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Mathematics colloquia series

Engaging with universities and industry bodies around the world, the Department of Mathematics and Statistics presents a series of regular colloquiums and seminars during the year. The programs enhance the learning environment, linking students and staff with academic and industry experts.

For information on the mathematics colloquium series at Bundoora, please contact Dr Jane Pitkethly on 03 9479 3958 or by sending an email to J.Pitkethly@latrobe.edu.au.

Upcoming programs

Title: Hopf algebras and solutions to the Yang–Baxter equation

• Speaker: Dr Karen Dancer (University of Queensland).
• Time & date: 11:00am, Tuesday 3 November.
• Venue: Access Grid Room, 310 Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract:

  A Hopf algebra is a structure that is simultaneously an algebra and a co-algebra. In the 1980's V.G. Drinfeld developed a double construction, which embeds any Hopf algebra in a larger Hopf algebra that is inherently quasi-triangular. A consequence of this is that the double algebra provides a solution to the Yang–Baxter equation. In this talk I will introduce Hopf algebras and outline the Drinfeld double construction, focusing on the example of finite group algebras. I will briefly discuss their representation theory before using them to construct solutions to the Yang–Baxter equation.

Title: Computing in infinite groups

• Speaker: Dr Murray Elder (University of Queensland).
• Time & date: 1:00pm, Tuesday 3 November.
• Venue: Access Grid Room, 310 Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract:

  I will define the notion of a growth function for (infinite) groups, and discuss my work on computing these functions for different groups. I will also mention some variations, such as geodesic growth and cogrowth functions, and what these functions can tell us about groups.

Past programs

Title: The Banach-Tarski Paradox

• Speaker: Dr Ian Hawthorn (University of Waikato).
• Time & date: 2pm Friday 23 October.
• Venue: Access Grid Room, 310 Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract:

  In a paper published in 1924, S. Banach and A. Tarski proved a truly remarkable theorem, namely that it is possible to disassemble a 3-D ball like a jigsaw puzzle into a small finite number of pieces and to reassemble these using only rigid motion to make two balls each the same size as the original. This result is known as the Banach-Tarski paradox.

  Banach and Tarski suggested that their paradox calls into question the appropriateness of using the axiom of choice which is required at one point in the proof. However the modern perspective is that the root of the paradox lies not in the axiom of choice, but in the algebraic properties of the rotation group, and that the use of the axiom of choice is only incidental.

  The Banach-Tarski paradox is one of those beautiful pieces of mathematics which deserves to be taken out and treasured regularly. This talk which is aimed at a general audience aims to present a proof of this remarkable result at a level comprehensible to an undergraduate student.

Title: Dynamics in some simple differential delay models

• Speaker: Anatoli F. Ivanov (Pennsylvania State University and the University of Ballarat).
• Time & date: 2pm Friday 16 October.
• Venue: Access Grid Room, 310 Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract:

  We discuss the problem of global dynamics in some differential delay equations of simple form which serve as mathematical models of several real life processes. One of the equations of interest reads as

  x’(t) = F(x(t - \tau)) - G(x(t)),

  where  F  and  G  are continuous real valued functions and  \tau > 0  is the delay. Among others, this equation has applications in physiology and economics which we briefly reflect on.

  A larger part of the presentation is a general overview of known results on basics of differential delay equations and their specific features as compared with corresponding ordinary differential equations. This part is intended for a general mathematical audience including students. The second smaller part of the talk discusses more specific and recent results including those of my own research.

Title: Integrals of open 2D lattices

• Speaker: Dmitry Demskoi.
• Time & date: 2pm Friday 25 September.
• Venue: Access Grid Room, 310 Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: Open (finite, non-periodic) 2D Toda lattices are the most well known representatives of the class of exactly solvable hyperbolic systems. These lattices have long been known to possess the complete sets of integrals, however, the explicit formulas for them have never been presented apart from a few particular cases. I will discuss a solution of this problem for A_n-type lattices and their reductions which include in particular B_n and C_n lattices. I will also consider a few other lattices that are shown to have integrals and explicit solutions.

Title: Some exact solutions of the fin problem with a power law temperature-dependent thermal conductivity

• Speaker: Joel Moitsheki (University of the Witwatersrand).
• Time & date: 1pm Monday 21 September 2009.
• Venue: Access Grid Room, 310 Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: This study investigates the exact solutions of nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient. Both the conduction and the heat transfer terms are given by the same power law in one case and the distinct power law in the other. Classical Lie symmetry techniques are employed to construct the exact solutions which satisfy the realistic boundary conditions. The effects of the physical applicable parameters such as thermo-geometric fin parameter and the fin efficiency are analyzed.

Title: A survey of Tutte-Whitney polynomials

• Speaker: Graham Farr (Monash University).
• Time & date: 2pm Friday 18 September 2009.
• Venue: Access Grid Room, 310 Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: The Tutte-Whitney polynomial of a graph is a two-variable polynomial that contains a lot of interesting information about the graph. It includes, for example, the chromatic, flow and reliability polynomials of a graph, the Ising and Potts model partition functions of statistical mechanics, the weight enumerator of a linear code, and the Jones polynomial of an alternating link.
  This talk is a survey of this polynomial, including a generalisation to arbitrary codes and arbitrary real-valued functions on the power set of a set. We also describe an extension that includes the partition function of the Ashkin-Teller model, one of the oldest models in statistical mechanics.

