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Today's date is Sunday, September 20, 2020
Centre for the Mathematics of Symmetry and Computation
 February 2019
Tuesday 12
16:00 - SEMINAR - Groups and Combinatorics Seminar: Mariagrazia Bianchi, 4pm Feb 12 More Information
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.
Friday 15
16:00 - SEMINAR - Groups and Combinatorics Seminar: Gabriel Verret, 4pm Feb 15 More Information
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.
Thursday 21
16:00 - SEMINAR - Mathematics and Statistics colloquium : Particle modelling applied to industrial and biophysical problems More Information
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.

 March 2019
Friday 08
16:00 - SEMINAR - Groups and Combinatorics Seminar: Calin Borceanu, 4pm March 08 in Weatherburn LT More Information
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.

Tuesday 12
16:00 - SEMINAR - Groups and Combinatorics Seminar: Sabina Pannek, 4pm March 12 More Information
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.

 April 2019
Friday 12
16:00 - SEMINAR - Groups and Combinatorics Seminar: Michael Giudici, 4pm Apr 12 in Weatherburn LT More Information
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.
Wednesday 17
16:00 - SEMINAR - Groups and Combinatorics Seminar: Martin Liebeck, 4pm Apr 17 in Woolnough LT More Information
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.

 May 2019
Friday 03
16:00 - EVENT - Groups and Combinatorics Seminar: Emilio Pierro, 4pm May 3 in Weatherburn LT More Information
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 https://staffhome.ecm.uwa.edu.au/~00059629/GroupsAndCombinatoricsSeminar/S19.html
Friday 10
16:00 - SEMINAR - Groups and Combinatorics Seminar: Gordon Royle, 4pm May 10 in Weatherburn LT More Information
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.
Friday 17
16:00 - SEMINAR - Groups and Combinatorics Seminar: Cheryl Praeger, 4pm May 17 in Weatherburn LT More Information
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.
Friday 24
16:30 - SEMINAR - Groups and Combinatorics Seminar: Alex Bors, 4pm May 24 in Weatherburn LT More Information
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.

 June 2019
Thursday 20
16:00 - VISITING SPEAKER - Mathematics and Statistics Colloquium : Penalizing control volatility in nonlinear optimal control More Information
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.

 July 2019
Thursday 18
16:00 - SEMINAR - Mathematics and Statistics colloquium : Hyperspheres in hyperspace More Information
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.

 August 2019
Friday 02
16:00 - SEMINAR - Groups and Combinatorics Seminar: Phill Schultz, 4pm Aug 2 in Weatherburn LT More Information
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.
Thursday 22
16:00 - SEMINAR - Mathematics and Statistics colloquium : Convex surfaces of constant Gauss curvature and partially free boundaries More Information
Consider a pair of parallel planes in and a strictly convex closed curve laying on one of the planes. Is there a convex hypersurface of constant Gauss curvature such that it is trapped between the planes, the curve is on its boundary, and the surface strikes the other plane at given constant angle? This problem can be viewed as a generalisation of the Alt-Caffarelli problem for the Gauss curvature case. In this talk we will discuss the existence of weak solutions and the regularity of the free boundary, which is the unknown part of the boundary of surface.
Friday 23
16:00 - SEMINAR - Groups and Combinatorics Seminar: Hassan Alavi, 4pm Aug 23 in Weatherburn LT More Information
Speaker: Hassan Alavi (Buali Sina University, Iran)

Title: On transitive automorphism groups of 2-designs

Time and place: 4pm Friday 23 Aug 2019, Weatherburn LT

Abstract: A 2-design with parameters (v,k,lambda) is an incidence structure consisting of a set of v points and a set of b blocks with the incidence relation such that every block is incident with exactly k points, and every pair of points is incident with exactly lambda; blocks. An automorphism group of a 2-design is a group of permutations on points of the design which maps blocks to blocks and preserves the incidence and non-incidence. The main part of this talk is devoted to giving a survey on recent study of 2-designs admitting a flag-transitive automorphism group. I also present some recent results on block-transitive automorphism groups of 2-designs.
Friday 30
16:00 - SEMINAR - Groups and Combinatorics Seminar: Carlisle King, 4pm Aug 30 in Weatherburn LT More Information
Speaker: Carlisle King (University of Western Australia)

Title: Edge-primitive 3-arc-transitive graphs

Time and place: 4pm Friday 30 Aug 2019, Weatherburn LT

Abstract: Let Gamma be a finite simple graph with G = Aut(Gamma). We say that Gamma is edge-primitive if G acts primitively on the edges of Gamma. An s-arc in Gamma is an ordered path of length s. We say that Gamma is s-arc-transitive if G is transitive on the set of s-arcs of Gamma.

