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Today's date is Wednesday, November 25, 2020
Centre for the Mathematics of Symmetry and Computation
 February 2010
Tuesday 16
12:00 - SEMINAR - Groups and Combinatorics Seminar: Some results on constant maps in the transition monoid of an automaton More Information
Groups and Combinatorics Seminar

Tim Boykett (Johannes Kepler Universitat Linz, Austria)

will speak on

Some results on constant maps in the transition monoid of an automaton

at 12 noon in Maths Lecture Room 2 on Tuesday 16 February.

Abstract: We investigate finite state automata and are interested in the situation when certain words can map any given state to a fixed terminal state. These so called reset words correspond to constant maps or right zeroes in the transition monoid of the automaton. A conjecture of J. Cerny states that if an automaton has a reset word, then it has one of length less than or equal to (n-1)^2, where n is the number of states. This bound is reached by a class defined by Cerny and 8 known sporadic examples.

In this talk I will present an outline of some results including some classes with much smaller minimal reset words. I will also outline several open areas of work.

All welcome.
Tuesday 23
12:00 - SEMINAR - Groups and Combinatorics Seminar: On a problem of Janko, a conjecture of Isbell and a £50 prize of Cameron. More Information
Groups and Combinatorics Seminar

Pablo Spiga (UWA)

will speak on

On a problem of Janko, a conjecture of Isbell and a £50 prize of Cameron.

at 12 noon in MLR2 on Tuesday 23 February.

Abstract: Recently Janko asked whether it is possible to classify the 2-generated p-groups whose maximal subgroups are 2-generated. In this talk we present some data and some partial results in order to show that a possible classification seems feasible for p>4. Also, we show how the groups arising in this classification are useful for obtaining a partial result towards a conjecture of Isbell (and hopefully a £50 prize of Cameron). All required definitions are given during the talk.

All welcome.

 March 2010
Tuesday 02
12:00 - SEMINAR - Groups and Combinatorics Seminar:Hemisystems of generalised quadrangles More Information
Groups and Combinatorics Seminar

John Bamberg (UWA)

will speak on

Hemisystems of generalised quadrangles

at 12 noon in MLR2 on Tuesday 2 March

Abstract: (Joint work with Michael Giudici and Gordon Royle). A hemisystem of a generalised quadrangle is a set of half the lines of the generalised quadrangle such that every point is on a constant number m of lines. (So necessarily, m is half the number of lines on any point). The generalised quadrangles of interest are those which meet the Higman-Sims bound, whereby a hemisystem naturally produces a partial quadrangle and strongly regular graph. Segre (1965) showed that there exists a hemisystem of the classical generalised quadrangle H(3,3^2), and it was conjectured by J. A. Thas in 1995 that no hemisystem of H(3,q^2) exists for q>3. Ten years later, Cossidente and Penttila proved that for every q odd, there exists a hemisystem of H(3,q^2), and their examples arise from an embedding of a sub-geometry into H(3,q^2) (namely, an elliptic quadric Q-(3,q)). The only known generalised quadrangles with the same parameters of H(3,q^2), q odd, are the flock generalised quadrangles. We will present in this talk an improvement of Cossidente and Penttila's results to flock generalised quadrangles.

All welcome
Tuesday 09
12:00 - SEMINAR - Groups and Combinatorics Seminar: The generalised Curtis-Tits system and black box groups More Information
Groups and Combinatorics Seminar

Sukru Yalcinkaya (UWA)

will speak on

The generalised Curtis-Tits system and black box groups

at 12 noon in MLR2 on Tuesday 9th of March.

Abstract: The Curtis-Tits presentation of groups of Lie type is the main identification theorem used in the classification of the finite simple groups. I will describe the most general form of the Curtis-Tits presentation of finite groups of Lie type where the Phan's presentation for the twisted groups appears as a special case. I will also talk about a beautiful application of this result to the recognition of finite black box groups.

If time permits I will briefly talk about how Curtis-Tits and its Phan variations link two theories: Theory of black box groups and groups of finite Morley rank.

