Applied Mathematics / General Seminars
MAHLER LECTURE SERIES 2013 | Professor Akshay Venkatesh (Stanford University) | Friday 11 October 2013, 1:30PM
Torsion in the homology of arithmetic groups
Take a Bianchi group - e.g, invertible 22 matrices with entries in the Gaussian integers - and abelianize it. The result is often a very large torsion group. I will discuss this phenomenon and how it relates to number theory.
MAHLER LECTURE SERIES 2013 | Professor Akshay Venkatesh (Stanford University) | Friday 11 October 2013, 5:00PM
How to stack oranges in three dimensions, 24 dimensions, and beyond
How can we pack balls as tightly as possible? In other words: to squeeze as many balls as possible into a limited space, what's the best way of arranging the balls? It’s not hard to guess what the answer should be — but it’s very hard to be sure that it really is the answer! I'll tell the interesting story of this problem, going back to the astronomer Kepler, and ending almost four hundred years later with Thomas Hales. I will then talk about stacking 24-dimensional oranges: what this means, how it relates to the Voyager spacecraft, and the many things we don’t know beyond this.
About the speaker
Akshay received his PhD from Princeton University in 2002. His research is pure mathematics specifically, in number theory and related areas. His research interests are in the fields of counting, equidistribution problems in automorphic forms and number theory, in particular representation theory, locally symmetric spaces and ergodic theory. In 2008 he won the SASTRA Ramanujan Prize. This annual prize is for outstanding contributions to areas of mathematics influenced by the genius Srinivasa Ramanujan. The Mahler Lectureship is awarded every two years to a distinguished mathematician who preferably works in an area of mathematics associated with the work of the late Professor Mahler. It is usually expected that the Lecturer will speak at one of the main Society Conferences and visit as many universities as can be reasonably managed.
Graduate Colloquium Series | Ray Withers (ANU) | Friday 26 April 2013, 3:30pm
Order and ‘disorder’, a chemist's view: what we know, what we don’t know and what we (often wrongly!) assume.
In 1912 von Laue, Friedrich & Knipping first exposed a crystal to a beam of X-rays. The experiment was initially carried out in order to understand the nature of the radiation itself; instead, its real importance was the discovery of X-ray diffraction. In the same year, 1912, WL Bragg developed his famous law thereby making it possible to calculate the positions of atoms within crystals from the intensities of diffracted beams. The “diffraction” of X‑rays thus changed from the status of being a physical phenomenon to that of a tool for exploring the arrangement of atoms within crystals. The extraordinary success of X-ray crystallography ever since has led to the now largely “mature” science of crystallography. Like all successful disciplines, however, its very success inevitably led to the imposition of rigid “rules” as to what constitutes crystalline order and what doesn’t e.g. “ .. A unit cell of a crystal is a .. parallel-sided region .. from which the entire crystal can be built up by purely translational displacements ..” Shriver and Atkins, P66, 2009!! Wrong, as demonstrated by the (eventual) widespread acceptance of aperiodic/quasicrystalline order! Likewise, the direct observation of curved graphite planes in Transmission Electron Microscope (TEM) images of carbon support films in the 1960’s was ignored because “ .. lattice planes can’t curve ..”. Bye-bye the chance to discover bucky balls and bucky tubes much earlier than they were! In this contribution, a range of other fundamentalist type structural notions will be discussed ranging from the strange use of “nodal planes” when describing the molecular orbitals of 1-D “crystalline”, periodic ring molecules such as benzene or cyclopentadienyl to the question of why there is extensive notation describing “real”, but not “reciprocal”, space to the notion that a crystal structure refined from ISIS or synchrotron data with a good R‑factor is necessarily correct. The need to always keep thinking and to extend our ideas of what constitutes order to encompass whatever we experimentally encounter is still with us and continues to separate thoughtful structural chemists from handle‑turners. ‘Ordered’ crystalline materials are often far more subtle than the straitjackets imposed by crystallographic or chemical fundamentalism. Functionally useful materials (piezoelectrics, relaxor ferroelectrics, ionic conductors, solid solutions etc.), for example, are often modulated and frequently inherently flexible [1-3]. A detailed understanding of the structure, both average as well as local (on the relevant length and time scales) of such materials, is essential for an understanding of their properties and of methods to optimize and manipulate them. In this contribution, the results obtained from several such systems will be described including inherently Pb-free polar functional materials and the Li3xLn2/3‑xTiO3, 0.047 < x < 0.147, family of Li ion conductors. The local crystal chemistry underlying the inherent structural flexibility of these materials will be discussed along with the characteristic diffraction signatures of such behaviour.
Graduate Colloquium Series | Anne Thomas (University of Sydney) | Friday 19 April 2013, 3:30pm
3-manifolds, cube complexes and lattices
A recent spectacular result in low-dimensional topology is that every closed 3-manifold has a finite cover which fibres over a circle. This was conjectured by Thurston in the 1970s, and proved by Agol in 2012, using geometric group theory, in particular group actions on cube complexes. I will explain some of the key ideas, and then give some applications to the study of lattices in locally compact groups.
Zhiyuan Liu (Monash University) | Friday 8 March 2013, 3:30pm
Optimal Toll Design Problem of Urban Congestion Pricing
In a road transport network, the drivers’ route choice behaviour is un-cooperative, which would lead to an unwise use of the network and severe traffic congestions in some areas. Congesting pricing is one of the few instruments used by the transport authorities to properly adjust drivers’ route choice decisions. Based on given pricing locations, the Toll Design Problem aims to obtain the optimal toll pricing rate such that the total level of congestions in the network can mitigated. This presentation will first briefly review some congestion pricing practices in Singapore, London and Scandinavia. Then discuss about the modelling skills for the optimal toll deign problem. Subsequently, modelling for the Toll Design Problem for some newly proposed pricing schemes will be covered.
Lisa Clark (University of Otago) | Friday 22 February 2013, 3:30pm
Spectral properties of C*-algebras associated to groupoids
Groupoids appear in a number of different branches of pure mathematics. In operator algebras, we associate a C*-algebra to a grouoid so that properties of the algebra can be seen in propoerties of the groupoid. In this talk, I will begin by describing how a groupoid is a generalisation of the action of a group on a set. Then, I will describe how to associate a C*-algebra to a groupoid and demonstrate how spectral properties of the algebra correspond with topological properties of the groupoid. This talk should be accessible to a general math audience.