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Today's date is Thursday, October 29, 2020
Centre for the Mathematics of Symmetry and Computation
 March 2011
Tuesday 08
12:00 - SEMINAR - Groups and Combinatorics Seminar: The Cayley Isomorphism (CI) problem More Information
Groups and Combinatorics Seminar

Joy Morris (University of Lethbridge, Canada)

will speak on

The Cayley Isomorphism (CI) problem

at 12 noon in MLR2 on Tuesday 8th of March

Abstract: In a perfect world in which all isomorphisms between graphs must be in some sense ``natural," it would be possible to attack the problem of determining whether or not given graphs are isomorphic, simply by checking a (hopefully small) class of ``natural" isomorphisms. For Cayley graphs, ``natural" isomorphisms between the graphs Cay(G;S) and Cay(G;S') on the group G, would consist exclusively of automorphisms of the group G. Alas, our world is not perfect. However, there are some Cayley graphs X=Cay(G;S) for which the isomorphism problem can be solved in this manner. That is, for such a graph X, the Cayley graph Cay(G;S') is isomorphic to X if and only if there is an automorphism of G that takes S to S' (and hence acts as a graph isomorphism). Such a graph is said to have the Cayley Isomorphism, or CI, property. Furthermore, there are some groups G for which every Cayley graph Cay(G;S) has the CI property; these groups are said to have the CI property. The Cayley Isomorphism problem is the problem of determining which graphs, and which groups, have the CI property.

In this talk, I will discuss the motivation and background of the CI problem (which stems from a 1977 paper by Laszlo Babai), and survey some of the results that have been obtained on this problem. Traditionally, the problem has been confined to finite Cayley graphs. Towards the end of the talk, I will discuss the extension of this problem to infinite graphs, and some results I have obtained on locally finite graphs in joint work with Babai.

All welcome
Friday 11
14:15 - SEMINAR - DECRA Information Seminar : Guidance on applying for an ARC Discovery Early Career Researcher Award Website | More Information
The Research Development Office will be presenting a seminar regarding the rules and how to write a winning application for an ARC Discovery Early Career Researcher Award.
Tuesday 15
12:00 - SEMINAR - Groups and Combinatorics Seminar: Does every Cayley graph have a hamiltonian cycle? More Information


Groups and Combinatorics Seminar

Dave Morris (University of Lethbridge, Canada)

will speak on

Does every Cayley graph have a hamiltonian cycle?

at 12 noon in MLR2 on Tuesday 15th of March

Abstract: It was conjectured 40 years ago that every connected Cayley graph has a hamiltonian cycle, but there is very little evidence for such a broad claim. The talk will describe some of the progress that has been made, and present a few of the many open problems. Almost all of the talk will be understandable to anyone familiar with the fundamentals of graph theory and group theory.

All welcome
Tuesday 22
12:00 - SEMINAR - Groups and Combinatorics Seminar: Normal coverings of finite symmetric and alternating groups More Information
Groups and Combinatorics Seminar



Daniela Bubboloni (University of Firenze)

will speak on

Normal coverings of finite symmetric and alternating groups

at 12 noon on Tuesday 22nd of March in MLR2

Abstract: In this talk we consider the symmetric and alternating finite groups G=S_n, A_n and investigate the minimum number gamma(G) of maximal subgroups H_i, i=1 ... k of G such that each element in G lies in some G-conjugate of H_i. The number gamma(G) lies between a*phi(n) and b*n for certain constants a, b, where phi(n) is the Euler phi-function and depends on the arithmetical complexity of n. In the case where n is divisible by at most two primes, we determine the exact value for gamma(S_n) when n is odd and for gamma(A_n) when n is even.

All welcome
Tuesday 29
12:00 - SEMINAR - Groups and Combinatorics Seminar: A duality for mixed abelian groups More Information
Groups and Combinatorics Seminar

Phill Shultz (UWA) will speak on

A duality for mixed abelian groups

at 12 noon on Tuesday 29th of March in MLR2

Abstract: An abelian group is mixed if it contains non-zero elements of finite and of infinite order, and well mixed if it is not the direct sum of a torsion and a torsion-free group. The classification of well mixed groups is an intractable problem.

