September 2013

Friday 06 
15:00  SEMINAR  Groups and Combinatorics Seminar, Connections between Graph Theory & Combinatorics on Words

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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  socalled "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 Noncommuting Graph

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Abstract:
In this talk, we will consider the noncommuting graph of a nonabelian finite group G; its vertex set is the set of noncentral 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 noncommuting graph such as connectivity, regularity, etc., and we show that, for many groups G, if H is a group which has the same noncommuting graph of G, then they have the same order. We determine the structure of any finite nonabelian group G (up to isomorphism) for which its noncommuting graph is a complete multipartite graph. We also show that a noncommuting 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

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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 CohenLenstra heuristics: from arithmetic to topology and back again

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Akshay Venkatesh (Stanford) Mahler Lecturer and IAS ProfessoratLarge
will speak on
The CohenLenstra 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 CohenLenstra 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, Straightline programs with memory and applications to computational group theory

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Abstract:
Straightline 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 straightline 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

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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 ktiling. We will first give a historical survey that includes the investigations of classical tilings by translations (which we call 1tilings 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 1tile Euclidean space.
Today we know that ktilings 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 ktile is much wider than the collection of objects that 1tile, and there is currently no complete knowledge of the polytopes that ktile, even in 2 dimensions. We will cover both ``ancient'', as well as very recent, results concerning 1tilings and more generally ktilings. 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

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Abstract:
Given a finite group G and a faithful irreducible FGmodule 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

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Abstract:
Let H be a finite linear group acting completely reducibly on a finite vector space V. Gabriel Navarro asked: if the Horbits containing vectors a and b have coprime lengths m and n, is there an Horbit of length mn?
We answered, by showing that the Horbit 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

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Abstract:
Codes arising from algebraic geometry, first introduced by Goppa, gained attention when Tsfasman–Vladut–Zink used them to improve the GilbertVarshamow 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

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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 
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 socalled 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 nonergodic 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 Kortewegde 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 
Selfavoiding 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

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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

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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 worstcase 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 2geodesics

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Alice Devillers (UWA)
will speak on
Graphs and transitivity on 2geodesics
at 3pm Friday January the 17th in Blakers Lecture Theatre.
Abstract :
Joint work with Wei Jin, Cai Heng Li, Cheryl Praeger, Akos Seress.
An sgeodesic in a graph is a shortest path connecting two vertices at distance s. We say that a graph is locally transitive on sgeodsics if the stabiliser of any vertex is transitive on the sgeodesics starting at that vertex. Being locally transitive on sgeodesic is not a monotone property: if an automorphism group G of a graph is locally transitive on sgeodesics, it does not follow that G is locally transitive on shorter geodesics. For instance, (local) transitivity on 2geodesics does not imply local
transitivity on arcs (1geodesics).
In this talk, I will first show a nice characterisation of all graphs that are locally transitive on 2geodesics, but not locally transitive on 1geodesics.
Then I will describe graphs that are (locally) transitive on 2geodesics and on arcs, in terms of their local structure.

Friday 24 
15:00  SEMINAR  CMSC Technical Seminar, An introduction to IPE

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Irene Pivotto (UWA)
will speak on
An introduction to IPE
at 3pm Friday January the 24th in Blakers Lecture Theatre.
Abstract:
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 firstorder logic

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André Nies (University of Auckland)
will speak on
Groups and firstorder logic
at 3pm Friday January the 31st in Blakers Lecture Theatre.
Abstract :
We study the expressive power of firstorder logic for groups. A finitely generated group is called quasifinitely 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 quasifinitely 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 firstorder 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

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Phill Schultz (UWA)
will speak on
Row echelon matrices, flags and Grassmannians
at 3pm Friday February the 7th in Blakers Lecture Theatre.
Abstract:
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

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Dugald MacPherson (University of Leeds)
will speak on
Metrically homogeneous graphs
at 3pm Friday February the 21st in Blakers Lecture Theatre.
Abstract:
A `homogeneous structure' is a countably infinite relational structure M (e.g. graph, kuniform 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 graphdistance 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

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Eric Swartz (UWA)
will speak on
Subgroup covering numbers of symmetric groups
at 3pm Friday February the 28th in Weatherburn Lecture Theatre.
Abstract:
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 LuiseCharlotte 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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