Summer School on Differential Equations in Geometry and Physics

Dates: December 8 - 11, 1998
Location: University of Adelaide
Organisers: Adam Harris, Nalini Joshi

Programme: In addition to individual lectures on specialist topics, a three-lecture mini-course at an introductory level will be contributed by each of the following speakers.

Prof. Michael Eastwood (Tues, Wed, Fri 9:00 am)
"The Einstein-Weyl equations"
Abstract: The Einstein equations arise naturally in differential geometry but have only trivial solutions in three-dimensions. Slightly more general are the Einstein-Weyl equations. These have interesting solutions in three dimensions. I'll explain what these equations are and say something about their three-dimensional solutions. Nothing much will be assumed.

Assoc. Prof. Michael Murray (Wed 11:30 am, Thurs, Fri 10:00 am)
"Instantons, monopoles, and rational maps"
Abstract: The course will review a number of topics in the general area of Bogomolny monopoles. The topics include, instantons, monopoles, twistor constructions, the Nahm transform, rational maps and spectral curves. Anyone interested in a bit of pre-lecture reading should have a look at the book 'The Geometry and Dynamics of Magnetic Monopoles' M.F. Atiyah and N.J. Hitchin. The book is concerned principally with things we are not going to cover but Chapter 16 on Background Material is what it claims to be.

Dr Nalini Joshi (Tues, Fri 2:00 pm, Wed 4:30pm)
"How to exclude chaos from non-linear differential equations"
Abstract:Differential equations that exclude chaos from their manifold of solutions are rare. They include "completely integrable" systems that admit solitons as solutions. Integrability is closely related to the singularity structure of general solutions. In these lectures, I will introduce and review methods for analysing singularity structure for differential equations (both ordinary and partial). If time permits I hope to talk about difference equations as well.

Dr Mathai Varghese (Tues 11:30 am, Wed 2:00 pm)
"Magnetic Schroedinger operators"
Abstract: Magnetic Schrodinger operators turn out to be the Hamiltonians occuring in the theoretical model for the (integer) quantum Hall effect. Note that the experimental and theoretical Physicists who discovered the Quantum Hall effect and its generalizations were awarded Nobel Prizes in 1985 and 1998. I will discuss several of the basic properties of Magnetic Schrodinger operators on compact manifolds, including self-adjointness and some spectral properties.

Dr Siye Wu (Tues 4:30 pm, Wed 10:00 am, Thurs 9:00 am)
"The geometry and physics of the Seiberg-Witten equations"
Abstract:Lecture 1: classical and quantum aspects of N=1 supersymmetric gauge theories
Lecture 2: monopoles in supersymmetric gauge theories and electro-magnetic duality
Lecture 3: the Seiberg-Witten solution of N=2 supersymmetric gauge theories and applications to 4-manifold theory

Dr Peter Bouwknegt(Tues, Wed, Fri 3:00 pm)
"The Knizhnik-Zamolodchikov equations"
Abstract:In these lectures I will discuss various aspects of the Knizhnik-Zamolodchikov equations. In lecture 1 I will introduce the KZ-equations and discuss the associated monodromy representations of the braid group (Drinfeld-Kohno theorem). In lecture 2 I will discuss solutions by hypergeometric integrals and the homology of the associated local system (Schechtman-Varchenko). In lecture 3 I will explain the operator approach to finding solutions of the KZ-equations.

Lecture venue: Room 102, Mathematics Building, University of Adelaide (located opposite the Barr-Smith library, and between Engineering and the Union Hall). Lectures begin at 9:00 am on Tuesday, and will finish at 4:00 pm Friday. No lectures are scheduled on Thursday afternoon.

Accomodation: There are two locations.
(1) Kathleen Lumley College is at 51 Finniss St, North Adelaide, with a main entrance on Mc Kinnon Parade, next to the University Gym. Participants staying there must go to the common room on the first floor when they arrive. Pigeon holes just inside the door are marked alphabetically, and inside the box marked with the participant's second initial will be an envelope with room key and number. The college is a ten-minute walk along Frome street from the university campus.
(2) The residential wing of the Royal Adelaide Hospital is just off Frome Street, and is across the road from the university campus. Participants staying there should check in at the receptionist's desk inside the ground-floor entrance, where they will be given a room number and key.

Transport: The most convenient transport from Adelaide airport is by taxi. These are easily hailed from in front of the ticketing hall entrance, and take about 20-25 minutes to get to the city centre. The fare is about $15.

There will be no registration fee. Funding will be available to support travel and/or accomodation for interstate participants.

Further Information: Adam Harris
Department of Pure Mathematics
University of Adelaide