In this seminar, I will use a duality to describe the structure of well mixed groups G satisfying two finiteness conditions:

1. G has finite rank, that is, G contains a maximal free subgroup of finite dimension;

2. G is self-small, that is, the image of every homomorphism from G into an infinite direct sum of copies of G is contained in the direct sum of finitely many of these copies.

All welcome

 April 2011
Tuesday 05
12:00 - SEMINAR - Groups and Combinatorics Seminar: On the lambda-design conjecture More Information
Groups and Combinatorics Seminar

Akos Seress (UWA)

will speak on

On the lambda-design conjecture

at 12 noon in MLR2 on Tuesday 5th of April.

Abstract:

A lambda-design is a set system consisting of v sets on a v-element underlying set so that any two distinct members intersect in exactly lambda points and not all sets are of the same size.

Ryser and Woodall's lambda-design conjecture states that all lambda-designs can be obtained from symmetric designs a certain block-complementation procedure. We report about recent efforts toward the solution of this conjecture.

All welcome

 May 2011
Tuesday 03
12:00 - SEMINAR - Groups and Combinatorics Seminar: On the vanishing graph of finite groups More Information
Groups and Combinatorics Seminar

Pablo Spiga (UWA)

will speak

On the vanishing graph of finite groups

at 12 noon Tuesday 3rd of May in MLR2

Abstract: In this talk, we investigate a graph Gamma encoding some information on the zeros of the irreducible complex characters of a finite group G. (We say that the element g of G is a zero, if there exists an irreducible complex character $ hi$ with $ hi(g)=0$.) This graph Gamma is called "the vanishing graph" of G. The vertices of Gamma are the primes p such that G contains a zero whose order is divisible by p. Moreover, the edges of Gamma are the pairs {p,q} such that G contains a zero whose order is divisible by pq.

Clearly, Gamma is a subgraph of the prime graph. In this talk, we study the density of the graph Gamma and we show how some geometrical properties of Gamma generalise two classical character theory results of Thompson and of Ito-Michler.
Tuesday 10
12:00 - SEMINAR - Groups and Combinatorics Seminar: The automorphisms of McCullough-Miller space More Information
Groups and Combinatorics Seminar

Adam Piggott (Bucknell University)

will speak on

The automorphisms of McCullough-Miller space

at 12 noon 10th of May in MLR2

Abstract: McCullough-Miller's space X=X(W) is a topological model for the outer automorphism group of a free product of groups W. We will discuss the question of just how good a model it is. In particular, we consider circumstances under which Aut(X) is precisely Out(W).

All welcome
Tuesday 17
12:00 - SEMINAR - Groups and Combinatorics Seminar: FinInG: A GAP package for Finite Incidence Geometry More Information
Groups and Combinatorics Seminar

John Bamberg (UWA)

will speak on

FinInG: A GAP package for Finite Incidence Geometry

at 12 noon on Tuesday 17th of May in MLR2

Abstract: FinInG is a GAP-package under development for computation in finite incidence geometry, based on the computer algebra package GAP (Groups, Algorithms and Programming). The algebraic power of GAP is employed, particularly in its facility with matrix and permutation groups. The aim of the package FinInG is to provide users with the basic tools to work in various areas of finite incidence geometry from the realms of projective spaces to the flat lands of generalised polygons. In this talk we will give some examples and explain some of the main features of the package.