All welcome
Tuesday 16
12:00 - SEMINAR - Groups and Combinatorics Seminar: On arc-transitive graphs with large arc-stabilisers More Information
Groups and Combinatorics Seminar

Gabriel Verret (University of Ljubljana)

will speak

On arc-transitive graphs with large arc-stabilisers

at 12 noon Tuesday 16 March in Maths Lecture Room 2

Abstract : A classical result of Tutte is that the order of the arc-stabiliser in a cubic arc-transitive graph is bounded above by 16. This surprising result is both interesting from a theoretical point of view and it also has applications, for example in enumerating small cubic arc-transitive graphs. We wish to understand under what hypothesis Tutte's result generalizes to other valencies and what can be said when it does not.

All welcome.
Tuesday 23
12:00 - SEMINAR - Groups and Combinatorics Seminar: A classification of regular maps of square-free order More Information
Groups and Combinatorics Seminar

Cai Heng Li (UWA)

will speak on

A classification of regular maps of square-free order

at 12 noon on Tuesday 23 March in MLR2

Abstract: I shall describe a classification of rotary and regular maps of square-free order (ie, square-free number of vertices). Basically (and a bit surprisingly), the underlying graphs of such maps have simple form.

All welcome

 April 2010
Tuesday 13
12:00 - SEMINAR - Groups and Combinatorics Seminar: Infinite Veronesean Caps More Information
Groups and Combinatorics Seminar

Jeroen Schillewaert (University of Canterbury, NZ)

will speak on

Infinite Veronesean Caps

at 12 noon in MLR 2 on Tuesday 13 April.

Abstract: Veronesean varieties are fundamental objects in geometry, be it classical algebraic geometry or modern finite geometry. In the finite case both quadric Veroneseans and Hermitian Veroneseans, which are defined algebraically, were characterized geometrically as Veronesean caps. Replacing counting arguments by a more synthetic approach we were able to extend these results to arbitrary division rings. Moreover, we consider a much larger class of Veronesean caps, and their relation with Jordan algebras. (joint work with H. van Maldeghem).
Tuesday 20
12:00 - SEMINAR - Groups and Combinatorics Seminar:Serendipity, involutions and regular semisimple matrices More Information
Groups and Combinatorics Seminar

Cheryl Praeger

will speak on

Serendipity, involutions and regular semisimple matrices

at 12noon in MLR2 on Tuesday April 20.

Abstract: (joint work with Akos Seress)

Key to studying finite simple groups is to study their involution centralisers, and this is true also in a computational setting. In a seminal paper, Chris Parker and Rob Wilson present and analyse a practical algorithm for computing the centraliser of an involution z in a finite classical group of odd characteristic. Fundamental to their approach, using Bray’s theorem and an observation of Richard Parker, is the estimation of the proportion of pairs of conjugates of z whose product has odd order -- and is regular semisimple. Their analysis estimates that O(n) random selections are needed to find a suitable random conjugate of z.

Motivated by the wish to improve on Parker and Wilson's algorithm by exploiting the pairs of conjugates of z whose product has even order, we were led by experimental evidence to estimate the proportion of such pairs whose product is regular semisimple. Serendipity came to our rescue: we recognised a probability generating function for estimating such pairs as similar to one analysed by Jason Fulman, Peter Neumann and me for computing the proportion of separable matrices in unitary groups. A similar analysis enabled us find the limiting proportion, depending only on the field size q, for general linear groups and to prove exponentially fast convergence.

Tuesday 27
12:00 - SEMINAR - Groups and Combinatorics Seminar: Proportions of elements of certain orders in classical groups More Information
Groups and Combinatorics Seminar

Simon Guest (UWA) will speak on

Proportions of elements of certain orders in classical groups

at 12 noon Tuesday 27 April in MLR2

Abstract: (joint work with Cheryl, and Tomasz Popiel)

Let G be a finite group. We say that an element g in G has 2-part order 2^j if 2^j is the largest power of 2 dividing the order of g. To analyze recognition algorithms for classical groups, we are sometimes presented with the following question. Take the direct product of two classical groups A x B and choose a random element (a,b); what is the probability that (a,b) powers up to an element of the form (z,1), where z is an involution in A? We require that the 2-part order of a be greater than the 2-part order of b. In order to estimate this probability, we first establish lower bounds on the proportion of elements in the symmetric group with a given 2-part order. We will describe the relationship between maximal tori in a classical group and its Weyl group. Since the Weyl group of a classical group involves the symmetric group, we can use the lower bounds for the symmetric group, together with this relationship, to obtain corresponding lower bounds for classical groups of odd characteristic. In fact, if A and B have dimension m and n-m, and m is contained in the interval [n/3,n/2] (for example if A x B is the centralizer of a strong involution), then we show that the probability that (a,b) powers up to (z,1) is at least an explicit constant.

