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 Melbourne, please contact Dr Yury Nikolayevsky on 03 9479 1644 or by sending an email to Y.Nikolayevsky@latrobe.edu.au.
Future programs
Snippets from Soil-Water Infiltration: Where Approximate Becomes Exact
Joint La Trobe University and RMIT University Colloquium
Water flow in unsaturated soil is well described by a nonlinear diffusion-convection equation whose transport coefficients increase strongly with water content. Basic understanding of the dynamics has come mostly from analysis of the Dirichlet problem with constant-concentration boundary conditions; in the 1950s J.R. Philip showed by symmetry arguments that the early-time cumulative infiltration is proportional to 
, the first term of a 
power series. Since the 1980s, several research groups have used an integrable one-dimensional version of Richards’ equation, with realistic nonlinear transport coefficients, to predict experimentally verifiable quantities. Neat expressions emerge for time to incipient surface ponding under constant-rainfall (Robin-type) boundary conditions, for the dependence of sorptivity (leading-order infiltration coefficient) on pond depth and for the second and higher infiltration coefficients. These exact results are at odds with those of the traditional Green-Ampt model, whose shortcomings have been exposed after 101 years.
The integrable model under Dirichlet boundary conditions transforms to a perturbed classical Stefan problem that can be solved by a formal series after separation of variables. This leads to general questions about when such a construction is possible. The best known approaches to separation are the canonical coordinates of a symmetry, and the Staeckel matrix construction.
Speaker
Prof Phil Broadbridge (La Trobe University).
Time and Date
3:30pm, Friday 1 June, 2012.
Venue
Building 8, Level 9, Room 67, RMIT University, Melbourne City Campus.
Past programs
Residuated Lattices for the Working Mathematician
A residuated lattice is an algebra which combines a lattice and a monoid by means of residuation operations (left and right divisions). For example, the set of two-sided ideals in any ring forms a residuated lattice (this lattice was studied by Birkhoff in 1934, and I believe it was the very first residuated lattice to be discovered). In 1930s residuated lattices were investigated in a series of papers by Ward and Dilworth, but then not much happened until about 20 years ago, when residuated lattices were rediscovered by two very different groups of researchers:
- algebraists (as a generalisation of lattice-ordered groups) and
- logicians (as algebraic semantics for proof systems known as Gentzen calculi).
The theory of residuated lattices is to a large extent a result of interactions between these two groups. I will present some of that theory, trying to meet two largely incompatible conditions: (a) to be non-technical, (b) to focus on open problems.
Speaker
A/Prof Tomasz Kowalski (La Trobe University).
Time and Date
2pm, Friday 27 April, 2012.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
A tale of two G2
Exceptional Lie group G2 is a beautiful 14-dimensional continuous group, having relations with such diverse notions as triality, 7-dimensional cross product and exceptional holonomy. It was found abstractly by Killing in 1887 (complex case) and then realized as a symmetry group by Engel and Cartan in 1894 (real split case). Later in 1910 Cartan returned to the topic and realized split G2 as the maximal finite-dimensional symmetry algebra of a rank 2 distribution in R5. In other words, Cartan classified all symmetry groups of Monge equations of the form y’=f(x,y,z,z’,z’’). I will discuss the higher-dimensional generalization of this fact, based on the joint work with Ian Anderson. Compact real form of G2 was realized by Cartan as the automorphism group of octonions in 1914. In the talk I will also explain how to realize this G2 as the maximal symmetry group of a geometric object.
Speaker
Prof Boris Kruglikov (Tromsø University, Norway).
Time and Date
2pm, Friday 20 April, 2012.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Splittings, commutators and the linear Schrödinger equation
In this talk I report recent work, joint with Karolina Kropielnicka, on the solution of the linear Schrödinger equation by exponential splitting in a manner that separates different scales of frequency. I demonstrate that, contrary to expectations, once the equation is discretized appropriately, the size of iterated commutators becomes very small: appropriate discretizations include finite difference and spectral collocation methods, but not the standard spectral method usually employed in this context. This rapid decay in the size of commutators allows us to use high-order symmetric Zassenhaus splittings which separate frequencies very well and allow large time steps. The talk will bring together themes in PDEs, numerical analysis, free Lie algebras and even some approximation theory.
Speaker
Prof Arieh Iserles (University of Cambridge, UK).