Title: Moving frames, calculus of variations and Noether's theorem

• Speaker: Prof. Elizabeth Mansfield (University of Kent).
• Time & date: 2pm Friday 11 September 2009.
• Venue: Access Grid Room, 310 Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: "Moving frames" or "repere mobile" are associated with the name of Elie Cartan, although the ideas are older, and were used to solve a variety of equivalence problems in differential geometry. Recent reformulations, in particular the seminal work of Fels and Olver, have freed the ideas to allow for a wider variety of applications, for example in computer vision and Lie group invariant numerical schemes, and also to be able to compute effectively with differential invariants and their differential relations in symbolic computation.
  In this talk, I will give an overview of moving frames as they are now conceived. The application discussed will be to the Calculus of Variations, specifically, the derivation and structure of Euler Lagrange equations arising from variational problems having a Lie group symmetry. I will also discuss the interplay between the moving frame and the conservation laws that arise via Noether's theorem.

Title: Faces of the scl norm ball (scl = stable commutator length)

• Speaker: Prof. Danny Calegari (California Institute of Technology).
• Time & date: 2pm Friday 14 August 2009.
• Venue: Hooper Lecture Theatre, La Trobe University, Bundoora Campus.
• Abstract: It often happens that a solution of an extremal problem in geometry has more regularity and nicer features than one has an a priori right to expect. I will show how a simple topological problem—when does an immersed curve on a surface bound an immersed subsurface?—is unexpectedly related to linear programming, geometric and dynamical rigidity, and symplectic representations.
• This talk is part of the Clay–Mahler Lecture Tour 2009.

Title: Poles, strings, braids and lattices

• Speaker: Prof. Arun Ram (University of Melbourne).
• Time & date: 2pm Friday 1 May 2009.
• Venue: Seminar Room 1, Room 212, Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: The double affine braid group has important applications to Macdonald polynomials, group representations, mathematical physics and combinatorics. The classical type double affine braid groups have nice pictorial presentations which exhibit the tantalizing symmetries at play. In this talk I'll draw some of these pictures and explain their role in topology, harmonic analysis, combinatorics and the study of symmetry.

Title: An introduction to arc-transitive graphs

• Speaker: Sanming Zhou (University of Melbourne).
• Time & date: 12pm Monday 20 April 2009.
• Venue: Access Grid Room, 310 Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: Exploring symmetry of geometric objects has always been one of the most fascinating aspects of mathematics. In the theory of graphs, since Tutte's seminal paper on highly arc-transitive cubic graphs published in 1947, there has been extensive research on symmetries of graphs, which are measured in terms of the automorphism group of the graph. In the case where the automorphism group is transitive on the set of arcs, the graph is said to be arc-transitive, where an arc is an ordered pair of adjacent vertices. This talk will be a gentle introduction to arc-transitive graphs. If time allows, I will explain some recent results on imprimitive arc-transitive graphs.

Title: Forecasting the incidence of cancer in regional Victoria

• Speaker: Prof. Terry Mills (Loddon Mallee Integrated Cancer Service).
• Time & date: 2pm Friday 3 April 2009.
• Venue: Access Grid Room, 310 Physical Sciences 2, La Trobe University, Bundoora Campus.
• Abstract: Loddon Mallee Integrated Cancer Service is responsible for planning the delivery of cancer services in the Loddon Mallee Region of Victoria. A key part of strategic planning for these services is forecasting the incidence of cancer in the region, and this leads to some interesting mathematical problems.
  The presentation is based on joint work with T. Barber, N. Brown, R. Hamilton-Keene, and P. Hartney (all at LMICS).

Title: Abelian groups: their role in the pathological behaviour of braids and the downfall of democracy!

• Speaker: Prof. Brian Davey.
• Time & date: 2pm Friday 27 March 2009.
• Venue: Hooper Lecture Theatre, La Trobe University, Bundoora.
• Abstract: A famous theorem in social choice theory, due to Kenneth Arrow, says that there is no "reasonable" system for combining preference orders. For example, in an election we may want a system for taking the voters' various rankings of the candidates in order of preference and obtaining a single ranking of the candidates.
  I will discuss an unexpected connection between Arrow's Theorem and my research into braids (a kind of partially ordered set). Indeed, my study of braids led to a theorem about finite abelian groups that can be applied to simplify the proof of Arrow's Theorem; a surprising interconnection between three apparently quite different parts of mathematics.

Title: Puzzle-based learning

• Speaker: Zbigniew Michalewicz, University of Adelaide.
• Time & date: 2pm Friday 12 December 2008.

Title: The convergence of binomial trees for pricing European and American options

• Speaker: Mark Joshi, University of Melbourne.
• Time & date:2pm Friday 28 November 2008.

Title: Primal algebras, electronic circuits and evolution

• Speaker: David Clark, SUNY New Paltz, USA.
• Time & date: 2pm Friday 7 November 2008.

Title: Arithmetic on elliptic curves

• Speaker: Daniel Delbourgo, Monash University.
• Time & date: 2pm Friday 17 October 2008.