What's On at UWA

Speaker: Saul Freedman (University of Western Australia)

Title: p-groups related to exceptional Chevalley groups

Time and place: 4pm Friday 19 Oct 2018, Weatherburn LT

Abstract: In 1978, Bryant and Kovács proved that if H is a subgroup of the general linear group GL(d,p), with d > 1 and p prime, then there exists a p-group P such that Aut(P) induces H on the Frattini quotient of P. However, it is not known in general when we can choose P to be small, in terms of its exponent-p class, exponent, nilpotency class and order. In this talk, we consider the representation theory of the (finite) simply connected versions of the exceptional Chevalley groups, and their overgroups in corresponding general linear groups, in order to construct small related p-groups. This follows on from the recent work of Bamberg, Glasby, Morgan and Niemeyer, who constructed small p-groups related to maximal subgroups of GL(d,p).

The Monty Hall problem is a probability puzzle based on an American television game show. Some generalisations of the original problem are considered here: - The probability distribution is generalised from equal likelihood to an arbitrary known prior distribution, with the number of doors changed to a general n. - Optimal decision rule among a class of randomised strategies is derived. - The behaviour of the host and its consequences are taken into account. - Variations and further generalisations are considered.

Speaker: Kyle Rosa (University of Western Australia)

Title: Bounds for Semiprimitive Permutation Groups

Time and place: 4pm Friday 2 Nov 2018, Weatherburn LT

Abstract: Semiprimitive groups are permutation groups for which every normal subgroup is either transitive, semiregular, or both. These groups naturally generalise primitive and quasiprimitive groups, and inherit many of their structural properties. I will present the results from my master's thesis, in which I establish bounds on quantities such as order, element degree, and base size.

For abstracts and titles of seminars (upcoming and past), see http://www.maths.uwa.edu.au/~glasby/GroupsAndCombinatoricsSeminar/S18.html

In this talk, we introduce some basics of nonlocal equations, with some applications in mind coming from physics and material sciences. In particular, the equation taken into account comes from a model, developed by Rudolf Peierls and Frank Nabarro, that describes the edge dislocation of atoms in an ideal crystal. Moreover, we present the construction of multibump, heteroclinic, homoclinic and chaotic trajectories, providing a symbolic dynamics in this framework.

Speaker: Marvin Krings (RWTH, Aachen)

Title: The p-part of the order of an almost simple group of Lie type

Time and place: 4pm Friday 23 Nov 2018, Robert Street LT (not Weatherburn LT)

Abstract: Primitive permutation groups are fundamental building blocks in the sense that every finite permutation group can be built from the primitive ones. Apart from the alternating group A_n and the symmetric group S_n of degree n, the primitive subgroups G of S_n are small. For example, in 1980 Praeger and Saxl showed that |G| e 4^n, which is much smaller than n!/2. Since this time, powerful results such as the O’Nan-Scott Theorem, which classifies the primitive permutation groups, and the Classification of the Finite Simple Groups, have become available. We will bound the p-part |G|_p of |G| for some prime p. This is, the largest p-power p^{ u_p(G)} that divides |G|. The bound |G| e 4^n implies nu_p(G) e n og_p(4). We prove the stronger bound nu_p(G) e frac{2sqrt{n}}{(p-1)}+1 (with five exceptions). For several cases, we even obtain a bound that is logarithmic in n. Our proof uses the O'Nan-Scott theorem to reduce to simple groups. The hardest case, and the one I will discuss, is when the simple group is of Lie type.

Speaker: Cai Heng Li (Southern University of Science and Technology, Shenzhen, PRC)

Title: Erdös-Ko-Rado Problem for Permutation Groups

Time and place: 4pm Friday 01 Feb 2019, Weatherburn LT

Abstract: A classical result of Erdös-Ko-Rado in extremal set theory is about intersections of subsets of a set, leading to the so-called Erdös-Ko-Rado problem in various versions. I will explain the problem for permutation group version, and then address a conjecture about the upper-bound for the numbers of intersecting sets.

Speaker: Mariagrazia Bianchi (Università degli Studi di Milano)

Title: Conjugacy class sizes in finite groups: variations on the theme

Time and place: 4pm Tuesday 12 Feb 2019, Weatherburn LT

Abstract: The study of conjugacy class sizes goes back to the beginning of the last century with Burnside's result, Miller's studies on groups with few conjugacy classes and Ito who began the study of the structure of a group in terms of the number of conjugacy classes. Over the last 30 years the subject has become fashionable and many papers have been written on this topic. In this talk I try to summarize some results obtained by using two particular graphs: the common divisor graph and the prime graph in relationship with conjugacy class sizes and finally the variation regarding the so-called vanishing classes, that are those classes of elements g for which there exists an irreducible non linear character chi such that chi(g)=0.

Speaker: Gabriel Verret (University of Auckland)

Title: An update on the Polycirculant Conjecture

Time and place: 4pm Friday 15 Feb 2019, Robert Street LT

Abstract: One version of the Polycirculant Conjecture is that every finite vertex-transitive digraphs admits a non-trivial semiregular automorphism. I will give an overview of the status of this conjecture, as well as describe some recent progress with Michael Giudici.