Time and Date
2pm, Friday 30 March, 2012.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Metrical musings on Littlewood and friends
The celebrated Littlewood conjecture in Diophantine approximation concerns the simultaneous approximation of two real numbers by rationals with the same denominator. A cousin of this conjecture is the mixed Littlewood conjecture of de Mathan and Teulié, which is concerned with the approximation of a single real number, but where some denominators are preferred to others.
In the talk, we will derive a metrical result extending work of Pollington and Velani on the Littlewood conjecture. Our result implies the existence of an abundance of numbers satisfying both conjectures.
Speaker
Prof Simon Kristensen (Aarhus University, Denmark).
Time and Date
2pm, Friday 23 March, 2012.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Dispersive Quantization – the Talbot Effect
The evolution, through linear dispersion, of piecewise constant periodic initial data leads to surprising quantized structures at rational times, and fractal, non-differentiable profiles at irrational times. Similar phenomena have been observed in optics and quantum mechanics, where it is known as the Talbot effect after an optical experiment by one of the founders of photography, and lead to intriguing connections with exponential sums arising in number theory. Ramifications of these observations for linear and nonlinear wave models and numerics will be discussed.
Speaker
Prof Peter J Olver (University of Minnesota, USA).
Time and Date
2pm, Tuesday 13 March 2012.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Exploring Spatial Data in R
A non-mathematical talk on visualizing and analysing spatial data in R. I'll discuss the different types of spatial data, the main R packages needed for the analysis of those types, and present a selection of examples from a wide range of application areas. I'll briefly illustrate how R can be used to visualize data in other software such as Google Earth and Quantum GIS. We'll look at geological data, rainforests, cancer cases, biological cells, and maps of our favourite Australian state.
Speaker
Dr Alec Stephenson, Swinburne University of Technology.
Time and Date
11am, Friday 4th November 2011.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Crossings and colourings: beyond the 5 colour theorem
A graph is planar, if it can be drawn in the plane without edge crossings. It follows from the Euler formula, that any planar graph has a vertex of degree at most 5. Therefore, using a greedy colouring algorithm, we can trivially colour the vertices of a planar graph by 6 colours. It is still easy to prove that 5 colours suffice. What happens if we allow some edge crossings in the drawing? Can we still 5 colour the vertices? Is there any connection between the minimum number of crossings in a drawing of a graph G and its chromatic number? Both of these graph parameters are hard to compute. Still, Mike Albertson formulated a conjecture that for any r-chromatic G, the crossing number of G is at least the crossing number of K_r, the complete graph on r vertices. This conjecture is a weakening of Hajos conjecture and it includes the 4 colour theorem as a subcase. We will show that this conjecture follows from some known results up to a factor of 4. It is also true for r at most 16, despite the fact that we only know the crossing number of K_r for r at most 12.
This is joint work with Geza Toth (Renyi Institute, Budapest, Hungary).
Speaker
Dr János Barát, Monash University.
Time and Date
2:30pm, Friday 21 October, 2011.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Geodesically equivalent metrics: on the crossroad of differential geometry, integrable systems and mathematical physics
Can two different metrics have the same geodesics? Yes! The first examples were constructed already by Lagrange, and different versions of the question were actively studied by virtually all differential geometers 100 years ago. During my talk I will explain the solution of the Lie Problem which is the infinitesimal version of the question above; this is a joint result with R. Bryant and G. Manno), of the Beltrami Problem (which is presicely the question above, my contribution is to solve it on closed manifolds), and of the Lichnerowicz-Obata conjecture (which suggests an answer to Schouten problem). There are three main tools of the proof: integrable systems, geometric theory of partial differential equations and singularity theory.
Speaker
Prof Vladimir Matveev, University of Jena, Germany.
Time and Date
1:00pm, Monday 26 September, 2011.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
How to reconstruct a metric by its unparameterised geodesics
We discuss whether it is possible to reconstruct a metric by its unparameterized geodesics, and how to do it effectively. We explain why this problem is interesting for general relativity. We show how to understand whether all curves from a sufficiently big family are unparameterized geodesics of a certain affine connection, and how to reconstruct algorithmically a generic 4-dimensional metric by its unparameterised geodesics. The algorithm works most effectively if the metric is Ricci-flat. We also prove that almost every metric does not allow nontrivial geodesic equivalence, and construct all pairs of 4-dimensional geodesically equivalent metrics of Lorenz signature. If the time allows, I will also explain how this theory helped to solve two problems explicitly formulated by Sophus Lie in 1882, and the semi-Riemannian two-dimensional version of the projective Lichnerowicz-Obata conjecture. The new results of the talk are based on the papers arXiv:1010.4699, arXiv:1002.3934, arXiv:0806.3169, arXiv:0802.2344 and arXiv:0705.3592; joint with Bryant, Bolsinov, Kiosak, Manno and Pucacco.
