August 2012

Tuesday 21 
13:00  SEMINAR  Groups and Combinatorics Seminar: Compositions of n and an application to the covering number of the symmetric groups

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Groups and Combinatorics Seminar
Pablo Spiga (University of MilanoBicocca)
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 kcomposition 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 kcompositions satisfying some "coprimeness" condition. As an application we give a Classificationfree proof of some results on the covering number of the symmetric group.
All welcome

Tuesday 28 
13:00  SEMINAR  Groups and Combinatorics Seminar: Are Three Squares Impossible?

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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

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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 NashWiliams and Tutte gives necessary and sufficient conditions for a graph to
have k pairwise edgedisjoint 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 edgedisjoint spanning trees is a
considerably easier problem that finding edgedisjoint 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 edgedisjoint 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

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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 
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 3dimensional space with `no holes’ is equivalent to the 3dimensional 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 3dimensional 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 metaprimitive permutation groups

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Groups and Combinatorics Seminar
Cai Heng Li (UWA)
will speak on
Finite metaprimitive permutation groups
at 1pm Tuesday 16th of October in MLR2
Abstract: A transitive permutation group is called metaprimitive if its any imprimitive quotient action is primitive, namely, each of the block systems is maximal. I will discuss the structural properties of metaprimitive groups.
All welcome.

Tuesday 23 
13:00  SEMINAR  Groups and Combinatorics Seminar: Clifford theory and Hecke algebras

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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 Ktheory of
Steinberg varieties following KazhdanLusztig). Despite the
mathsphysics 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 sGeodesic Transitive Graphs

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Groups and Combinatorics Seminar
Wei Jin (UWA)
will speak on
Finite sGeodesic 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 sgeodesic if the distance between u and v is s.
A graph is said to be sgeodesic transitive if, for each i less than or equal to s, all
igeodesics are equivalent under the group of graph automorphisms. In this talk, I will show the relationship of 2geodesic transitive graphs with a certain family of partial linear spaces. I will also compare sgeodesic transitivity of graphs with two other wellknown transitivity properties, namely sarc transitivity and sdistance 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

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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 (rankones, semisimple, nonderogatory) in terms of a commuting graph.
All welcome.


December 2012

Tuesday 04 
Groups and Combinatorics Seminar
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 nonantipodal 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 nonantipodal 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, codegroup 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

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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.
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.


March 2013

Friday 01 
15:00  SEMINAR  Groups and Combinatorics Seminar, The Wall and Guralnick conjectures: history and legacy

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Abstract:
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

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Jiping Zhang (Peking University)
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.

Friday 08 
15:00  SEMINAR  Groups and Combinatorics Seminar, Generalised ngons and the FeitHigman theorem

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Name: Jon Xu (University of Melbourne/University of Western Australia)
will speak on
Generalised ngons and the FeitHigman 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 ngons.
In my talk, generalised ngons will be defined as a certain class of bipartite graphs, so as to skip the (rather abstruse) buildingtheoretic definition. I will also state and outline a proof of the FeitHigman theorem, which states that the majority of generalised ngons 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, ErdösKoRado sets in finite classical polar spaces

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Abstract:
ErdösKoRado sets (EKR sets) are a family of ksets 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

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Abstract:
Let G be a group, let H be a subgroup of G and let V be an irreducible KGmodule over a field K. We say that (G,H,V) is an irreducible triple if V is an irreducible KHmodule. 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

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Andrew Francis (University of Western Sydney)
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.


April 2013

Friday 05 
15:00  SEMINAR  Groups and Combinatorics Seminar, Redeipolynomials in finite geometry

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Abstract:
L. Redei has studied in a detailed way socalled "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 "Redeipolynomials" in finite geometry. We will discuss some cases to explain the principle of using Redeipolynomials finite projective spaces and some particular generalized quadrangle. Then we discuss a problem on maximal partial ovoids, that has been partially solved using Redeipolynomials, 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 ErdosStone Theorem for finite geometries

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Abstract:
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 ErdosStone 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 HalesJewett Theorem of Furstenberg and Katznelson.

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

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Abstract:
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)


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