What's On at UWA

The Postgraduate and Honours Expo showcases a host of opportunities for further study, including honours and postgraduate coursework and research possibilities.

Discover the courses each faculty has to offer, learn about postgraduate scholarships, attend information sessions and talk to staff, honours and postgraduate students.

For more information about the Expo along with details on the presentations being held throughout the evening please go uwa.edu.au/postgradexpo

Groups and Combinatorics Seminar

Akos Seress (UWA)

will speak

On the diameter of permutation groups

at 1pm Friday 7th of October in MLR2

Abstract: Joint work with H. Helfgott. Given a finite group G and a set A of generators, the diameter diam(Gamma(G,A)) of the Cayley graph Gamma(G,A) is the smallest number x such that every element of G can be expressed as a word of length at most x in A and the inverse of A. We are concerned with the diameter under the worst-case generators: diam(G):= max(diam(Gamma(G,A))).

It has long been conjectured that the diameter of the symmetric group of degree n is polynomially bounded in n, but the best previously known upper bound was exponential in the square root of n (Babai, Seress, 1988). We give a quasipolynomial upper bound for diam(Sym(n)). The same bound applies to the alternating groups.

This addresses a key open case of Babai's conjecture that the diameter of all nonabelian finite simple groups G is bounded by a polylogarithmic function of the group order. The first class of groups for which the conjecture was verified was PSL(2,p), p prime (Helfgott, 2008). This has been generalised to all simple groups of Lie type of bounded rank (Pyber, Szabo, 2011 and Breuillard, Green, Tao, 2011); the unbounded-rank cases are likely to raise combinatorial problems of the type studied in this paper.

By a theorem of Babai, Seress (1992), our result implies a quasipolynomial upper bound on the diameter of all transitive permutation groups of degree n.

Our approach combines ideas on growth in groups (as in (Helfgott, 2008), (Helfgott, 2011)) with an adaptation of older techniques on permutation groups -- most notably (Babai, 1982) and (Pyber, 1993) -- to sets of permutations.

Groups and Combinatorics Seminar

Simon Guest (UWA)

will speak on

The spread of a finite group

at 1pm on Friday 14th of October in MLR2

Abstract: Let G be a finite group. We say that G has spread at least k if for any k distinct nontrivial elements x_1, ... , x_k in G, there exists y in G such that x_i and y generate G for each i. If G does not have spread at least 1 then G is said to have spread 0. Using elementary methods we can prove that if G has a non-trivial normal subgroup N such that G/N is non-cyclic then G must have spread 0. It has been conjectured by Guralnick and Kantor that the converse is true. They can prove that the converse holds in many cases. We will discuss some recent joint work with Tim Burness involving the remaining cases.

Groups and Combinatorics Seminar

-------------------------------------------------

Brian Corr

"Transferring Proportions of Invertible Matrices to the Full Matrix Algebra M(n,q)"

Friday 21st October (2011), 1pm, MLR2

Abstract: Many questions in Computational Group Theory revolve around the counting of objects inside groups and other structures. Monte Carlo and Las Vegas algorithms depend on a random search for objects with desirable properties, and knowing how abundant these objects are is the key to knowing how long our search should take.

The Quokka Theory of Niemeyer and Praeger is a powerful tool for counting desirable matrices in subgroups of the General Linear Group GL(n,q), but it depends on the invertibility of the matrices. We present a new theory which allows us, under certain conditions, to transfer results about subsets of GL(n,q) into results about subsets of the full matrix algebra M(n,q), and present an application of how this applies to results obtained through Quokka Theory.

Early Childhood, Primary and Secondary teachers are invited to share this event.

Professor Ramagge is a member of the ACARA Mathematics Advisory Panel and of the Educational Advisory Committee of the Australian Mathematical Sciences Institute.

She has helped to develop support materials for teachers that are available free to all Australian Schools as part of The Improving Mathematics Education in Schools project funded by the Australian Government.

RSVP ONLINE

Cost: free

Registrations are essential.

Places are limited!