Speaker
Prof Vladimir Matveev, University of Jena, Germany.
Time and Date
2:00pm, Monday 26 September, 2011.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Geometric approach to partial differential equation
Speaker
Naghmana Tehseen, La Trobe University.
Time and Date
3:00pm, Friday 16 September, 2011.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
The geometric study of differential equations aims to describe the local and global structure of the solutions as integral submanifolds of an equation manifold.
The study of partial differential equations (PDEs) leads to the concept of Vessiot distributions, which provides the convenient formal framework to investigate PDEs on an appropriate jet bundle. In this talk I will give a brief exposition of Vessiot distributions of PDEs and then go on to describe some recent work.
The Sprague-Grundy function for the real game Euclid
Speaker
Bao Ho, La Trobe University.
Time and Date
2:30pm, Friday 16 September, 2011.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
The game Euclid, introduced and named by Cole and Davie, is played with a pair of nonnegative integers. The two players move alternatively, each subtracting a positive integer multiple of one of the integers from the other integer without making the result negative. The player who reduces one of the integers to zero wins. The name Euclid has also been used for a subtle variation of this game due to Grossman in which the game stops when the two entries are equal. For that game, Straffin showed that the losing positions (a, b) with a < b are precisely the same as those for Cole and Davie's game. Nevertheless, the Sprague-Grundy functions are not the same for the two games. We give an explicit formula for the Sprague-Grundy function for the original game of Euclid and we explain how the Sprague-Grundy functions of the two games are related.
Totally geodesic subalgebras of 
-graded filiform Lie algebras
Speaker
Ana Hinić-Galić, La Trobe University.
Time and Date
2:00pm, Friday 16 September, 2011.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
A metric Lie algebra 
is a Lie algebra equipped with an inner product. A subalgebra 
of a metric Lie algebra 
is said to be totally geodesic if a Lie subgroup corresponding to 
is a totally geodesic submanifold relative to the left-invariant Riemannian metric on the simply connected Lie group associated to 
defined by the inner product.
-graded metric filiform Lie algebras (the question is motivated by an earlier result of Kerr and Payne).
Dirichlet analogues of q-series and some tiling patterns from quasicrystals
Speaker
Dr Geoffrey Campbell, La Trobe University.
Time and Date
2:00pm, Friday 2 September, 2011.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
In writing new analogues for classical q series connected with Dirichlet series generating functions, the issue of natural interpretation of coefficients arose. Happily, many such interpretations have recently arisen in enumerative tilings that occur in the theory of quasicrystals. To be able to explain the concepts in a simple fashion, we shall mainly use two examples, discussed previously by Baake and Grimm, namely the triangular lattice, written as
, 
,
the set (ring) of Eisenstein integers, and the vertex set of the twelvefold symmetric shield tiling, see Fig. 1. We concentrate on explicit results for these examples, and refer to original sources for a more general exposition and for details.
Chaos, quantum mechanics and number theory
Mahler Lecture 2011
Speaker
Prof Peter Sarnak, Princeton University.
Time and Date
1:00pm, Tuesday 16 August, 2011.
Venue
Szental Lecture Theatre, La Trobe University, Melbourne Campus.
Abstract
The correspondence principle in quantum mechanics is concerned with the relation between a mechanical system and its quantization. When the mechanical system are relatively orderly ("integrable"), then this relation is well understood. However when the system is chaotic much less is understood. The key features already appear and are well illustrated in the simplest systems which we will review. For chaotic systems defined number-theoretically, much more is understood and the basic problems are connected with central questions in number theory.
Class groups of cyclotomic fields
Speaker
Prof Alexander Stolin, University of Göteborg, Sweden.
Time and Date
2:00pm, Friday 12 August, 2011.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
Class groups of cyclotomic fields.
Introduction to Yang-Baxter equation and quantum groups
Speaker
Prof Alexander Stolin, University of Göteborg, Sweden.
Time and Date
2:30pm, Friday 5 August, 2011.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
In my talk, I will introduce classical and quantum Yang-Baxter equations and the algebraic tools to solve them – Lie bialgebras and Hopf algebras. No preliminary knowledge of Lie theory is needed.