Particle methods have capabilities that particularly suit numerical simulation of complex phenomena involved in industrial and biophysical application domains. The two core methods used in this talk are DEM (Discrete Element Method) and SPH (Smoothed Particle Hydrodynamics). Coupling of these methods also provides powerful capabilities to model multiphase behaviour. Industrial application to crushing and grinding, mixing and water cooling will be presented. Coupling to biomechanical models allows simulation of humans interacting with their environment. Examples of elite swimming, diving, kayaking and skiing will be shown. The use of these methods to simulate digestion (from breakdown in the mouth through stomach) and intestines will also be discussed.

Speaker: Calin Borceanu (UWA)

Title: Searching for partial congruence partitions in groups of order p^8

Time and Place: 4pm Friday 08 Mar 2019, Weatherburn LT

Abstract: A partial congruence partition (or PCP) of a group is defined simply as a set of pairwise disjoint subgroups which pairwise factorise the whole group. This project has focused on finding examples of PCP that are 'large' in the sense that they are close the best known theoretical bounds on the maximum number of subgroups comprising a PCP. In particular, we focused on the non-elementary abelian groups of order 2^8 and 3^8, where no large examples were previously known. These groups, with one exception, were exhaustively searched for large PCP. In this talk I will discuss some of the theoretical bounds with a focus on deriving results that are useful computationally, before moving on to some details of the computational enumeration.

Speaker: Sabina Pannek

Title: Elements with large irreducible submodules contained in maximal subgroups of the general linear group

Time and place: 4pm Tuesday 12 Mar 2019, Blakers LT

Abstract: We refer to an element of the finite general linear group GL(V) as being fat if it leaves invariant, and acts irreducibly on, a subspace of dimension greater than dim(V)/2. Fat elements generalise the concept of ppd-elements, which are defined by the property of having orders divisible by certain primes called primitive prime divisors. In 1997, Guralnick, Penttila, Praeger and Saxl classified all subgroups of GL(V) containing ppd-elements. Their work has had a wide variety of applications in computational group theory, number theory, permutation group theory, and geometry. Our overall goal is to carry out an analogous classification of all subgroups of GL(V) containing fat elements.

During my PhD candidature I examined the occurrence of fat elements in GL(V) and various of its maximal subgroups. I showed that, often, this problem can be handled in a uniform way by considering "extremely fat" elements and counting certain irreducible polynomials. In my talk, I will present this method for groups belonging to Aschbacher's C2 class. The results we obtain significantly differ from the findings of the ppd-classification.

Speaker: Michael Giudici (University of Western Australia)

Title: Arc-transitive bicirculants

Time and place: 4pm Friday 12 Apr 2019, Weatherburn LT

Abstract: A graph on 2n vertices is a bicirculant if it admits an automorphism that is a permutation with two cycles of length n. For example, the Petersen and Heawood graphs. Arc-transitive bicirculants of valencies three, four and five have previously been classified by various authors. In this talk I will discuss recent joint work with Alice Devillers and Wei Jin that characterises all arc-transitive bicirculants and provides a framework for their complete classification.

Speaker: Martin Liebeck (Imperial College London)

Title: Girth, words and diameters of Cayley graphs

Time and place: 4pm Wednesday 17 Apr 2019, Woolnough LT

Abstract: The girth of a graph is the minimal length of a cycle in the graph. Finding regular graphs with large girth relative to their diameter is the subject of much interest, and a fruitful source of examples has been found in Cayley graphs of various families of finite classical groups. I shall discuss some new results in this area on the girth of Cayley graphs of finite classical groups G on random sets of generators. The main tool is a new bound on the probability that a given word w takes the value 1 when evaluated in G, in terms of the length of w.

Speaker: Emilio Pierro (University of Western Australia)

Title: The 2-transitive permutation representation of the small Ree groups

Time and place: 4pm Friday 03 May 2019, Weatherburn LT

Abstract: Given a group G, the question of which subsets S of G generate G is of natural interest. One approach to this question is determining the Möbius function of G, introduced by Hall in 1936. In order to determine the Möbius function, it is necessary to have a good understanding of the subgroup structure of G, which is of course of interest in its own right. In this talk we discuss how the Möbius function is determined in practice, using the specific case of the small Ree groups.