Register online at www.education.uwa.edu.au/maths-event

All registrations received on,or before, September 30 will be in the draw to win an iPad.

Groups and Combinatorics Seminar

Cheryl Praeger (UWA)

will speak on

Conditions for solubility of finite groups

at 1pm Friday 4th of November in Maths Lecture Room 2

Abstract: In John Thompson's famous N-groups paper, he proved that a finite group is soluble if and only if all of its 2-generated subgroups are soluble. This influenced several recent explorations to understand how much local "soluble information" is actually needed to confirm global solubility, including work by Simon Guest discussed last year. I will report on joint work with Marcel Herzog, Silvio Dolfi and Bob Guralnick. It exploits local information different from other approaches.

All welcome

Groups and Combinatorics Seminar

Alan Woods (UWA)

will speak on

Combinatorial counting principles in axiomatic number theory

at 1pm on Friday 11th of November in Maths Lecture Room 2

Abstract: Proofs of even simple statements in number theory are notorious for being ``tricky''. They often seem to involve concepts that go well beyond what is needed to state the theorem. Is there some way of systematically characterising the assumptions (``axioms'', if you will) required?

It has long been recognised that full Peano Arithmetic (as sometimes briefly encountered in first or second year courses) is actually significantly stronger than what is needed for everyday number theoretic practice (or for that matter, the typical finite combinatorics or finite group theory seminar). An alternative is the axiom system similar to Peano Arithmetic, but with induction hypotheses restricted to being arithmetic formulas having only bounded quantifiers, i.e., ``there exists'' occurs only in the form ``there exists y < x'', and ``for every'' occurs only in the form ``for every y < x''. Such a bounded formula can only ``talk about'' numbers bounded above by the free variables appearing in it. There is a long list of theorems of elementary number theory for which no proof from this weaker axiom system is known, and yet which are provable if, for a suitably chosen bounded arithmetic formula A(x), one adds a census function c(n)=|{x: x < n and A(x)}| counting how many numbers x less than n have the property A(x), and allows induction on bounded quantifier formulas containing c.

Using census functions one can also prove certain combinatorial principles, notably versions of the pigeonhole principle and the equipartition principle - a phenomenon which facilitates some of the applications to number theory.

Examples surveyed briefly should include: the existence of arbitrarily large primes (including recently published joint work with Cornaros), the index of the subgroup of squares in the multiplicative group (mod p) for p prime, Lagrange's four squares theorem, and Jerabek's recent proof of the Quadratic Reciprocity Law which is arguably the logically simplest known proof of this theorem of Gauss.

Groups and Combinatorics Seminar

Sukru Yalcinkaya (UWA)

will speak

On probabilistic generation of classical groups

at 1pm Friday 18th of November in MLR2

Abstract: Let G be a classical group and V be the underlying vector space of dimension 2n over a field of size q. Let ppd(q,n) denote a prime number which divides q^n-1 but not q^i-1 for i<n. Generically, such primes exist by Zsigmondy's result. We consider the elements of order ppd(n,q) which act irreducibly on a unique n-dimensional subspace and fix pointwise an n-dimensional subspace of V. We call these elements good elements. I will discuss the probabilistic generation of G by two random conjugate good elements and present an application in the computational group theory. This is a joint work with Cheryl Praeger and Akos Seress.

Groups and Combinatorics Seminars

Eric Swartz (Binghamton)

will speak on

Locally 2-arc transitive graphs admitting an almost simple group of Suzuki type

at 3pm in MLR2

and

Jan Saxl (Cambridge)

