Instructional Workshop on Analysis and Geometry

Canberra, Jan 23 to Feb 10, 1995

Volume 34 in the Proceedings of the Centre for Mathematics and its Applications

Introduction

An Instructional Workshop on Analysis and Geometry was held at the Australian National University between the 23rd January and the 10th February, 1995. Over one hundred people attended, with more than twenty universities represented. The majority of participants were Australian PhD students, but there were also students from Germany, New Zealand, Japan and Taiwan. Twenty-five Australian and over-seas speakers participated.

The rationale behind the Workshop was to help rectify an incompleteness in the coursework material available to most mathematics research students in Australia. It was intended that someone attending the introductory first week of the workshop would be exposed to many of the ideas and techniques fundamental to modern analysis and geometry, while those who attended all three weeks would gain an overview up to current research levels. The Workshop was designed with research students in mind, but was also intended for established mathematicians working in other fields.

The workshop consisted of a number of broad streams, each starting at an introductory level and becoming more specialised in the later weeks. There were five one-hour lectures each day.

Notes were prepared by the speakers and were distributed prior to the lectures. The strong demand for these notes has led us to collect them here in three volumes, which we have organised on a subject basis. Volume One deals primarily with Partial Differential Equations, Volume Two with Differential Geometry and Geometric Measure Theory, and Volume Three covers topics from Harmonic Analysis and Nonlinear Analysis.

The Workshop was sponsored by:

1. The School of Mathematical Sciences at the ANU, in conjunction with its 1994-5 Special Program on Nonlinear Analysis.
2. The Australian Mathematical Soceity through the miniconference fund.
3. The Graduate School at the ANU.

We would like to thank the speakers for their efforts, both at the Workshop and in the preparation of these lecture notes.

Tim Cranny and John Hutchinson
(Editors)

Contents

Part I - Partial Differential Equations
Measure Theory and Function Spaces John Hutchinson	1
Second Order Linear Partial Differential Equations John Urbas	39
Finite Element Method for Numerically Solving PDE's Steve Roberts	77
The Dirichlet Problem for the Minimal Surface Equation Graham Williams	91
Elliptic Systems John Hutchinson	111
Nonclassical Solutions of Fully Nonlinear Elliptic PDE's Tim Cranny	121
Inverse Spectral Problems Richard Melrose	137
An additional series of talks, "Nonlinear Elliptic Equations of Second Order" was presented by Neil Trudinger and has been published separately in Lectures in Mathematical Sciences, Univ. Tokyo, 9, (1995).

Part II - Geometric Analysis
Geometric Measure Theory Marty Ross	1
Geometric Measure Theory Frank Morgan	51
Berstein's Theorem Maria Athanassenas	67
Geometric Evolution Equations Klaus Ecker	79
Minimal Surfaces in R3 Yi Fang	109
Several Complex Variables Alex Isaev	121
Differential Geometry Robert Bartnik	125
Reimannian Geometry and Mathematical Physics Michael Murray	165
Hyperbolic Equations and General Relativity Robert Bartnik	195

Part III - Operator Theory and Nonlinear Analysis
Basic Semigroup Theory Derek Robinson	1
Harmonic Analysis Tony Dooley	35
Operator Theory and Harmonic Analysis David Albrecht, Xuan Duong and Alan McIntosh	77
Introduction to the Theory of Critical Points Jan Chabrowski	137
Degree Theory Norm Dancer	183