of GL(V) containing ppd-elements. Their work has had a wide variety of applications in computational group theory, number theory, permutation group theory, and geometry. Our overall goal is to carry out an analogous classification of all subgroups of GL(V) containing fat elements.

During my PhD candidature I examined the occurrence of fat elements in GL(V) and various of its maximal subgroups. I showed that, often, this problem can be handled in a uniform way by considering "extremely fat" elements and counting certain irreducible polynomials. In my talk, I will present this method for groups belonging to Aschbacher's C2 class. The results we obtain significantly differ from the findings of the ppd-classification.

Title: Arc-transitive bicirculants

Time and place: 4pm Friday 12 Apr 2019, Weatherburn LT

Abstract: A graph on 2n vertices is a bicirculant if it admits an automorphism that is a permutation with two cycles of length n. For example, the Petersen and Heawood graphs. Arc-transitive bicirculants of valencies three, four and five have previously been classified by various authors. In this talk I will discuss recent joint work with Alice Devillers and Wei Jin that characterises all arc-transitive bicirculants and provides a framework for their complete classification.

Title: Girth, words and diameters of Cayley graphs

Time and place: 4pm Wednesday 17 Apr 2019, Woolnough LT

Abstract: The girth of a graph is the minimal length of a cycle in the graph. Finding regular graphs with large girth relative to their diameter is the subject of much interest, and a fruitful source of examples has been found in Cayley graphs of various families of finite classical groups. I shall discuss some new results in this area on the girth of Cayley graphs of finite classical groups G on random sets of generators. The main tool is a new bound on the probability that a given word w takes the value 1 when evaluated in G, in terms of the length of w.

Title: The 2-transitive permutation representation of the small Ree groups

Time and place: 4pm Friday 03 May 2019, Weatherburn LT

Abstract: Given a group G, the question of which subsets S of G generate G is of natural interest. One approach to this question is determining the Möbius function of G, introduced by Hall in 1936. In order to determine the Möbius function, it is necessary to have a good understanding of the subgroup structure of G, which is of course of interest in its own right. In this talk we discuss how the Möbius function is determined in practice, using the specific case of the small Ree groups.

For upcoming seminars see https://staffhome.ecm.uwa.edu.au/~00059629/GroupsAndCombinatoricsSeminar/S19.html

Title: From Lehman Matrices To (Im)Perfect Graphs

Time and place: 4pm Friday 10 May 2019, Weatherburn LT

Abstract: A pair (A,B) of square 0/1 matrices is called a Lehman pair if AB^T = J + k I where J is the all-ones matrix, I is the identity matrix and k is a positive integer, and an individual square 0/1 matrix is called a Lehman matrix if it belongs to a Lehman pair. The study of such matrices arose independently in the work of Lehman on problems in operations research, and the work of Bridges and Ryser who viewed them as generalisations of certain combinatorial designs. A number of authors have given methods of constructing Lehman matrices, including several recursive constructions that generate larger Lehman matrices from smaller ones, but always with the same value of k. In joint work, Dillon Mayhew, Irene Pivotto and I discovered a curious construction that transforms certain Lehman matrices with k=1 into “Lehman-like” matrices with k=-1 (and vice versa). Although barely mentioned in the literature on Lehman matrices, solutions to the matrix equation AB^T = J - I are essentially equivalent to a class of graphs known as "partitionable graphs", which were the central object of study in the decades-long effort to prove Berge’s Strong Perfect Graph Conjecture by a direct characterisation of minimal imperfect graphs.

In this talk, I will introduce all the necessary background concepts, and describe how such an innocuous definition leads quite naturally to such disparate areas of combinatorics.

Title: Limited geodesic transitivity for finite regular graphs

Time and place: 4pm Friday 17 May 2019, Weatherburn LT

Abstract: Joint work with Wei Jin.

For vertex transitive graphs, transitivity on t-arcs, t-geodesics, or distance t vertex pairs, for t leq s, all give symmetry measures of the graph in balls of radius s about a vertex. If the graph has girth g, and s leq g/2, then the sets of t-arcs and t-geodesics are the same for each t leq s, and so the conditions of s-arc transitivity and s-geodesic transitivity are equivalent. The next cases where s= (g+1)/2 and s=(g+2)/2 are interesting. There are s-geodesic transitive examples that are not s-arc transitive. Those which have s=2 and g=3 are collinearity graphs of point-line incidence geometries. However there is no nice general description for the cases where s= 3 and g is 4 or 5. Our approach has required us to classify, as a bye product, all 2-arc transitive strongly regular graphs, and to examine their normal covers. We have lots to describe, as well as open problems to pose.