will speak on

Conjugation characters of simple groups

at 4pm in MLR2

Abstract 1: A graph Gamma is said to be locally (G,2)-arc transitive for G a subgroup of Aut(Gamma) if Gamma contains a 2-arc, and, for any vertex alpha of Gamma, G is transitive on the 2-arcs of Gamma starting at alpha. In this talk, we will discuss general results involving locally (G,2)-arc transitive graphs and recent progress toward the classification of the locally (G,2)-arc transitive graphs, where G is a subgroup of Aut(Sz(q)) with soc(G) = Sz(q), q = 2^m for some odd natural number m. In particular, we will discuss seven families of vertex-intransitive locally (G,2)-arc transitive graphs. Furthermore, for any graph Gamma in one of these families, the full automorphism group of Gamma is a subgroup of Aut(Sz(q)), and the only locally 2-arc transitive graphs admitting an almost simple group of Suzuki type whose vertices all have valency at least three are (i) graphs in these seven families, (ii) (vertex transitive) 2-arc transitive graphs admitting an almost simple group of Suzuki type, or (iii) double covers of the graphs in (ii). Since the graphs in (ii) have been classified by Fang and Praeger (``Finite two-arc transitive graphs admitting a Suzuki simple group," Comm. Alg., 27(8):3727-3754, 1999), this completes the classification of locally 2-arc transitive graphs admitting a Suzuki simple group.



Abstract 2: In this talk we consider two problems concerning ordinary representations of simple groups: characters of conjugation actions, and squares of irreducible characters. These turn out to be closely related. This is joint work with Heide, A. Zalesskii, and Tiep.

Groups and Combinatorics Seminar

Gabriel Verret (University of Primorska, Slovenia)

will speak on

A census of cubic vertex-transitive graphs

at 11am Wednesday 22nd of February in MLR2

Abstract: We explain how some of our recent results have allowed us to compute a census of all cubic vertex-transitive graphs of order at most 1280. We present some data about the census and discuss some interesting related questions. This is joint work with Primoz Potocnik and Pablo Spiga.

Groups and Combinatorics Seminar

Neil Gillespie (UWA)

will speak on

Neighbour transitive codes and connections with power line communication

at 11am in MLR2 on Wednesday 29th of February

Abstract: Power line communication has been proposed as a possible solution to the "last mile" problem in telecommunications i.e. providing economical high speed telecommunications to millions of end users. In addition to the usual background noise, there are two other types of noise that must be considered for any successful implementation of power line communication. In this talk we characterise a family of neighbour transitive codes, and show that such codes have the necessary properties to be useful in power line communication.

Groups and Combinatorics Seminar

Cheryl Praeger(UWA)

will speak on

Coprime subdegrees for primitive permutation groups and completely reducible linear groups

at 11am on Wednesday 7th of March in MLR2



Abstract. This work was inspired by a question of Gabriel Navarro about orbit lengths of groups acting on finite vector spaces, and is joint work with Pablo Spiga, Silvio Dolfi and Bob Guralnick. If a finite group H acts irreducibly on a finite vector space V, then we proved that for every pair of non-zero vectors, their orbit lengths a, b have a non-trivial common factor.

This could be interpreted in the context of permutation groups. The group VH is an affine primitive group on V and a, b are orbit lengths of the point stabiliser H, that is, a and b are subdegrees of VH. This raises a question about subdegrees for more general primitive permutation groups. Coprime subdegrees can arise, but (we show) only for three of the eight types of primitive groups. Moreover it is never possible to have as many as three pairwise coprime subdegrees. All proofs depend on the finite simple group classification.

Groups and Combinatorics Seminar

John Bamberg (UWA)

will speak on

Pentagonal geometries

at 11am on Wednesday 14th of March in MLR2

Abstract: Jacques Tits introduced generalised polygons in order to geometrically describe groups of Lie type, and as their name suggests, they are generalisations of ordinary regular polygons. In this talk, we look at the concept of a "pentagonal geometry" as a generalisation of the pentagon and the Desargues configuration, in the same vein that the generalised polygons share the fundamental properties of ordinary polygons. (This is joint work with Simeon Ball, Alice Devillers and Klara Stokes).

Groups and Combinatorics Seminar

Gordon Royle (UWA)

will speak on

Minor-closed classes of graphs and matroids

at 11am in MLR2 on Wed 21st of March

Abstract: This talk will give a general survey of results and problems relating to characterising classes of graphs and matroids that are closed under taking minors.

