July to December 2000
Organisers: VI Arnold (Moscow and Parix IX), JW Bruce (Liverpool), V Goryunov (Liverpool), D Siersma (Utrecht)
Organising Committee: D Siersma - Director (Utrecht), VM Zakalyukin - Co-Director (Moscow State Aviation Institutute), JP Brasselet, (CNRS-Mathematical Institute Luminy), V Vassiliev (Russian Academy of Science), CTC Wall (Liverpool)
Introduction and Background
Singularities arise naturally in a huge number of different areas of Mathematics and Science. As a consequence Singularity Theory lies at the crossroads of the paths connecting the most important areas of applications of mathematics with its most abstract parts. For example it connects the investigation of optical caustics with simple Lie algebras and the theory of regular polyhedra, and also relates hyperbolic PDE wavefronts to knot theory and the theory of the shape of solids to commutative algebra.
The main goal in most problems of singularity theory is to understand the dependence of some objects of interest in analysis and geometry, or physics, or in some other science (functions, varieties, mappings, vector or tensor fields, differential equations, models and so on) on parameters. For generic points in the parameter space their exact values influence only the quantitative aspects of the phenomena, their qualitative features remaining stable under small changes of parameter values.
However for certain exceptional values of the parameters the qualitative topological features of the phenomena may suddenly change under a small variation of the parameter. This change is called a perestroika, bifurcation, metamorphosis or catastrophe in different branches of science.
In spite of its fundamental character, and the central position it now occupies in mathematics, singularity theory is a surprisingly young subject. In one sense it can be viewed as the modern equivalent of the differential calculus. In its current form the subject started with the fundamental discoveries of Whitney, Thom, Arnold, Mather and Brieskorn. Substantial results and exciting new developments within the subject have continued to flow in the intervening years, while the theory has encompassed more and more applications.
We mention here the Vassiliev theory of discriminant spaces which introduced eg new knot invariants; the connections with mirror symmetry; applications to image analysis (Damon) and recently connections between Stokes sets and combinatorics.
The primary purpose of the ASI is the introduction of new developments in singularity theory to a broader audience, thus establishing new contacts and hopefully advancing a broad front of research. The importance of pedagogical skills has been borne in mind in choosing the ASI lecturers, as the instructional aspect is regarded as central to the success of the meeting. Also, tutorial sessions should contribute to this.
The meeting will be held at the Isaac Newton Institute for Mathematical Sciences in Cambridge. The Newton Institute views this area as of sufficient importance to have funded a six month programme on Singularity Theory from July to December 2000 which should start with the proposed ASI. The Newton Institute has excellent lecture facilities with imaginatively designed discussion areas which will facilitate interaction between the participants, and a well planned library. Housing and the main University library are nearby. Some financial support would be available through the Newton Institute.