What's On at UWA

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.

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.

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...

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.

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.

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.

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.

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.

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.

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.

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?

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, 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.

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 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.

Binzhou Xia (Peking University)

will speak on

Factorizations of almost simple groups with a soluble factor

at 3pm Friday March the 7th in Weatherburn Lecture Theatre.

Abstract:

Factorizations of almost simple groups arise in many contexts. I will talk about the factorizations with one factor soluble, including construction of examples in classical groups and the classification result.

John Bamberg (UWA)

will speak on

The brilliant career of Frédéric Vanhove

at 3pm Friday March the 14th in Weatherburn Lecture Theatre.

Pablo Spiga (Universita' di Milano-Bicocca)

will speak on

Have we ever tried to count Cayley graphs?

at 3pm Friday March the 21st in Weatherburn Lecture Theatre.

Abstract:

In this talk we give some elementary upper bounds on the number of finite Cayley graphs. The asymptotic number of Cayley graphs is much harder to pin down and we give a brief outline of the main technique and the main ingredients needed for this counting. On the way we leave some problems on oriented Cayley graphs and tournaments.

Michael Zieve (University of Michigan)

will speak on

Primitive groups, diophantine equations, and functional equations

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

Abstract:

I will explain how results about primitive groups play a crucial role in proving results about diophantine equations and functional equations. A sample application is that, for any polynomial f(X) with rational coefficients, the function Q-->Q defined by c --> f(c) is (<=6)-to-1 over all but finitely many values.

Cai-Heng Li (UWA)

will speak on

Edge transitive dessins d'enfant

at 3pm Friday April the 4th in Weatherburn Lecture Theatre.

Abstract:

A 2-cell embedding of a bipartite graph in an orientable surface is called a dessin d'enfant. Thus a dessin d'enfant is an orientable bipartite map. I will present an explicit representation of an edge transitive dessin in terms of a group with two distinguished generators, and apply it to study the dessin.

Steven Noble (Brunel University)

will speak on

Graphs are to matroids what ribbon graphs are to ...?

at 3pm Friday April the 11th in Weatherburn Lecture Theatre.

Abstract:

Much of the combinatorial structure of an abstract connected graph is encoded in its set of spanning trees. These form one of the canonical examples of the bases of a matroid. Ribbon graphs contain extra topological information on the embedding of the underlying abstract graph in a surface. The structures playing the role of spanning trees are the subgraphs having one boundary component. The combinatorial structure which they form is a delta-matroid, which roughly speaking is like a matroid except that the bases do not need to have the same size. We will discuss three classes of delta-matroids, some natural operations on ribbon graphs which correspond to natural operations on delta-matroids, give an excluded minor theorem for one of the classes of delta-matroids and, time-permitting describe the Bollobas-Riordan polynomial which forms one way of generalizing the Tutte polynomial to ribbon graphs.

Michael Giudici (UWA)

will speak on

Some recent results on elusive groups

at 3pm Friday May the 2nd in Weatherburn Lecture Theatre.

Abstract:

Every transitive permutation group has a derangement of prime power order but not necessarily a derangement of prime order. A transitive permutation group is called elusive if it has no derangements of prime order. I will talk about some recent results on this topic including elusive groups of automorphisms of graphs of small valency and a new construction of elusive groups.