Title: Words, permutations, and the nonsolvable length of a finite group

Time and place: 4pm Friday 24 May 2019, Weatherburn LT

Abstract: In group theory, the term “word” denotes any concatenation of variables and their formal inverses, such as xxyx^{-1}zy^{-1}. Words are to group theorists what polynomials are to ring theorists: formal expressions into which elements from a concrete structure (a ring resp. group) can be substituted and which can be used to formulate equations over those structures. One type of question commonly studied in this context is the following: Given a word w(X_1,...,X_d) and a number rho in (0, 1], what can one say about finite groups G in which for some g in G, the equation w(X_1,...,X_d) = g has at least rho|G|^d solutions (g_1,...,g_d) in G^d? In this talk, I will discuss recent results of this form which were achieved in collaboration with Aner Shalev from the Hebrew University of Jerusalem.

Title: Actions of Aut(V) on linearly independent subsets of a vector space V

Time and place: 4pm Friday 02 Aug 2019, Weatherburn LT

Abstract: Let V be a finite dimensional rational vector space. The action of the general linear group Aut(V) on the set of bases of V is the main content of First Year Linear Algebra, so you might think there can be nothing new to say about it; but the subject still holds a few surprises.

For example, if G is a subgroup of V, the stabilizer of G in Aut(V) acts on the set of maximal linearly independent subsets of V contained in G. The orbits of this action determine the indecomposable decompositions of G.

Furthermore, the stabilizer of the set of subgroups of V containing G as a subgroup of finite index acts transitively on a larger set of linearly independent subsets, and this action determines a coarse structure theorem for G.

Title: On transitive automorphism groups of 2-designs

Time and place: 4pm Friday 23 Aug 2019, Weatherburn LT

Abstract: A 2-design with parameters (v,k,lambda) is an incidence structure consisting of a set of v points and a set of b blocks with the incidence relation such that every block is incident with exactly k points, and every pair of points is incident with exactly lambda; blocks. An automorphism group of a 2-design is a group of permutations on points of the design which maps blocks to blocks and preserves the incidence and non-incidence. The main part of this talk is devoted to giving a survey on recent study of 2-designs admitting a flag-transitive automorphism group. I also present some recent results on block-transitive automorphism groups of 2-designs.

Title: Edge-primitive 3-arc-transitive graphs

Time and place: 4pm Friday 30 Aug 2019, Weatherburn LT

Abstract: Let Gamma be a finite simple graph with G = Aut(Gamma). We say that Gamma is edge-primitive if G acts primitively on the edges of Gamma. An s-arc in Gamma is an ordered path of length s. We say that Gamma is s-arc-transitive if G is transitive on the set of s-arcs of Gamma.

In 1981, Weiss proved that there exists no finite s-arc-transitive graph of valency at least 3 for s geq 8. Since then, there has been considerable effort to characterise s-arc-transitive graphs for s leq 7. One interesting family of graphs is that of edge-primitive graphs. Many famous graphs are edge-primitive, such as the Heawood graph and the Higman-Sims graph. In 2011, Li and Zhang classified finite edge-primitive s-arc-transitive graphs for s geq 4. We study the problem of classifying finite edge-primitive 3-arc-transitive graphs. This is joint work with Michael Giudici.

Title: Automorphism groups of simple graphs with few vertex-orbits

Time and place: 4pm Friday 13 Sep 2019, Weatherburn LT

Abstract:In this talk we will investigate how to construct automorphism groups of graphs with few vertex orbits. The base case is to construct automorphism groups with 2 vertex orbits. We will see how to describe their group theoretic structure and give hints towards an algorithm for constructing these groups and towards generalizations to more orbits.

If time permits, we will see how to count isomorphism classes of graphs with the same automorphism group using the table of marks.

Title: Automorphism orbits and element orders in finite groups

Time and place: 4pm Friday 27 Sep 2019, Weatherburn LT

Abstract: Joint with Michael Giudici and Cheryl E. Praeger.

In contrast to other kinds of structures (such as graphs), for groups G, the assumption that the automorphism group Aut(G) acts transitively on G is not interesting to study, as only the trivial group satisfies it. Various weakenings of this condition have been proposed and studied, though. For example, in a paper from 1992, Zhang extensively studied finite groups G with the property that for every element order o in G, the action of Aut(G) on order o elements in G is transitive. He called such finite groups AT-groups. Zhang’s ideas and methods also spurred some interest in the graph-theoretic community, due to a connection with CI-groups (groups G such that any two isomorphic Cayley graphs over G are “naturally isomorphic” via an automorphism of G).