In 1981, Weiss proved that there exists no finite s-arc-transitive graph of valency at least 3 for s geq 8. Since then, there has been considerable effort to characterise s-arc-transitive graphs for s leq 7. One interesting family of graphs is that of edge-primitive graphs. Many famous graphs are edge-primitive, such as the Heawood graph and the Higman-Sims graph. In 2011, Li and Zhang classified finite edge-primitive s-arc-transitive graphs for s geq 4. We study the problem of classifying finite edge-primitive 3-arc-transitive graphs. This is joint work with Michael Giudici.

 September 2019
Friday 13
16:00 - SEMINAR - Groups and Combinatorics Seminar: Dominik Bernhardt, 4pm Sep 13 in Weatherburn LT More Information
Speaker: Dominik Bernhardt (RWTH Aachen)

Title: Automorphism groups of simple graphs with few vertex-orbits

Time and place: 4pm Friday 13 Sep 2019, Weatherburn LT

Abstract:In this talk we will investigate how to construct automorphism groups of graphs with few vertex orbits. The base case is to construct automorphism groups with 2 vertex orbits. We will see how to describe their group theoretic structure and give hints towards an algorithm for constructing these groups and towards generalizations to more orbits.

If time permits, we will see how to count isomorphism classes of graphs with the same automorphism group using the table of marks.
Thursday 19
16:00 - SEMINAR - Mathematics and Statistics colloquium : Mathematics and Suicide More Information
The Young Lives Matter Foundation (https://www.uwa.edu.au/institutes/young-lives-matter/home)  aims to leverage research expertise across UWA to address the leading cause of death among 15- to 24-year olds in Western Australia. The foundation will address this aim by developing improved predictors of risk of self-harm and by better understanding the ways in which individuals interact with a myriad of health services. A deliberate and explicit focus of YLM is to tackle these goals through doing research differently. A key component of this is through new approaches in mathematics. I will provide an overview of some of the pilot work we have conducted over the last 18 months. Through direct observational study at Sir Charles Gairdner Hospital we have developed complex systems models of information transmission and patient flow within the health system. This has allowed us to evaluate system performance and identify key bottlenecks in the delivery of health services. A separate, data-driven, pilot study at Perth Clinic has developed machine learning algorithms which out-perform admission-based psychiatric evaluation for risk of self-harm.  This is joint work with Michael McCullough, Sean Hood, Andrew Page, David Lawrence, Binu Jayawardena, and Geoff Hooke.
Friday 27
16:00 - SEMINAR - Groups and Combinatorics Seminar: Alex Bors 4pm Sep 27 in Weatherburn LT More Information
Speaker: Alexander Bors (University of Western Australia)

Title: Automorphism orbits and element orders in finite groups

Time and place: 4pm Friday 27 Sep 2019, Weatherburn LT

Abstract: Joint with Michael Giudici and Cheryl E. Praeger.

In contrast to other kinds of structures (such as graphs), for groups G, the assumption that the automorphism group Aut(G) acts transitively on G is not interesting to study, as only the trivial group satisfies it. Various weakenings of this condition have been proposed and studied, though. For example, in a paper from 1992, Zhang extensively studied finite groups G with the property that for every element order o in G, the action of Aut(G) on order o elements in G is transitive. He called such finite groups AT-groups. Zhang’s ideas and methods also spurred some interest in the graph-theoretic community, due to a connection with CI-groups (groups G such that any two isomorphic Cayley graphs over G are “naturally isomorphic” via an automorphism of G).

In this talk, we present results on finite groups G that are “close to being AT- groups”, essentially showing that such groups are “almost soluble” (i.e., they have a soluble normal subgroup of bounded index). A finite group G is an AT-group if and only if the numbers of Aut(G)-orbits on G and of distinct element orders in G respectively are equal. Hence we measure the “closeness of G to being an AT-group” by comparing those two numbers, considering both their difference and quotient. Along the way, we obtain a curious quantitative characterisation of the Fischer-Griess Monster group M.

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