What's On at UWA

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.

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.

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.

Abstract:

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.

Abstract:

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.

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.

Abstract:

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).

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 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.

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)

Abstract:

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.

Abstract:

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.

Metacirculants were introduced by Alspach and Parsons in 1982 and have been a rich source of various topics since then, including the Hamiltonian path problem of vertex-transitive graphs. A metacirculant has a vertex-transitive metacyclic subgroup of automorphisms, and a long-standing curious question in the area is if the converse statement is true, namely, whether a graph with a vertex-transitive metacyclic automorphism group is a metacirculant. We shall answer this question in the negative.

Abstract:

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.

Abstract:

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.

Abstract:

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.

Abstract:

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.

Abstract:

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.

Abstract:

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.

This week the seminar will consist of three 20 minute talks, starting at 3pm Friday 21st of June in Blakers Lecture Theatre.

--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.

Abstract:

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.

Abstract:

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.