All welcome

Groups and Combinatorics Seminar

Phill Schultz (UWA)

will speak on

The subspace lattice

at 11am on Wed 28th of March in MLR2

Abstract: I describe algebraically, combinatorially and anatomically the lattice of subspaces of a finite dimensional vector space V. The description allows you to calculate, from given subspaces U and W of V, bases for U + W and UW and for all subspaces and complements of U. Also, you can describe the Hasse Diagram of the lattice of subspaces of V.

Groups and Combinatorics Seminar

Wei Jin (UWA)

will speak on

s-geodesic transitive graphs

at 11am on Wednesday 4th of April in MLR2.

Abstract: In a finite graph Gamma, a geodesic from a vertex u to a vertex v is one of the shortest paths from u to v, and this geodesic is called an i-geodesic if the distance between u and v is i. The graph Gamma is said to be s-geodesic transitive if the graph automorphism group is transitive on the set of s-geodesics. In this talk, I will compare the s-geodesic transitivity with other two well-known transitive properties, namely s-arc transitivity and s-distance transitivity, and determine the local structure of 2-geodesic transitive graphs, and also give some results about the family of locally disconnected 2-geodesic transitive but not 2-arc transitive graphs.

All welcome

Groups and Combinatorics Seminar

Anne Thomas (Sydney)

will speak on

Polyhedral complexes, locally compact groups and lattices

at 10:30am on Wednesday 11th of April in MLR2

***Note earlier time than usual*******

Abstract: This will be an introduction to the part of geometric group theory which is concerned with polyhedral complexes, their automorphism groups and lattices in their automorphism groups. We will show via key examples how finite geometries and groups are used to construct and understand infinite polyhedral complexes and groups which act on them.

All welcome

Groups and Combinatorics Seminar

Tomasz Popiel (UWA)

will speak on

A class of abundant p-singular elements in finite classical groups

at 11am on Wednesday 18th of April in MLR2

Abstract: Elements with order divisible by certain primes have underpinned many algorithms for computing in finite classical groups G. For a prime p dividing the order of G and not dividing q, where G is defined over a field with q elements, we introduce a subfamily of the p-singular elements of G (elements with order divisible by p) comprising elements that leave invariant certain "large" subspaces of the natural G-module. We determine the exact asymptotic value of the proportion of these elements in G, which turns out to be a constant multiple of the best known lower bound for the proportion of all p-singular elements, the latter having been obtained in a 1995 paper of Issacs, Kantor and Spaltenstein. We also present an efficient algorithm for testing whether a given element of G belongs to our new subfamily of p-singular elements. This talk represents joint work with Alice Niemeyer and Cheryl Praeger.

All welcome

Groups and Combinatorics Seminar

Steven Noble (Brunel University, UK)

will speak on

The Merino-Welsh Conjecture

at 11am Wednesday 2nd of May in MLR2

Abstract: The Merino-Welsh conjecture states that for any loopless, bridgeless graph G, the maximum of the number of acyclic orientations and the number of totally cyclic orientations of G is at least the number of spanning trees of G.

Each of these invariants is an evaluation of the Tutte polynomial, which is a two-variable graph polynomial with positive coefficients. Computational evidence hints that for bridgeless, loopless graphs, the Tutte polynomial might be convex along the portion of lines x+y = constant lying in the positive quadrant. The conjecture is a first small step towards resolving this question.

We will explain the motivation behind the conjecture and discuss some generalizations, in particular to matroids. We will then show that the most general convexity conjecture holds for a large class of matroids and that something much weaker holds for all loopless, bridgeless matroids. Almost no knowledge of matroids will be assumed!

All welcome

Groups and Combinatorics Seminar

Alice Devillers (UWA)

will speak on

Automorphisms and opposition in twin buildings.

at 11am on Wednesday 9th of May in MLR2.

Abstract: Opposition in twin buildings generalises the notion of opposition in spherical buildings. With James Parkison and Hendrik Van Maldeghem, we looked at automorphisms mapping some/all residues to opposite residues.For instance we proved that an automorphism of a thick twin building (swapping the two halves) always maps at least one spherical residue to an opposite. However, if the building is also locally finite and 2-spherical, then not every residue of a given type can be mapped to an opposite.