All Welcome

 May 2010
Tuesday 04
12:00 - SEMINAR - Lectures on Growth in Groups and Graphs More Information
Nick Gill (University of Bristol) will be giving a series of three talks on the topic of Growth in Groups and Graphs as follows:

* Tuesday 4 May, 12 noon, MLR2, I: Sum-Product

* Friday 7 May, 11am, MLR3, II: Growth in Groups of Lie Type

* Friday 14th May 11am MLR3, III: Escape

These are part of a UWA research collaboration award funding visits to Perth by Nick Gill and Harald Helfgott. The aim of the talks is to give the necessary background and introduction to research in the area. Nick will give a further three talks later in the year. Nick has prepared a page of supporting material at https://www.maths.bris.ac.uk/~manpg/austlit.html

The titles and abstracts for the first three talks are as follows.

I: SUM-PRODUCT We introduce the idea of growth in groups, before focussing on the abelian setting. We take a first look at the sum-product principle, with a brief foray into the connection between sum-product results and incidence theorems.

We then focus on Helfgott’s restatement of the sum-product principle in terms of groups acting on groups.

II: GROWTH IN GROUPS OF LIE TYPE Since Helfgott first proved that “generating sets grow” in SL_2(p) and SL_3(p), our understanding of how to prove such results has developed a great deal. It is now possible to prove that generating sets grow in any finite group of Lie type; what is more the most recent proofs are very direct – they have no recourse to the incidence theorems of Helfgott’s original approach.

We give an overview of this new approach, which has come to be known as a ”pivotting argument”. There are five parts to this approach, and we outline how these fit together.

III: ESCAPE The principle of “escape from subvarieties” is the first step in proving growth in groups of Lie type. We give a proof of this result, and its most important application (for us) – the construction of regular semisimple elements.

We then examine other related ideas from algebraic geometry, in particular the idea of non-singularity.
Tuesday 11
12:00 - SEMINAR - Groups and Combinatorics Seminar: Three Hamilton Decomposition Problems More Information
Groups and Combinatorics Seminar

Brian Alspach (University of Newcastle)

will speak on

Three Hamilton Decomposition Problems

at 12 noon Tuesday 11 May in MLR2

Abstract: This talk deals with three middle-aged problems on decomposing graphs into Hamilton cycles. There will be sokmething old, something new, something borrowed, and something blue.

All welcome
Tuesday 18
12:00 - SEMINAR - Groups and Combinatorics Seminar:Symmetry properties of subdivision graphs, and serendipity More Information
Groups and Combinatorics Seminar

Alice Devillers (UWA)

will speak on

Symmetry properties of subdivision graphs, and serendipity

at 12 noon in MLR2 on Tuesday 18 May

Abstract: (Joint work with Ashraf Daneshkhah and Cheryl E. Praeger)

The subdivision graph $S( igma)$ of a graph $ igma$ is obtained from $ igma$ by `adding a vertex' in the middle of every edge of $ igma$. In other words, it is the incidence graph of $ igma$ if $ igma$ is seen as a vertices-edges geometry.

In work with Ashraf and Cheryl, we studies various symmetry properties of $S( igma)$. In particular we tried to characterise when $S( igma)$ is locally s-arc transitive, and locally s-distance transitive. I will explain the results we obtained.

Subdivision graphs also appeared by surprise on work with Cheryl about locally s-distance transitive graphs admitting a $K_{1,2}$ quotient, so we have an application to our subdivision graph results. Generalised polygons also make an unexpected appearance.

All welcome

18:00 - PUBLIC LECTURE - The Unreasonable Effectiveness of Mathematics Website | More Information
An IAS public lecture by Winthrop Professor Cheryl Praeger AM FAA, Director of the Centre for Mathematics of Symmetry and Computation at UWA and 2009 WA Scientist of the Year.

