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Today's date is Tuesday, November 24, 2020
Centre for the Mathematics of Symmetry and Computation
 June 2013
Friday 14
15:00 - SEMINAR - Groups and Combinatorics Seminar, Recognising the Symmetric and Alternating Squares More Information
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.
Friday 21
15:00 - SEMINAR - Groups and Combinatorics Seminar More Information
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.

 July 2013
Friday 05
15:00 - SEMINAR - Groups and Combinatorics Seminar, On Pseudo-Injective Group Algebra More Information
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.

 August 2013
Friday 02
15:00 - SEMINAR - Groups and Combinatorics Seminar, Maximal arcs that contain regular hyperovals More Information
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.
Friday 09
15:00 - SEMINAR - Groups and Combinatorics Seminar, Semisymmetric graphs of prime valency More Information
Abstract:

A graph is called X-semisymmetric if the group X acts transitively on edges, but not on vertices. Such graphs are necessarily bipartite and bi-regular, of valencies k and l say. There is a natural relationship between semisymmetric graphs and amalgams of groups. This leads us to consider the "universal" example: the bi-regular tree T of valencies k and l. The conjecture of Goldschmidt says that when k and l are primes there are (up to isomorphism) finitely many locally finite groups X such that T is X-semisymmetric (X is locally finite if it acts with finite vertex stabilisers). For k=l=3 it was shown by Goldschmidt that there are 15 such groups. In the talk I will give an overview of some results which bear relevance to the conjecture, and report on some recent progress with respect to certain small primes.
Sunday 11
10:00 - OPEN DAY - 2013 Open Day : Join us for our Centenary Open Day and experience all that UWA has to offer Website | More Information
Come and find out about our undergraduate and postgraduate courses, career options, scholarship opportunities, our valuable research, community programs and facilities.

There's also residential college tours, hands-on activities, live music, entertainment, and plenty of fun activities for the whole family as we celebrate our 100th birthday.
Friday 16
15:00 - SEMINAR - Groups and Combinatorics Seminar, Decomposing tensor products over fields of small characteristic More Information
Abstract:

One motivation for this talk comes from representation theory: decomposing a tensor product of irreducible (or indecomposable) representations as a sum of smaller degree irreducible (or indecomposable) representations. Other motivations come from quantum mechanics and Frobenius algebras.

Consider an $r imes r$ matrix $K_r$ over a field $F$ with 1s on the main diagonal and first upper diagonal (positions $(i,i)$ and $(i,i+1)$) and zeros elsewhere. The tensor product $K_r times K_s$ is a unipotent matrix whose Jordan canonical form is determined by some partition of $rs$. We will show that this partition enjoys surprising symmetries: duality, periodicity, regularity. Our original motivation was to study this partition when the characteristic $p$ of $F$ is small (i.e. $p<r+s-1$). The large characteristic case ($p e r+s-1$) was solved recently by Iima and Iwamatsu.

This is joint work with Cheryl E. Praeger and Binzhou Xia.
Friday 23
15:00 - SEMINAR - Groups and Combinatorics Seminar, Breaking symmetries of infinite graphs More Information
Abstract:

Symmetry breaking involves colouring the elements of a combinatorial structure so that the resulting structure has no nontrivial symmetries. In this talk I'll give an introduction to symmetry breaking, with a particular focus on infinite graphs. I'll also discuss a number of research directions that are opening up. Along the way I'll highlight some of the interesting open questions and conjectures that are being worked upon.

Many of these problems relate to very deep problems in group theory, but I'll try to make the group theory as accessible as possible.
Friday 30
15:00 - SEMINAR - Groups and Combinatorics Seminar, The structure of 3-separations of matroids More Information
Abstract:

In this talk, I will give an introduction to decomposition theory of 3-connected matroids. In order to make it more accessible I will also introduce some related definitions and examples.

 September 2013
Friday 06
15:00 - SEMINAR - Groups and Combinatorics Seminar, Connections between Graph Theory & Combinatorics on Words More Information
Abstract:

A string or word is usually thought of as a sequence of letters drawn from some alphabet. Applications to bioinformatics and other areas suggest the utility of defining strings on subsets of the alphabet instead -- so-called "indeterminate" strings. I describe recent work that connects such strings to ideas from graph theory, and wonder if graph theoretical concepts and knowledge might be still further applied to their analysis and use.
Friday 13
15:00 - SEMINAR - Groups and Combinatorics Seminar, On The Non-commuting Graph More Information
Abstract:

In this talk, we will consider the non-commuting graph of a non-abelian finite group G; its vertex set is the set of non-central elements of G, and two distinct vertices x and y are joined by an edge if they do not commute together. Actually, we study some properties of the non-commuting graph such as connectivity, regularity, etc., and we show that, for many groups G, if H is a group which has the same non-commuting graph of G, then they have the same order. We determine the structure of any finite non-abelian group G (up to isomorphism) for which its non-commuting graph is a complete multipartite graph. We also show that a non-commuting graph is a strongly regular graph if and only if it is a complete multipartite graph.
Friday 20
15:00 - SEMINAR - Groups and Combinatorics Seminar, Generalised quadrangles constructed from groups More Information
Abstract:

This is just a survey talk of various ways to construct all of the known finite generalised quadrangles starting with a group and a configuration of subgroups of that group. In particular, the speaker will give a summary of where one of the "retreat" problems is at.
Wednesday 25
11:00 - SEMINAR - Mathematics Colloquium: The Cohen-Lenstra heuristics: from arithmetic to topology and back again More Information
Akshay Venkatesh (Stanford) Mahler Lecturer and IAS Professor-at-Large

will speak on

The Cohen-Lenstra heuristics: from arithmetic to topology and back again.

at 11am in the Science Library Access Grid room.