All welcome
Tuesday 24
12:00 - SEMINAR - Groups and Combinatorics Seminar: Primitive generalised quadrangles More Information
Groups and Combinatorics Seminar

Michael Giudici (UWA)

will speak on

Primitive generalised quadrangles

at 12 noon Tuesday 25th of May in MLR2

Abstract: A generalised quadrangle is an incidence structure of points and lines such that the bipartite incidence graph has diameter 4 and girth 8. The classical examples are the low dimensional polar spaces associated with the classical groups PSp(4,q), PSU(4,q) and PSU(5,q). In this talk I will discuss recent work with John Bamberg, Joy Morris, Gordon Royle and Pablo Spiga aimed at characterising the classical examples in terms of the action of their automorphism group on points and lines.

All welcome
Tuesday 31
12:00 - SEMINAR - Groups and Combinatorics Seminar: Fat elements in quasi-simple groups of Lie type More Information
Groups and Combinatorics Seminar

Sabina Pannek (UWA/Aachen)

will speak on

Fat elements in quasi-simple groups of Lie type

at 12 noon on Tuesday 31st of May in MLR2

Abstract: We call an element of the general linear group GL(d,q) fat, or more precisely a fat(d,q;e)-element, if it leaves invariant and acts irreducibly on a subspace of dimension e > d/2 > 1. Equivalently, g in GL(d,q) is a fat(d,q;e)-element if the characteristic polynomial for g has an irreducible factor over GF(q) of degree e > d/2 > 1.

We wish to classify irreducible subgroups G of GL(d,q) which contain fat elements and are quasi-simple, that is where G is a covering group of an alternating group, a covering group of a sporadic simple group, or a quasi-simple group of Lie type. In my talk, I will discuss recent results in the latter case of G being a quasi-simple group of Lie type.

 June 2011
Tuesday 14
10:30 - SEMINAR - Groups and Combinatorics Seminar More Information
Groups and Combinatorics Seminar

There will be three 25 minute talks on Tuesday 14th of June in MLR2 starting at 10:30 am.

10:30am:

Sylvia Ozols (Adelaide)

will speak on

The Bruck-Bose Construction

11am:

Wei Jin (UWA)

will speak

On distance, geodesic and arc transitivity of graphs

11:30am

Carmen Amarra (UWA)

will speak on

Quotient-complete arc-transitive graphs



Abstract 1: The Bruck-Bose representation of a projective plane is a tool both for examining objects of any even dimensional projective space in the more familiar setting of a projective plane, and for 'magnifying' objects in certain projective planes by looking at them in a higher dimensional space. In this talk we will go through some finite projective geometry background, including Baer subplanes and partitions of odd dimensional projective spaces. We will define the Bruck-Bose construction and get a general understanding of how it works.



Abstract 2: We compare three transitivity properties of finite graphs, namely, for a positive integer s, s-distance transitivity, s-geodesic transitivity and s-arc transitivity. It is known that if a finite graph is s-arc transitive but not (s+1)-arc transitive then s<8 and s not equal to 6. We show that there are infinitely many geodesic transitive graphs with this property for each of these values of s, and that these graphs can have arbitrarily large diameter if and only if 0< s<4. Moreover, for a prime p we prove that there exists a graph of valency p that is 2-geodesic transitive but not 2-arc transitive if and only if p = 1 (mod 4), and for each such prime there is a unique graph with this property: it is an antipodal double cover of the complete graph K_{p+1} and is geodesic transitive with automorphism group PSL(2,p) x Z_2. This is joint work with A. Devillers, C.H. Li and C. E. Praeger.

Abstract 3: A graph Gamma is G-quotient-complete (for some G in Aut(Gamma)) if it has at least one nontrivial G-normal quotient which is a complete graph, and each of its other nontrivial G-normal quotients is either a complete graph or an empty graph. We define the parameter k to be the number of G-normal quotients of Gamma which are complete, and in this talk we consider the family of quotient-complete graphs with k>2. We construct all the graphs Gamma; in this family together with the corresponding automorphism groups G, and give upper bounds for k in terms of the order of Gamma.

