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Today's date is Friday, August 07, 2020
Centre for the Mathematics of Symmetry and Computation
 August 2012
Tuesday 28
13:00 - SEMINAR - Groups and Combinatorics Seminar: Are Three Squares Impossible? More Information
Groups and Combinatorics Seminar

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

 October 2012
Tuesday 02
13:00 - SEMINAR - Groups and Combinatorics: Packing Steiner trees More Information
Groups and Combinatorics Seminar

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.
Monday 08
13:10 - SEMINAR - Groups and Combinatorics Seminar: Commuting graphs of groups More Information
Groups and Combinatorics Seminar

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.
Monday 15
18:00 - PUBLIC LECTURE - The solution of the Poincare conjecture Website | More Information
A Public Lecture by Professor J. Hyam Rubinstein, Department of Mathematics & Statistics, University of Melbourne.

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 [email protected]
Tuesday 16
13:00 - SEMINAR - Groups and Combinatorics: Finite meta-primitive permutation groups More Information
Groups and Combinatorics Seminar

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.
Tuesday 23
13:00 - SEMINAR - Groups and Combinatorics Seminar: Clifford theory and Hecke algebras More Information
Groups and Combinatorics Seminar

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

 November 2012
Tuesday 20
13:00 - SEMINAR - Groups and Combinatorics Seminar: Finite s-Geodesic Transitive Graphs More Information
Groups and Combinatorics Seminar

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
Tuesday 27
13:00 - SEMINAR - Groups and Combinatorics Seminar: Graphs and general preservers of zero products More Information
Groups and Combinatorics Seminar

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.

 December 2012
Tuesday 04
13:00 - SEMINAR - Groups and Combinatorics Seminars More Information
Groups and Combinatorics Seminar

Neil Gillespie (UWA)

will speak on

Completely regular codes with large minimum distance


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

at 1pm Tuesday 4th of December in MLR2


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.

 February 2013
Friday 22
15:00 - SEMINAR - Groups and Combinatorics Seminar: Algebraic aspects of Hadamard matrices More Information
Our Groups and Combinatorics Seminar will resume this Friday.

Padraig Ó Catháin (The University of Queensland)

will speak on

Algebraic aspects of Hadamard matrices

at 3pm Friday 22nd of February in MLR2.


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.

 March 2013
Friday 01
15:00 - SEMINAR - Groups and Combinatorics Seminar, The Wall and Guralnick conjectures: history and legacy More Information

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.
Tuesday 05
13:00 - SEMINAR - Groups and Combinatorics Seminar, Control of fusions in fusion systems and applications More Information
Jiping Zhang (Peking University)

will speak on

Control of fusions in fusion systems and applications

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


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.
Friday 08
15:00 - SEMINAR - Groups and Combinatorics Seminar, Generalised n-gons and the Feit-Higman theorem More Information
Name: Jon Xu (University of Melbourne/University of Western Australia)

will speak on

Generalised n-gons and the Feit-Higman theorem

at 3pm on Friday 8th of March.


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.
Friday 15
15:00 - SEMINAR - Groups and Combinatorics Seminar, Erdös-Ko-Rado sets in finite classical polar spaces More Information

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.
Friday 22
15:00 - SEMINAR - Groups and Combinatorics Seminar, Irreducible subgroups of classical algebraic groups More Information

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.
Thursday 28
14:00 - SEMINAR - Groups and Combinatorics Seminar: Bacterial genome evolution with algebra More Information
Andrew Francis (University of Western Sydney)

will speak on

Bacterial genome evolution with algebra

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



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.

 April 2013
Friday 05
15:00 - SEMINAR - Groups and Combinatorics Seminar, Redei-polynomials in finite geometry More Information

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).
Tuesday 09
13:00 - SEMINAR - Groups and Combinatorics Seminar, The Erdos-Stone Theorem for finite geometries More Information

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.
Friday 12
15:00 - SEMINAR - Groups and Combinatorics Seminar, On the number of matroids More Information

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)
Friday 19
15:00 - SEMINAR - Groups and Combinatorics Seminar, Arc-transitive graphs with large automorphism groups More Information

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.

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