I will discuss some models of what a "random abelian group" is, and some conjectures (the Cohen-Lenstra heuristics of the title) about how they show up in number theory. I'll then discuss the function field setting and a proof of these heuristics, with Ellenberg and Westerland. The proof is an example of a link between analytic number theory and certain classes of results in algebraic topology ("homological stability").
Friday 27
15:00 - SEMINAR - Groups and Combinatorics Seminar, Straight-line programs with memory and applications to computational group theory More Information
Abstract:

Straight-line programs offer a method for encoding group computations in a "black box" sense, namely without using specifics of the group's representation or how the group operations are performed. We advocate that straight-line programs designed for group computations should be accompanied by comprehensive complexity analyses that take into account not only the number of group operations needed, but also memory requirements arising during evaluation. We introduce an approach for formalising this idea and discuss a fundamental example for which our methods can drastically improve upon existing implementations. This is joint work (in progress!) with Alice Niemeyer and Cheryl Praeger.

 October 2013
Friday 11
15:00 - SEMINAR - Groups and Combinatorics Seminar, Multiply tiling Euclidean space by translating a convex object More Information
Abstract:

We study the problem of covering Euclidean space R^d by possibly overlapping translates of a convex body P, such that almost every point is covered exactly k times, for a fixed integer k. Such a covering of Euclidean space by translations is called a k-tiling. We will first give a historical survey that includes the investigations of classical tilings by translations (which we call 1-tilings in this context). They began with the work of the famous crystallographer Fedorov and with the work of Minkowski, who founded the Geometry of Numbers. Some 50 years later Venkov and McMullen gave a complete characterization of all convex objects that 1-tile Euclidean space.

Today we know that k-tilings can be tackled by methods from Fourier analysis, though some of their aspects can be studied using purely combinatorial means. For many of our results, there is both a combinatorial proof and a Harmonic analysis proof. For k larger than 1, the collection of convex objects that k-tile is much wider than the collection of objects that 1-tile, and there is currently no complete knowledge of the polytopes that k-tile, even in 2 dimensions. We will cover both ``ancient'', as well as very recent, results concerning 1-tilings and more generally k-tilings. These results are joint work with Nick Gravin, Mihalis Kolountzakis, and Dmitry Shiryaev.
Friday 18
15:00 - SEMINAR - Groups and Combinatorics Seminar, Regular orbits of Sym(n) and Alt(n) on irreducible representations More Information
Abstract:

Given a finite group G and a faithful irreducible FG-module V where F is a field of prime order, we can ask whether G has a regular orbit on the vectors of V. This problem is related to determining which primitive permutation groups of affine type have a base of size 2, as well as the famous k(GV)-problem and a conjecture of Brauer concerning defect groups of blocks. We will consider the regular orbit problem for the symmetric and alternating groups.
Friday 25
15:00 - SEMINAR - Groups and Combinatorics Seminar, Coprime actions of finite linear groups More Information
Abstract:

Let H be a finite linear group acting completely reducibly on a finite vector space V. Gabriel Navarro asked: if the H-orbits containing vectors a and b have coprime lengths m and n, is there an H-orbit of length mn? We answered, by showing that the H-orbit containing a + b has length mn, and by showing, moreover, that in this situation H cannot be irreducible. That is to say, a stabiliser in an affine primitive permutation group does not have a pair of orbits of coprime lengths. I will make some comments, if time permits, about coprime orbit lengths for stabilisers in arbitrary primitive permutation groups. This is joint work with Silvio Dolfi, Bob Guralnick and Pablo Spiga.

 November 2013
Friday 01
15:00 - SEMINAR - Groups and Combinatorics Seminar, Algebraic geometry codes More Information
Abstract:

Codes arising from algebraic geometry, first introduced by Goppa, gained attention when Tsfasman–Vladut–Zink used them to improve the Gilbert-Varshamow bound. We will give a gentle introduction to some of the beautiful ideas from algebraic geometry used to build these codes. We will then show how to construct them, and then discuss the Tsfasman–Vladut–Zink bound. There will be an emphasis on examples.
Friday 15
15:00 - SEMINAR - Groups and Combinatorics Seminar, A miscellany of topics related to semiregular graph automorphisms More Information
Abstract :

I will discuss a few things, all related to semiregular graph automorphisms : the polycirculant conjecture, the abelian normal quotient method, an interesting class of graphs...
Monday 18
13:00 - SEMINAR - Integrable-like behavior in the Fermi-Pasta-Ulam model Website | More Information
In 1950’s Fermi, motivated by fundamental questions of statistical mechanics, started a numerical experiment in collaboration with Pasta and Ulam to test the ergodic properties of nonlinear dynamical systems. The chosen so-called FPU system was a one dimensional chain of N nonlinear coupled oscillators, described by a quadratic potential of nearby particle interactions plus a cubic perturbation. Fermi’s ergodic hypothesis states that a system under an arbitrarily small perturbing force becomes generically ergodic. Starting with the longest wavelength normal mode, the FPU system showed a non-ergodic behavior. Many pioneer works followed for the explanation of this paradox. The most prominent of them have been the work of Zabusky and Kruskal (1965), with evidence of connection between the FPU system in the thermodynamic limit and the pde Korteweg-de Vries, and the work of Flaschka et al. (1982), where the authors discovered a similar behavior of the FPU model in the Toda chain. Recent developments show a more complete picture of the problem and its explanation.

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