ICE-EM Australian Graduate School in Mathematics

The courses available at the ICE-EM Australian Graduate School in Mathematics this year are Algebraic Structures, Dynamical Systems, and Stochastic Processes. Each course will be presented by two internationally renowned researchers. Each researcher will present approximately 15 lectures.
* Please note that the information for the courses and timetabling may be varied slightly. Any changes will be posted and highlighted.

Stochastic Processes

Course:

Random Fields and Geometry

Lecturer:

Robert Adler

Duration:

Three weeks, 6 - 22 July 2005.

Content:

The "random fields" of the course title are simply stochastic processes defined over parameter spaces with dimension greater than one. (Think of random (sea) surfaces, air temperature in a room, etc.) The "geometry" is the study of the shapes that these fields generate, which is of both theoretical and applied interest. In this course I shall concentrate more on the theory than the application, but we shall never be too far away
from well motivated real world problems.

The following outline covers the entire fifteen lectures and is extremely optimistic. It is also very far from being cast in stone, and will undoubtedly change depending on who takes the course and what we turn out to be interested in, but it should give you an idea of where we are heading.

All of this material can be found in an (almost complete) book with Jonathan Taylor of Stanford, with the same title as the course and which can be downloaded from http://iew3.technion.ac.il/~radler/publications.html

1.	Why should we be interested in random fields, with answers from theory and practice
2.	Gaussian processes. Examples, including the Brownian family of processes, entropy.
3.	Representations of Gaussian processes, orthogonal expansions, spectral theory.
4.	Introduction to Integral Geometry
5.	Generalised Rice formulae, upcrossings, expected Euler characteristic for Gaussian processes
6.	Borell and Slepian inequalities and their variations
7.	Suprema distributions, tube formulae, the Euler-Poincare heuristic
8.	Introduction to Riemannian manifolds
9.	Curvature, critical point theory
10.	Tube formulae and their applications to statistics
11.	Gaussian fields on abstract manifolds
12.	Continuation
13.	Poincare's theorem, kinematic fundamental formulae
14.	Non-Gaussian processes
15.	Applications.

Prerequisites:

I shall assume a graduate level course in Probability at the level of any of the standard texts (Billingsley, Breiman, Chung, Durrett, Resnick, etc.). I shall also assume that you have played in the past with multivariate normal distributions at the level of a late undergraduate course.

It will be helpful to have taken previous courses in stochastic processes
and to know, for example, what Brownian motion is, but this is not at
all necessary. For parts of the course, an undergraduate course or two in
differential geometry will help a lot. However, since I doubt many will
have this I will develop all I need during the course.

Resources:

Virtually all the material can be found in the manuscript "Random Fields and Geometry"
at http://iew3.technion.ac.il/~radler/publications.html (Copies of this manuscript will be distributed to all students in the course.)

Course:

Statistical Methods in Genetics and Genomics

Lecturer:

Warren Ewens

Duration:

Two weeks, 4 - 15 July 2005 (15 lectures as well as tutorials and discussion classes).

Content:

The course will focus on three main areas:

1.	Evolutionary Population Genetics. Mathematical aspects of population genetics theory, focusing on stochastic processes describing changes in gene frequency as a result of selection and mutation.
2.	Bioinformatics. Statistical methods used to analyze large genetic and genomic data sets. The theory behind the BLAST procedure will be discussed in detail. The analysis of DNA sequences.
3.	Human genetics, especially statistical methods for finding disease genes.

Prerequisites:

A knowledge of standard stochastic process theory, including random walks and Markov chains. A knowledge of statistics up to the level of an honours degree. No prior knowledge of genetics is required.

Resources:

The following books will be available:
Ewens: Mathematical Population Genetics (2nd edition, Springer, 2004).
Ewens and Grant: Statistical Methods in Bioinformatics (2nd edition, Springer, 2005).
Thomas: Statistical Methods in Genetic Epidemiology (Oxford, 2004).

Dynamical Systems

Course:

Bifurcation Theory

Lecturer:

Jeroen Lamb

Duration:

Three weeks, 4 - 22 July 2005.

Content:

We develop techniques to describe how (locally, near equilibria and/or periodic solutions of ordinary differential equations) dynamics may change as external parameters are varied.

Hours:

1 hour lecture and discussion per morning Monday to Friday for three weeks.

Course:

Discrete Dynamical Systems and Chaos

Lecturer:

James Meiss

Duration:

To be advised

Content:

Discrete dynamical systems (mappings) arise quite naturally as models for many physical phenomena including population and disease dynamics in biology and impulsively or periodically forced systems in engineering and physics; however, they are also--from a mathematical point of view--quite general representations of dynamics. In this course we will study the behaviour of maps beginning with the one-dimensional case through multidimensional and conservative (symplectic) maps. We will introduce in a rigorous way the concepts of attractors, conjugacy and chaos. We will use computer experiments to investigate phenomena that are not amenable to rigorous analysis.

Topics include:

	Introduction to dynamical systems
	Stability, Invariance and Omega Limit Sets
	Attractors and Conjugacy
	Logistic Map, Period Doubling & Periodic Orbits, Invariant Cantor Sets
	Symbolic Dynamics
	Period 3 implies Chaos
	Chaos: Sensitive Dependence & Transitivity, Lyapunov Exponents
	Hamiltonian Systems: Action and the Poincaré Invariant
	Symplectic Maps
	The Symplectic Group: Eigenvalues of Symplectic Matrices
	Generating Functions & Twist Maps
	Variational Principles & Minimizing Periodic Orbits
	Aubry-Mather Theory: Tori and Cantori

Hours:

To be advised

Algebraic Structures

Course:

Representation Theory

Lecturer:

Stephen Bigelow

Duration:

Three weeks, 4 - 22 July 2005.

Content:

In the early days of group theory, a group was always a set of linear functions on a vector space, that is, a set of matrices. Over time the theory split into the study of abstract groups in their own right, and the study of different ways to represent a given group by matrices. The latter is representation theory.

This course will focus on the representation theory of the symmetric group. It will cover the Specht construction of irreducible representations of the symmetric group, and as much of the combinatorics of tableaux as time allows (eg. the Robinson-Schensted-Knuth algorithm, jeu-de-taquin, the hook-length formula).

Hours:

1 hour lecture and discussion per morning Monday to Friday for three weeks.

Course:

Modular Forms

Lecturer:

Mark Kisin

Duration:

Two weeks, 11 - 22 July 2005.

Content:

The course will be on modular forms. We will begin by studying modular forms in the most classical setting, as functions on the complex upper half plane. In this setting I will explain topics such as the relationship to quadratic forms, the complex q-expansion, L-series and their analytic continuation, and Hecke operators.

I will then explain the powerful point of view afforded by algebraic geometry, and the relationship with moduli of elliptic curves. As an application we will see how to define and prove results about modular forms in characteristic p.

Hours:

15 hours of morning lectures (with discussion time) over two weeks.