The Topology of Compact Groups
Speaker
Prof Sid Morris, Adjunct Professor, La Trobe University; Emeritus Professor, University of Ballarat.
Time and Date
2:00pm, Friday 29 July, 2011.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
This talk is aimed at mathematicians who are not experts in topological group theory but have at least a passing acquaintance with topology. Compactness is the most important topological property - it is regarded as the topological version of finiteness. Topological groups are both groups and topological spaces where the group operations of multiplication and inversion are continuous. Compact groups are a very important class of topological groups. The aim of this talk is to describe what compact groups look like as topological spaces.
Special solutions to ultradiscrete Painlevé equations
Speaker
Dr Christopher Ormerod, La Trobe University.
Time and Date
2:00pm, Friday 10 June, 2011.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
The ultradiscrete Painlevé equations are discrete dynamical systems over the max-plus semifield. We will consider special solutions of these systems. Many traditional methods used to analyze these systems cannot be applied in the setting of a semifield.
One way of overcoming these difficulties is to treat these equations as the image of a discrete dynamical system over a non-archimedean valuation field, such as the p-adics or various functions fields. As an application of this method, we are able to produce the first known examples of hypergeometric solutions of the ultradiscrete Painlevé equations.
Variational Principles for Discrete Integrable Systems
Speaker
Dr Sarah Lobb, La Trobe University.
Time and Date
2:00pm, Friday 20th May 2011.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
The conventional point of view is that the Lagrangian is a scalar object, which through the Euler-Lagrange equations provides us with one single equation. However, there is a key integrability property of certain discrete systems called multidimensional consistency, which implies that we are dealing with infinite hierarchies of compatible equations rather than just one single equation. Wanting this property to be reflected in the Lagrangian formulation, we arrive naturally at the construction of Lagrangian multiforms, i.e., Lagrangians which are the components of a form and satisfy a closure relation. Then we can propose a new variational principle for discrete integrable systems which brings in the geometry of the space of independent variables, and from this principle we can derive any equation in the hierarchy.
Many important examples fit into this framework; a nice example is the discrete potential Korteweg-de Vries equation.
The Five-fold Constellation and a Cryptographic Problem
Joint RMIT University and La Trobe University seminar
Speaker
Professor Kathy Horadam, RMIT.
Time and Date
3:30pm, Friday 13th May 2011.
Venue
Szental Lecture Theatre, La Trobe University, Melbourne Campus.
Abstract
Over the past 20 years de Launey, Flannery, Galati, Hughes, Perera, Horadam, Rao and others have investigated correspondences between generalised Hadamard matrices, group cohomology, relative difference sets, divisible designs and uncorrelated sequences. These have revealed a deep network of corresponding objects in combinatorics, algebra and information transmission I call the Five-fold Constellation. Different ideas of equivalence of these objects, arising naturally in each area, can propagate around the Constellation. I will outline these correspondences and explain how equivalence for relative difference sets assists in the search for good cryptographic functions.
A (gentle) introduction to Lie superalgebras.
Speaker
Dr David Ridout, ANU.
Time and Date
2:00pm, Friday 29 April 2011.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
Lie superalgebras are a mild generalisation of Lie algebras that are extremely important in mathematical physics. They arise when considering (super)symmetries which mix bosonic and fermionic objects. In this talk, we will first introduce the notion of a Lie algebra through two nice examples known as gl(2) and sl(3) before discussing their super-analogues, gl(1|1) and sl(2|1). Special attention will be paid to those features of Lie algebras which go horribly wrong for superalgebras. Such features are responsible for the fact that research on simple Lie superalgebras is still very active, whereas the classification of simple Lie algebras was completed in 1894.
Generalizing Integer Partition Theory to a Vector Partition Theory
Speaker
Dr Geoff Campbell, La Trobe University.
Time and Date
2:30pm, Friday 11 March 2011.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
In 2008 our La Trobe "Q Society Meeting" focused on the Theory of Integer Partitions. We now consider some assertions about vector partitions arising from the monograph newly submitted by the speaker. In particular, partitions over the set of visible points are considered, and it turns out that this may link to the Riemann Hypothesis, since the distribution of the visible points in the Cartesian plane is the reciprocal of the Riemann zeta function. We examine accessible and new ideas that arise with vector partitions; and display calculations for some of these, implying a generalization of the Euler Pentagonal number theorem, from integer partitions, to vector partitions.
