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Today's date is Sunday, October 25, 2020
Centre for the Mathematics of Symmetry and Computation
 September 2013
Friday 06
15:00 - SEMINAR - Groups and Combinatorics Seminar, Connections between Graph Theory & Combinatorics on Words More Information

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

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

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

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

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

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

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

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.
Friday 22
11:00 - SEMINAR - Self-avoiding walks—rigorous and non-rigorous results Website | More Information
Self-avoiding walks (SAWs) are widely studied as a problem in algebraic combinatorics by mathematicians, as a problem in algorithm design by computer scientists, as a model of phase transitions by mathematical physicists and as a model of polymers in dilute solution by chemists.

More recently biologists have used them as models of DNA folding, and to model experiments in which biological molecules are pulled from a surface. I will describe the rather short list of rigorous results, the longer list of what we "know" to be true but can't prove, and describe some numerical results that are of interest in applications. No prior knowledge is assumed.

 December 2013
Monday 09
9:00 - CONFERENCE - Australasian Conference on Combinatorial Mathematics and Combinatorial Computing : Held at UWA from the 9th to 13th of December Website | More Information
This year's Australasian Conference on Combinatorial Mathematics and Combinatorial Computing will be held here at UWA from the 9th to 13th of December. Put the date in your diary now and start looking for cheap flights to Perth. Visit the conference webpage to find the exciting lineup of invited speakers that we have lined up so far.

 January 2014
Friday 10
15:00 - SEMINAR - Groups and Combinatorics Seminar, Analysis and Implementation: the Two 'Editions' of a Matrix Group Algorithm More Information
Brian Corr (UWA)

will speak on

Analysis and Implementation: the Two 'Editions' of a Matrix Group Algorithm

at 3pm Friday January the 10th in Blakers Lecture Theatre.

Abstract :

The quality of an algorithm in computational mathematics is represented by two separate, yet equally important measures: the theoretical analyses which measure the runtime's growth for large input, and the implementations whose runtimes we can measure in seconds. This leads to different aspects of algorithm design being prioritised in different settings, and often two very different algorithms are produced: one is described in a journal article analysing the worst-case complexity, and a very different procedure is implemented in practice.

In this talk I present the motivation, overall structure, and details of a reduction algorithm for specific irreducible modules of a classical group G, and discuss issues specific to the implementation of the algorithm in the Magma computer algebra system.

This is joint work with Cheryl Praeger and Akos Seress, with special thanks to Eamonn O'Brien.
Friday 17
15:00 - SEMINAR - Groups and Combinatorics Seminar, Graphs and transitivity on 2-geodesics More Information
Alice Devillers (UWA)

will speak on

Graphs and transitivity on 2-geodesics

at 3pm Friday January the 17th in Blakers Lecture Theatre.

Abstract :

Joint work with Wei Jin, Cai Heng Li, Cheryl Praeger, Akos Seress.

An s-geodesic in a graph is a shortest path connecting two vertices at distance s. We say that a graph is locally transitive on s-geodsics if the stabiliser of any vertex is transitive on the s-geodesics starting at that vertex. Being locally transitive on s-geodesic is not a monotone property: if an automorphism group G of a graph is locally transitive on s-geodesics, it does not follow that G is locally transitive on shorter geodesics. For instance, (local) transitivity on 2-geodesics does not imply local transitivity on arcs (1-geodesics).

In this talk, I will first show a nice characterisation of all graphs that are locally transitive on 2-geodesics, but not locally transitive on 1-geodesics.

Then I will describe graphs that are (locally) transitive on 2-geodesics and on arcs, in terms of their local structure.
Friday 24
15:00 - SEMINAR - CMSC Technical Seminar, An introduction to IPE More Information
Irene Pivotto (UWA)

will speak on

An introduction to IPE

at 3pm Friday January the 24th in Blakers Lecture Theatre.


IPE is a drawing editor for creating figures in PDF or EPS format, which can then be inserted into LaTeX documents. I will show the basic features of IPE, as well as some of the more advanced ones.
Friday 31
15:00 - SEMINAR - Groups and Combinatorics Seminar, Groups and first-order logic More Information
André Nies (University of Auckland)

will speak on

Groups and first-order logic

at 3pm Friday January the 31st in Blakers Lecture Theatre.

Abstract :

We study the expressive power of first-order logic for groups. A finitely generated group is called quasi-finitely axiomatizable if a single sentence characterizes it within the class of finitely generated groups. I showed in 2005 that, for instance, the Heisenberg group is quasi-finitely axiomatizable. Recent work of Lasserre provides new examples, such as the Thompson groups.

A group is homogeneous if the orbit of every tuple under the action of automorphisms is described by its first-order properties. I proved (J. Algebra, 2003) that the free group F_2 has this property. Recent work of Perrin and Sklinos (Duke Math. J. 2013) extends this to F_n for larger n.

 February 2014
Friday 07
15:00 - SEMINAR - Groups and Combinatorics Seminar, Row echelon matrices, flags and Grassmannians More Information
Phill Schultz (UWA)

will speak on

Row echelon matrices, flags and Grassmannians

at 3pm Friday February the 7th in Blakers Lecture Theatre.


There is a well trodden path from finite dimensional vector spaces to algebraic geometry. How much progress along this path is possible if fields are replaced by rings?
Friday 21
15:00 - SEMINAR - Groups and Combinatorics Seminar, Metrically homogeneous graphs More Information
Dugald MacPherson (University of Leeds)

will speak on

Metrically homogeneous graphs

at 3pm Friday February the 21st in Blakers Lecture Theatre.


A `homogeneous structure' is a countably infinite relational structure M (e.g. graph, k-uniform hypergraph, digraph,...) with the property that every isomorphism between finite induced substructures of M extends to an automorphism of M. These are constructed by an amalgamation method developed by Fraisse (and independently Jonsson) in the 1950s. There are classification results, often very hard, in restricted contexts (e.g. graphs, partial orders, digraphs, totally ordered graphs) but in general classification seems out of reach.

I will discuss an attempt to classify `metrically homogeneous graphs', that is, countably infinite graphs M which become homogeneous when enriched by binary `distance relations' corresponding to graph-distance in M. This notion generalises distance transitivity for countably infinite graphs. Cherlin has produced a `catalogue' of metrically homogeneous graph and conjectures that it is complete. In joint work in progress with Amato and Cherlin, we verify the conjecture for metrically homogeneous graphs of diameter at most three.
Friday 28
15:00 - SEMINAR - Groups and Combinatorics Seminar, Subgroup covering numbers of symmetric groups More Information
Eric Swartz (UWA)

will speak on

Subgroup covering numbers of symmetric groups

at 3pm Friday February the 28th in Weatherburn Lecture Theatre.


Let G be a group. The subgroup covering number of G is defined to be the least integer m such that G is equal to the set theoretic union of m proper subgroups of G. In 2005, Maroti determined the subgroup covering number for the symmetric group S_n, when n > 9 is odd, and he provided bounds for sufficiently large even values of n. I will discuss these previous results, joint work with Luise-Charlotte Kappe and Daniela Nikolova towards filling in the gap for small values of n, and ongoing work to determine the exact value for large even values of n.

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