October 2011

Wednesday 05 
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

Friday 07 
13:00  SEMINAR  Groups and Combinatorics Seminar: On the diameter of permutation groups

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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 worstcase 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 unboundedrank 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.

Friday 14 
13:00  SEMINAR  Groups and Combinatorics Seminar: The spread of a finite group

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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 nontrivial normal subgroup N such that G/N is noncyclic 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.

Friday 21 
13:00  SEMINAR  Groups and Combinatorics Seminar: Transferring Proportions of Invertible Matrices to the Full Matrix Algebra M(n,q)

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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.


November 2011

Thursday 03 
10:00  EVENT  Teaching and Learning Mathematics in the Australian Curriculum : As part of UWA's Year of Mathematics, this event is a forum to discuss the teaching and learning of Mathematics in the Australian Curriculum.

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

Friday 04 
13:00  SEMINAR  Groups and Combinatorics Seminar: Conditions for solubility of finite groups

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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 Ngroups paper, he proved that a finite group is soluble if and only if all of its 2generated 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

Friday 11 
13:00  SEMINAR  Groups and Combinatorics Seminar: Combinatorial counting principles in axiomatic number theory

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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.

Friday 18 
13:00  SEMINAR  Groups and Combinatorics Seminar:On probabilistic generation of classical groups

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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^n1 but not q^i1 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 ndimensional subspace and fix pointwise an ndimensional 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.


January 2012

Wednesday 25 
Groups and Combinatorics Seminars
Eric Swartz (Binghamton)
will speak on
Locally 2arc 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 2arc, and, for any vertex alpha of Gamma, G is transitive on the 2arcs 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 vertexintransitive 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 2arc 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) 2arc 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 twoarc transitive graphs admitting a Suzuki simple group," Comm. Alg., 27(8):37273754, 1999), this completes the classification of locally 2arc 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.


February 2012

Wednesday 22 
11:00  EVENT  Groups and Combinatorics Seminar: A census of cubic vertextransitive graphs

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Groups and Combinatorics Seminar
Gabriel Verret (University of Primorska, Slovenia)
will speak on
A census of cubic vertextransitive 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
vertextransitive 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.

Wednesday 29 
11:00  SEMINAR  Groups and Combinatorics Seminar: Neighbour transitive codes and connections with power line communication

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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.


March 2012

Wednesday 07 
11:00  SEMINAR  Groups and Combinatorics Seminar: Coprime subdegrees for primitive permutation groups and completely reducible linear groups

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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 nonzero vectors, their orbit lengths a, b have a nontrivial 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.

Wednesday 14 
11:00  SEMINAR  Groups and Combinatorics Seminar:Pentagonal geometries

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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).

Wednesday 21 
11:00  SEMINAR  Groups and Combinatorics Seminar:Minorclosed classes of graphs and matroids

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Groups and Combinatorics Seminar
Gordon Royle (UWA)
will speak on
Minorclosed 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

Wednesday 28 
11:00  SEMINAR  Groups and Combinatorics Seminar: The subspace lattice

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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.


April 2012

Wednesday 04 
11:00  SEMINAR  Groups and Combinatorics Seminar: sgeodesic transitive graphs

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Groups and Combinatorics Seminar
Wei Jin (UWA)
will speak on
sgeodesic 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 igeodesic if the distance between u and v is i. The graph
Gamma is said to be sgeodesic transitive if the graph automorphism group is transitive on the set of sgeodesics. In this talk, I will compare the sgeodesic transitivity with other two wellknown transitive properties, namely sarc transitivity and sdistance transitivity, and determine the local structure of
2geodesic transitive graphs, and also give some results about the family of locally disconnected
2geodesic transitive but not 2arc transitive graphs.
All welcome

Wednesday 11 
10:30  SEMINAR  Groups and Combinatorics Seminar: Polyhedral complexes, locally compact groups and lattices

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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

Wednesday 18 
11:00  SEMINAR  Groups and Combinatorics Seminar: A class of abundant psingular elements in finite classical groups

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Groups and Combinatorics Seminar
Tomasz Popiel (UWA)
will speak on
A class of abundant psingular 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 psingular elements of G (elements with order divisible by p) comprising elements that leave invariant certain "large" subspaces of the natural Gmodule. 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 psingular 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 psingular elements. This talk represents joint work with Alice Niemeyer and Cheryl Praeger.
All welcome


May 2012

Wednesday 02 
11:00  SEMINAR  Groups and Combinatorics Seminar: The MerinoWelsh Conjecture

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Groups and Combinatorics Seminar
Steven Noble (Brunel University, UK)
will speak on
The MerinoWelsh Conjecture
at 11am Wednesday 2nd of May in MLR2
Abstract: The MerinoWelsh 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 twovariable 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

Wednesday 09 
11:00  SEMINAR  Groups and Combinatorics Seminar: Automorphisms and opposition in twin buildings

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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 2spherical, then not every residue of a given type can be mapped to an opposite.


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