Completeness in supergravity constructions
Speaker
Prof. Vicente Cortes, Hamburg University.
Time and Date
2:00pm, Friday 25 February, 2010.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
We prove that the supergravity r- and c-maps preserve completeness. As a consequence, any component H of a hypersurface {h=1} defined by a homogeneous cubic polynomial such that -d^2 h is a complete Riemannian metric on H defines a complete projective special Kahler manifold and any complete projective special Kahler manifold defines a complete quaternionic Kahler manifold of negative scalar curvature. We classify all complete quaternionic Kahler manifolds of dimension less or equal to 12 which are obtained in this way and describe some complete examples in 16 dimensions.
Modular invariant theory
Speaker
Wu Xinyuan
Time and Date
11am, Thursday 9 December 2010.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
This talk introduces extended symplectic Runge-Kutta-Nyström (ESRKN) integrators
for a system of oscillatory second-order differential equations.
LaTeX circa 2010: Teaching an old dog new tricks
Speaker
Mark Hickman.
Time and Date
2pm, Friday 3 December 2010.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
Over the past 20 years TeX, and in particular, LaTeX has become the lingua franca for communication in mathematics and mathematical based disciplines. Originally conceived for typesetting journal articles, LaTeX has been successfully extended to the world of electronic communication. In this talk I will discuss some of the more recent "extensions" of LaTeX to producing high quality graphics, database management and multi-screen presentations.
Convolution Surfaces Generated by Basic 1D and 2D Skeletons
Speaker
Evelyne Hubert.
Time and Date
2pm, Friday 26 November 2010.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
In this talk we present general closed form formulae for the convolution surfaces around sets of polygonal lines and planar polygons.
Convolution is a technique used in computer graphics to generate smooth 3D volumes around a skeleton of lower dimension. One-dimensional skeletons create tubular like volumes which are well suited for modeling organic shapes. For general shapes one needs to consider 2D skeletons as well.
Convolution surfaces are defined as level set of a function obtained by integrating a kernel function along this skeleton. To allow for interactive modeling, the technique has relied on closed form formulae for integration obtained through symbolic computation software.
We consider families of kernels indexed by an integer that controls either the smoothness or the sharpness of the shape created. Generality is achieved by exhibiting the recurrence relationship for the convolution functions generated by line segments. The convolution functions for polygons are then expressed in terms of the convolution functions generated by the bounding polygonal line by application of Green's theorem. This approach does not require prior triangulation and simplifies a great deal the geometrical computations previously needed when dealing with compact support kernels.
We believe that the material presented in this talk can serve as visual motivation for students to learn about symbolic integration as well as Green and Stokes theorem in differential geometry.
The Crossing Number of a Graph
Speaker
David R Wood.
Time and Date
2pm, Friday 19 November 2010.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
The "crossing number" of a graph G is the minimum number of crossings in a drawing of G in the plane. This talk will introduce the crossing number and explain why it is interesting. In particular, I will describe applications of the crossing number in combinatorial geometry and number theory. Finally, I will discuss connections between the crossing number and graph minors. No background in graph theory will be assumed.
Modular invariant theory
Speaker
Dr R James Shank, University of Kent.
Time and Date
2pm, Friday 23 July 2010.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
Let S := F[x1,...,xn] denote the ring of polynomials in n variables with coefficients in the field F. A subgroup G of the general linear group GLn(F) acts on the span of the xi by linear transformations. This action extends to an action on S by ring automorphisms. The subring consisting of those polynomials fixed by this action, SG, is known as the ring of invariants. Invariant theorists are interested in the structure of SG, in relating properties of G and SG, and in algorithms for constructing generators for SG. The subject has a rich history with natural links to representation theory, algebraic geometry and algebraic topology. I will discuss a variety of results and examples paying particular attention to the case when G is finite and the characteristic of F is a prime number dividing the order of G, i.e., modular invariant theory.
Optical superoscillations
Speaker
Prof Sir Michael Berry, University of Bristol.
Time and Date
2pm, Friday 9 July 2010.
Venue
HUED Lecture Theatre, La Trobe University, Melbourne Campus.