As the vital role of technology in modern society increases, the mathematical sciences are becoming indispensable.

The title, borrowed from Eugene Wigner, is re-proved daily as mathematicians find that the same fundamentals of mathematics can apply to vastly different problems, giving applications from physics to finance, to fast secure communication.

The supply of mathematics graduates and mathematics teachers in Australia is unable to meet education and employer demand.

What must Australia do to properly equip itself for the future?
Tuesday 25
12:00 - SEMINAR - Groups and Combinatorics Seminar: Point regular automorphism groups of generalised quadrangles More Information
Groups and Combinatoric Seminar

Michael Giudici (UWA)

will speak on

Point regular automorphism groups of generalised quadrangles

at 12 noon on Tuesday 25 May in MLR2

Abstract: (joint work with John Bamberg) Studying regular automorphism groups of combinatorial objects has had a long history. Recently there has been some focus on groups of automorphisms of generalised quadrangles that act regularly on the set of points. In this talk I will give a classification of all point regular automorphism groups of the thick classical generalised quadrangles. I will also construct point regular automorphism groups of some nonclassical generalised quadrangles. The constructions show that the class of groups which can act regularly on the set of points of a generalised quadrangle contains nonabelian 2-groups, groups of exponent 9 and nonspecial p-groups, and is thus wilder than previously thought.

All welcome.

 June 2010
Tuesday 22
12:00 - SEMINAR - Groups and Combinatorics Seminar: Symmetric graphs of diameter 2 arising from the classical groups More Information
Groups and Combinatorics Seminar

Carmen Amarra (UWA)

will speak on

Symmetric graphs of diameter 2 arising from the classical groups

at 12noon Tuesday 22nd June in MLR2

Abstract: Let V be a vector space of dimension d over a field of q elements, and let H be an irreducible subgroup of GL(V). Let S be an H-orbit consisting of nonzero vectors, satisfying -S = S, and let Gamma= Cay(V,S). Then Gamma is G-symmetric, where G = V times H. Our goal is to find the graphs Cay(V,S) which have diameter 2, that is, those for which V is contained in S+S. In particular, we shall consider the case where H is a classical group.



All welcome.

 July 2010
Friday 16
14:00 - SEMINAR - Groups and Combinatorics Seminar: Several open problems in graph theory More Information
Groups and Combinatorics Seminar

Mehdi Behzad, Sharif University, Tehran

will speak on

Several open problems in graph theory

at 2pm Friday 16th July in MLR2

(**Note the different time to usual***)

The speaker is also giving a Colloquium at 12 noon on Tuesday.

Abstract: The author's Ph.D. thesis [1] contains the Total Chromatic Number Conjecture which is still unsettled. Reference [3] introduces the Total Ramsey Numbers R"(r,s) for positive integers r and s, and provides the exact values of this function for all the lattice points outside the region {(r,s): r > 4, s > r 2-5r+8}.

In this informal talk, we first elaborate on the above old problems. Then, we introduce several classes of new problems in relation to a fiction entitled, `The Story of the King and the Mathematician.' Each new problem, presented in the context of a puzzle, is about dismantling, displacing, and reconstructing an arbitrary graph G [2].

Based on the fiction, the first class of the new problems is generalized to obtain the second class. Each member of the latter class introduces several new puzzles with different levels of difficulties. Next, we assign a weight function to the vertex set of the graph involved in each puzzle. One such weight function produces the second class. Hence, this third class is a generalization of the second as well as the first class. Finally, we dismantle, displace, and reconstruct the weighted graph G with prescribed conditions.

In order to solve each puzzle, weneed to obtain the values of a few parameters, provide a procedure to find all the annihilating subsets of the vertex set of the graph G, and find an algorithm to transfer G with pre-assigned conditions.

REFERENCES:

[1] M. Behzad, Graphs and their Chromatic Numbers, Doctoral Thesis, Michigan State University (1965).

[2] M. Behzad, Dismantling, Displacing and Reconstructing Graphs, Manuscript (2009).

[3] M. Behzad and H. Radjavi, Another analogue of Ramsey Numbers, Math. Annal.186, 228 - 232 (1970).