For upcoming seminars see http://staffhome.ecm.uwa.edu.au/~00059629/GroupsAndCombinatoricsSeminar/S19.html

Speaker: Gordon Royle (University of Western Australia)

Title: From Lehman Matrices To (Im)Perfect Graphs

Time and place: 4pm Friday 10 May 2019, Weatherburn LT

Abstract: A pair (A,B) of square 0/1 matrices is called a Lehman pair if AB^T = J + k I where J is the all-ones matrix, I is the identity matrix and k is a positive integer, and an individual square 0/1 matrix is called a Lehman matrix if it belongs to a Lehman pair. The study of such matrices arose independently in the work of Lehman on problems in operations research, and the work of Bridges and Ryser who viewed them as generalisations of certain combinatorial designs. A number of authors have given methods of constructing Lehman matrices, including several recursive constructions that generate larger Lehman matrices from smaller ones, but always with the same value of k. In joint work, Dillon Mayhew, Irene Pivotto and I discovered a curious construction that transforms certain Lehman matrices with k=1 into “Lehman-like” matrices with k=-1 (and vice versa). Although barely mentioned in the literature on Lehman matrices, solutions to the matrix equation AB^T = J - I are essentially equivalent to a class of graphs known as "partitionable graphs", which were the central object of study in the decades-long effort to prove Berge’s Strong Perfect Graph Conjecture by a direct characterisation of minimal imperfect graphs.

In this talk, I will introduce all the necessary background concepts, and describe how such an innocuous definition leads quite naturally to such disparate areas of combinatorics.

Speaker: Cheryl E Praeger (University of Western Australia)

Title: Limited geodesic transitivity for finite regular graphs

Time and place: 4pm Friday 17 May 2019, Weatherburn LT

Abstract: Joint work with Wei Jin.

For vertex transitive graphs, transitivity on t-arcs, t-geodesics, or distance t vertex pairs, for t leq s, all give symmetry measures of the graph in balls of radius s about a vertex. If the graph has girth g, and s leq g/2, then the sets of t-arcs and t-geodesics are the same for each t leq s, and so the conditions of s-arc transitivity and s-geodesic transitivity are equivalent. The next cases where s= (g+1)/2 and s=(g+2)/2 are interesting. There are s-geodesic transitive examples that are not s-arc transitive. Those which have s=2 and g=3 are collinearity graphs of point-line incidence geometries. However there is no nice general description for the cases where s= 3 and g is 4 or 5. Our approach has required us to classify, as a bye product, all 2-arc transitive strongly regular graphs, and to examine their normal covers. We have lots to describe, as well as open problems to pose.

Speaker: Alex Bors (University of Western Australia)

Title: Words, permutations, and the nonsolvable length of a finite group

Time and place: 4pm Friday 24 May 2019, Weatherburn LT

Abstract: In group theory, the term “word” denotes any concatenation of variables and their formal inverses, such as xxyx^{-1}zy^{-1}. Words are to group theorists what polynomials are to ring theorists: formal expressions into which elements from a concrete structure (a ring resp. group) can be substituted and which can be used to formulate equations over those structures. One type of question commonly studied in this context is the following: Given a word w(X_1,...,X_d) and a number rho in (0, 1], what can one say about finite groups G in which for some g in G, the equation w(X_1,...,X_d) = g has at least rho|G|^d solutions (g_1,...,g_d) in G^d? In this talk, I will discuss recent results of this form which were achieved in collaboration with Aner Shalev from the Hebrew University of Jerusalem.

Abstract: Change is necessary in any dynamic environment, but there is always a cost incurred when implementing change; one of the most obvious is wear and tear on the physical components in a system. In the optimal control field, the cost of change is almost always ignored, and this can lead to “optimal” control strategies that are volatile and impractical to implement. This talk will introduce a class of non-smooth optimal control problems in which the cost of change is incorporated via an objective term that penalizes the total variation of the control signal. We describe a discretization method, based on nonlinear programming and a novel transformation scheme, for converting this class of problems into a sequence of smooth approximate problems, each of which can be solved efficiently. Convergence results for this discretization scheme are discussed. The talk will conclude with examples in fisheries and crane control.

What would a beach look like in a different universe? How would sand flow in 4 or 5 dimensions? At the crossroad of mathematics, physics, and engineering, this talk will develop the numerical and technological tools required to answer those questions. Following the Discrete Element Method, widely used to simulate  particulate materials in physics and engineering, the higher dimensional equations of motion for rigid bodies will be solved. Specific properties of granular packing and flow will be extracted, and compared with well-established results in two and three dimensions. As higher dimensional simulations are challenging to visualise, we will also present complementary visualisation methods based on virtual reality technology, which are crucial to develop a good understanding of the overall behaviour of higher dimensional granular media.

Speaker: Phill Schultz (University of Western Australia)

Title: Actions of Aut(V) on linearly independent subsets of a vector space V

Time and place: 4pm Friday 02 Aug 2019, Weatherburn LT

Abstract: Let V be a finite dimensional rational vector space. The action of the general linear group Aut(V) on the set of bases of V is the main content of First Year Linear Algebra, so you might think there can be nothing new to say about it; but the subject still holds a few surprises.

For example, if G is a subgroup of V, the stabilizer of G in Aut(V) acts on the set of maximal linearly independent subsets of V contained in G. The orbits of this action determine the indecomposable decompositions of G.

Furthermore, the stabilizer of the set of subgroups of V containing G as a subgroup of finite index acts transitively on a larger set of linearly independent subsets, and this action determines a coarse structure theorem for G.