Abstract
Band-limited functions can oscillate arbitrarily faster than their fastest Fourier component over arbitrarily long intervals. Where such ‘superoscillations’ occur, functions are exponentially weak. In typical monochromatic optical fields, substantial fractions of the domain (one-third in two dimensions) are superoscillatory. Superoscillations have implications for signal processing, and raise the possibility of sub-wavelength resolution microscopy without evanescent waves. In quantum mechanics, superoscillations correspond to weak measurements, suggesting the observation of momenta far outside the range represented in the quantum state, for which a general statistical theory is given.
Correlation inequalities for the Tutte polynomial of a graph
Speaker
Arun Mani, Monash University.
Time and Date
2pm, Friday 2 July 2010.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
The Tutte polynomial of a graph is a polynomial in two variables, x,y, that is of central importance in the study of many combinatorial counting problems. However evaluating these polynomials for general graphs at most points in the two dimensional rational plane is a notoriously difficult computational problem. Correlation inequalities for Tutte polynomials are motivated both by a theoretical interest in explaining special properties of these polynomials, and a practical interest in obtaining efficiently computable approximate evaluations of these polynomials. In this talk we will discuss a family of such inequalities for Tutte polynomials of graphs in the region x,y ≥1 along with a new tool to prove them, and pose several open problems of significance in this area.
Opportunities through AMSI
Speaker
Prof Geoff Prince, Director, Australian Mathematical Sciences Institute.
Time and Date
2pm, Friday 18 June 2010
Venue
HUED Lecture Theatre, La Trobe University, Melbourne Campus.
Abstract
I will talk about the new postgraduate internship scheme which has just been announced with the Commonwealth’s Enterprise Connect agency. Internships are available for mathematics, statistics and cognate disciplines such as computer science, bioinformatics, physics and engineering. I will also give a briefing on AMSI’s latest business plan and the expansion of its scientific and higher education programs. These changes significantly increase opportunity for AMSI members including undergraduates, postgraduates and researchers.
Geometric calculus for second-order differential equations and applications
Speaker
Prof W Sarlet, Ghent University, Belgium.
Time and Date
11am, Tuesday 1 June 2010.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
We review the main features of the geometric calculus which has been introduced over the past 15 years in the study of second-order ordinary differential equations and then illustrate “the theory at work” in a recent generalization of the inverse problem of Lagrangian mechanics.
First, we recall the notion of forms and vector fields along the projection of a tangent bundle onto its base manifold, the general concept of derivation of such forms and the need for a connection for the classification of such derivations. A second-order system provides such a connection in a canonical way (Sode-connection). The main motivation for introducing such a calculus is that many geometrical objects of interest in applications arise from tensor fields along the tangent bundle projection via suitable lifting operations, and conversely, the decomposition of tensor fields on the full tangent bundle into horizontal and vertical components identifies the essential ingredients of the theory as being sections of an appropriate pullback bundle. We will try to explain these matters in simple terms.
An important class of derivations are the self-dual derivations of degree zero. Such derivations play a key role in the linearization of the Sode-connection, which gives rise to horizontal and vertical covariant derivatives. Other important ingredients of the resulting geometric calculus are the dynamical covariant derivative (also a self-dual derivation of degree zero) and the Jacobi endomorphism (a type (1,1) tensor field along the tangent bundle projection).
An interesting field of application is the inverse problem of the calculus of variations, which has many faces. We recall that the existence of a Lagrangian for a given Sode can be expressed by a single necessary and sufficient condition within the present framework and explain how one can derive from it a concise intrinsic formulation of the so-called Helmholtz conditions, which are necessary and sufficient for the existence of a variational multiplier for the given system. Finally, we show in more detail how exactly the same techniques still apply for a generalization of this inverse problem, which comes from Lagrangian equations with non-conservative forces of dissipative or gyroscopic type.
The mystery of chaos in the Lorenz equations
Speaker
Dr Hinke Osinga and Prof Bernd Krauskopf, University of Bristol.
Time and Date
2pm, Friday 12 March 2010.
Venue
Access Grid Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Abstract
The Lorenz system still fascinates many people because of the simplicity of the equations that generate such complicated dynamics on the famous butterfly attractor.
This talk addresses the role of the stable and unstable manifolds in organising the dynamics more globally. A main object of interest is the stable manifold of the origin of the Lorenz system, also known as the Lorenz manifold. This two-dimensional manifold and associated manifolds of saddle periodic orbits can be computed accurately with numerical methods based on the continuation of orbit segments, defined as solutions of suitable boundary value problems. This allows us to study bifurcations of global manifolds as the Rayleigh parameter is changed. We show how the entire phase space of the Lorenz system is organised and changes dramatically during the transition to chaotic dynamics.