 August 2011
Wednesday 03
11:00 - SEMINAR - Groups and Combinatorics Seminar: Perfect State Transfer More Information
Groups and Combinatorics Seminar

Murray Smith (UWA)

will speak on

Perfect State Transfer

at 11am Wednesday 3rd of August in MLR2



Abstract: In quantum information systems, it is desirable to transfer the quantum state of one qubit to another with high fidelity. This is known as perfect state transfer, and can be modelled as a graph theoretic problem. In this talk I will offer an introduction to the topic with types of graphs that can permit PST, specifically integral circulant graphs. I will also discuss my time spent at the University of Waterloo under the direction of Dr. Chris Godsil, and recent developments in characterising graphs with PST.

All welcome
Sunday 14
10:00 - EVENT - 2011 Open Day : Experience what's on offer at UWA Website | More Information
UWA opens up the whole campus to the public.

Come and find out about the courses on offer, valuable research, community programs, and facilities...all mixed with a day full of lots of fun activities for everyone!

 September 2011
Friday 09
13:00 - SEMINAR - Groups and Combinatorics Seminar: Algorithmic Generalisations of Small Cancellation Theory More Information
Groups and Combinatorics Seminar

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

Max Neunhöffer (University of St Andrews)

"Algorithmic Generalisations of Small Cancellation Theory"

Friday 9th September (2011), 1pm, MLR2

Abstract: In this talk I will report on joint work in progress with Stephen Linton, Richard Parker and Colva Roney-Dougal. We want to generalise classical small cancellation theory (SCT) in an algorithmic direction. SCT used to be a fixed set of conditions to test on a finite presentation of a group. If these conditions are fulfilled, SCT proves that the group is infinite, word-hyperbolic and provides a solution to the word problem. I will describe our ideas to generalise this beyond recognition. In the end, the computer will analyse a finite presentation of a group or other algebraic structure and come up with a SCT-like proof showing that the algebraic structure is infinite and providing a solution to the word problem.
Friday 16
13:00 - SEMINAR - Groups and Combinatorics Seminar:Generation and random generation: from simple groups to maximal subgroups More Information
Groups and Combinatorics Seminar

Tim Burness (University of Southampton)

will speak on

Generation and random generation: from simple groups to maximal subgroups

at 1pm on Friday 16th of September in MLR2

Abstract: Problems concerning the generation of finite simple groups have been studied since the early days of group theory in the 19th century. In this talk I will survey some of the main results and open problems in this area, and I will present some new results on the generation of maximal subgroups of simple groups. Applications and related open problems will also be discussed. This is joint work with Martin Liebeck and Aner Shalev.



All welcome
Friday 23
13:00 - SEMINAR - Groups and Combinatorics Seminar: Locally s-distance transitive graphs with a regular star normal quotient More Information
Groups and Combinatorics Seminar

Alice Devillers (UWA)

will speak on

Locally s-distance transitive graphs with a regular star normal quotient

at 1pm Friday 23rd of September in MLR2

Abstract: During the study of locally s-distance transitive graphs (Devillers-Giudici-Li-Praeger), it was found that an important case is the case of graphs with a star quotient. These graphs are bipartite, with a normal subgroup transitive on one bipart but not on the other. With Cheryl Praeger we studied in particular the case where the star has at least 3 branches, and the normal subgroup acts regularly on one bipart. I will explain our main result. Warning: there will be proofs.

All welcome
Friday 30
13:00 - SEMINAR - Groups and Combinatorics Seminar: The concept of p-deficiency and its applications More Information
Groups and Combinatorics Seminar

Anitha Thillaisundaram (Cambridge)

will speak on

The concept of p-deficiency and its applications.

at 1pm on Friday 30th September in MLR2

Abstract: We use Schlage-Puchta's concept of p-deficiency and Lackenby's property of p-largeness to show that a group having a finite presentation with p-deficiency greater than 1 is large. What about when p-deficiency is exactly one? We also generalise a result of Grigorchuk on Coxeter groups to odd primes.

All welcome.

 October 2011
Wednesday 05
16:00 - EXPO - UWA Postgraduate & Honours Expo 2011 Website | More Information
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 More Information
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.

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