All Welcome.
Tuesday 20
12:00 - SEMINAR - Groups and Combinatorics Seminar: Some wonderful conjectures (but almost no theorems) at the boundary between analysis, combinatorics and probability More Information
Groups and Combinatorics Seminar

Alan Sokal (New York University and University College, London)

will speak on

Some wonderful conjectures (but almost no theorems) at the boundary between analysis, combinatorics and probability

at 12 noon on Tuesday 20 July

in Maths Lecture Room 2.

Abstract: I discuss some analytic and combinatorial properties of the entire function $F(x,y) = um imits_{n=0}^ nfty rac{x^n}{n!} y^{n(n-1)/2}$. This function (or formal power series) arises in numerous problems in enumerative combinatorics, notably in the enumeration of connected graphs.

All welcome
Wednesday 21
11:00 - SEMINAR - Lectures on Growth in Groups and Graphs Part II More Information
Nick Gill (Bristol)

will continue his series of lectures on growth in groups and graphs

at the following times

Wednesday 21/7 at 11am

Friday 23/7 at 11am

and Tuesday 27/7 at 1pm

All seminars will be held in MLR2

I: Growth in soluble subgroups of GL_r(p)

Abstract: We show how to reduce the study of exponential growth in soluble subgroups of GL_r(p) to the nilpotent setting. We make use of ideas based on the sum-product phenomenon, as well as some machinery from linear algebraic groups. We will not assume any background from these areas.

II: An introduction to expanders

Abstract: This is a background seminar preparing the way for Seminar III, where we connect results on growth in simple groups to the explicit construction of families of expanders. In this seminar we will define what we mean by a family of expanders, stating (and sometimes even proving!!) background results that will be important later.

We will also try to explain why expanders are of such interest to so many different groups of people.

III: Using growth results to explicitly construct expanders

Abstract: We outline the method of Bourgain and Gamburd. They were the first to use results concerning growth in simple groups to explicitly construct expander graphs. Let S be a set in SL_2(Z) and define S_p to be the natural projection of S modulo p. Now let G_p be the Cayley graph of SL_2(F_p) with respect to the set S_p. Bourgain and Gamburd give precise results as to when the set of graphs {G_p : p a prime} forms a family of expanders. They make crucial use of the result of Helfgott (encountered in an earlier seminar) which states that "all generating sets in SL_2(p) grow".

All welcome

 August 2010
Tuesday 03
13:00 - SEMINAR - Groups and Combinatorics Seminar: Triple factorisations: geometric approach More Information
Groups and Combinatorics Seminar

Hassan Alavi (UWA)

will speak on

Triple factorisations: geometric approach

at 1pm Tuesday 3rd of August in MLR2

Abstract:

Triple factorisations G = ABA, for proper subgroups A and B of a group G, are fundamental in the study of Lie type groups, as well as in geometry. They correspond to flag-transitive collinearly connected point-line geometries (in which each pair of points lies on at least one line). Geometries satisfying the dual condition -where each pair of lines meets in at least one point- are called concurrently connected. In this talk, we consider a generalisation of point-line projective geometries called (m,k,j)-projective spaces and determine when they are collinearly and/or concurrently connected. These geometrical results arose from studying triple factorisations of general linear groups G with the subgroups A and B parabolic.



All welcome.
Tuesday 10
13:00 - SEMINAR - Groups and Combinatorics Seminar: Factorial Schur Polynomials from the viewpoint of the Six Vertex Model More Information
Groups and Combinatorics Seminar

Peter McNamara (Stanford)

will speak on

Factorial Schur Polynomials from the viewpoint of the Six Vertex Model

at 1pm Tuesday 10th August in MLR2

Abstract: The factorial Schur polynomials, also known as double Schubert polynomials for Grassmannian permutations are a family of polynomials in two alphabets of variables that generalise the classical Schur functions. Their biggest claim to fame is that they compute the equivariant cohomology of Grassmannians. In this talk, we present the appearance of factorial Schur polynomials combinatorially as the value of a partition function on a well chosen six vertex model from statistical mechanics. This combinatorial approach is suggestive of an underlying phenomenon waiting to be understood. This is joint work with Daniel Bump and Maki Nakasuji.

All welcome.

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