The fascination of the Lorenz system goes far beyond mathematics and as a bonus you will see how the Lorenz manifold was turned into a steel sculpture.
Hopf algebras and solutions to the Yang–Baxter equation
Speaker
Dr Karen Dancer, University of Queensland.
Time and Date
11am, Tuesday 3 November 2009.
Venue
Access Grid Room, 310 Physical Sciences 2, La Trobe University, Melbourne 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.
Computing in infinite groups
Speaker
Dr Murray Elder, University of Queensland.
Time and Date
1pm, Tuesday 3 November 2009.
Venue
Access Grid Room, 310 Physical Sciences 2, La Trobe University, Melbourne 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.
The Banach-Tarski Paradox
Speaker
Dr Ian Hawthorn, University of Waikato.
Time and Date
2pm, Friday 23 October 2009.
Venue
Access Grid Room, 310 Physical Sciences 2, La Trobe University, Melbourne 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.
Dynamics in some simple differential delay models
Speaker
Anatoli F Ivanov, Pennsylvania State University and University of Ballarat.
Time and Date
2pm, Friday 16 October 2009.
Venue
Access Grid Room, 310 Physical Sciences 2, La Trobe University, Melbourne 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.
Integrals of open 2D lattices
Speaker
Dmitry Demskoi.
Time and Date
2pm, Friday 25 September 2009.
Venue
Access Grid Room, 310 Physical Sciences 2, La Trobe University, Melbourne 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 An-type lattices and their reductions which include in particular Bn and Cn lattices. I will also consider a few other lattices that are shown to have integrals and explicit solutions.
Some exact solutions of the fin problem with a power law temperature-dependent thermal conductivity
Speaker
Joel Moitsheki, University of the Witwatersrand.
Time and Date
1pm, Monday 21 September 2009.
Venue
Access Grid Room, 310 Physical Sciences 2, La Trobe University, Melbourne 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 analysed.
A survey of Tutte-Whitney polynomials
Speaker
Graham Farr, Monash University.
Time and Date
2pm, Friday 18 September 2009.
Venue
Access Grid Room, 310 Physical Sciences 2, La Trobe University, Melbourne 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.
Moving frames, calculus of variations and Noether's theorem
Speaker
Prof Elizabeth Mansfield, University of Kent.
Time and Date
2pm, Friday 11 September 2009.
Venue
Access Grid Room, 310 Physical Sciences 2, La Trobe University, Melbourne 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.
Faces of the scl norm ball (scl = stable commutator length)
Speaker
Prof Danny Calegari, California Institute of Technology.
Time and Date
2pm, Friday 14 August 2009.
Venue
Hooper Lecture Theatre, La Trobe University, Melbourne 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.
Poles, strings, braids and lattices
Speaker
Prof Arun Ram, University of Melbourne.
Time and Date
2pm, Friday 1 May 2009.
Venue
Seminar Room 1, Room 212, Physical Sciences 2, La Trobe University, Melbourne 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.
An introduction to arc-transitive graphs
Speaker
Sanming Zhou, University of Melbourne.
Time and Date
12pm, Monday 20 April 2009.
Venue
Access Grid Room, 310 Physical Sciences 2, La Trobe University, Melbourne 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.
Forecasting the incidence of cancer in regional Victoria
Speaker
Prof Terry Mills, Loddon Mallee Integrated Cancer Service.
Time and Date
2pm, Friday 3 April 2009.
Venue
Access Grid Room, 310 Physical Sciences 2, La Trobe University, Melbourne 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).
Abelian groups: their role in the pathological behaviour of braids and the downfall of democracy!
Speaker
Prof Brian Davey
Time and Date
2pm, Friday 27 March 2009.
Venue
Hooper Lecture Theatre, La Trobe University, Melbourne Campus.
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.
Puzzle-based learning
Speaker
Zbigniew Michalewicz, University of Adelaide.
Time and Date
2pm, Friday 12 December 2008.
The convergence of binomial trees for pricing European and American options
Speaker
Mark Joshi, University of Melbourne.
Time and Date
2pm, Friday 28 November 2008.
Primal algebras, electronic circuits and evolution
Speaker
David Clark, SUNY New Paltz, USA.
Time and Date
2pm, Friday 7 November 2008.
Arithmetic on elliptic curves
Speaker
Daniel Delbourgo, Monash University.
Time and Date
2pm, Friday 17 October 2008.