In this talk, we present results on finite groups G that are “close to being AT- groups”, essentially showing that such groups are “almost soluble” (i.e., they have a soluble normal subgroup of bounded index). A finite group G is an AT-group if and only if the numbers of Aut(G)-orbits on G and of distinct element orders in G respectively are equal. Hence we measure the “closeness of G to being an AT-group” by comparing those two numbers, considering both their difference and quotient. Along the way, we obtain a curious quantitative characterisation of the Fischer-Griess Monster group M.

Title: Introduction to the GAP method selection for the working mathematician: Methods, Operations and Filters

Time and place: 4pm Friday 11 Oct 2019, Weatherburn LT

Abstract: GAP object can learn information about themselves that are stored as attributes and are present throughout the whole GAP session. Building on this, GAP has a method selection that dynamically changes applied algorithms depending on newly learned properties of the group. We will explore this on the example of computing the average order of elements in a collection and a group.

Title: On Gruenberg-Kegel graphs of finite groups

Time and place: 4pm Friday 18 Oct 2019, Weatherburn LT

Abstract: Let G be a finite group. The spectrum of G is the set of all its element orders. The Gruenberg-Kegel graph of G is a graph whose vertex set is the set of all prime divisors of the order of G, and two distinct vertices are adjacent in this graph if and only if their product is an element order of G. The spectrum is a very important invariant of a finite group. For example, many finite simple groups are defined by their spectra up to isomorphism. The concept of the Gruenberg-Kegel graph of a finite group widely generalizes the concept of the spectrum. In this talk we discuss some characterizations of finite groups by properties of their spectra and Gruenberg-Kegel graphs.

Title: On pronormality of subgroups of odd index in finite groups

Time and place: 4pm Friday 25 Oct 2019, Weatherburn LT

Abstract: A subgroup H of a group G is pronormal in G if for any element g from G, subgroups H and H^g are conjugate in the subgroup generated by H and H^g. Some problems in finite group theory, combinatorics, and permutation group theory were solved in terms of pronormality (see, for example, results by L. Babai, P. Palfy, Ch. Praeger, and others). Thus, the question of description of pronormal subgroups in finite groups is of interest. Well-known examples of pronormal subgroups in finite groups are normal subgroups, maximal subgroups, Sylow subgroups, Carter subgroups, Hall subgroups of solvable groups, and so on. An important problem is to describe families of pronormal subgroups in finite simple groups.

In 2012, E. Vdovin and D. Revin proved that the Hall subgroups are pronormal in finite simple groups and conjectured that the subgroups of odd index are pronormal in finite simple groups. This conjecture was disproved by A. Kondrat'ev, the speaker, and D. Revin in 2016. However, in many finite simple groups the subgroups of odd index are pronormal. In this talk we discuss a recent progress in the classification of finite simple groups in which the subgroups of odd index are pronormal and some connected questions.

Title: Subdegrees of primitive permutation groups

Time and place: 4pm Friday 01 Nov 2019, Robert Street LT

Abstract: A subdegree of a permutation group is the length of an orbit of a point stabiliser. The study of subdegrees of primitive permutation groups has a long history and has attracted the attention of many researchers. In this talk I will survey some results in this area and discuss some recent work on small subdegrees, constant subdgrees and coprime subdegrees. I will also discuss some applications to graph theory.

Cheese and wine to follow in the Maths common room.

Abstract: We all have our doubts off and on if life is really so wonderful. But that is not what I want to address here. Watching the Jimmy Stewart movie with this title, there was one scene which captured my imagination: the Guardian Angel shows George Bailey how the world would have been without him. Personally, I never had much need to know how the world would have looked without me. However, all other things equal, how would life have been if I had lived in a different time and place, would be something of interest to me! This is the stuff of movies and fairy tales. But at least it is possible to play this as an intellectual game. I was born and raised in Germany before WW II. After getting my Ph.D. in 1962, I married a fellow mathematician and we immigrated to the US one year later, where we taught at a university until our retirements, first at Ohio State and then at Binghamton University. What would life have been if I stayed in Germany, did not get married, were born fifty or one hundred years earlier, or were born in another country? Looking at actual and potential role models over the centuries helped me answer some of these questions. In essence, it got me back to the roots of what shaped my life.