MSI Weekly Bulletin - Week starting Monday 24 July, 2006

Throughout natural and engineered systems, the question of stability always arises. How does a species survive? How does a power network avoid blackouts? How does the Internet avoid traffic congestion? Mathematical system theory provides ways to analyse the stability of a complex system and design the system and its control to preserve stability under adverse circumstances. This talk will describe some basic ideas and techniques in this area.

In a recent joint paper of V. Bangert and Y. Long, we have proved that on every Finsler 2-sphere there exist at least two distinct closed geodesics. This answers positively a conjecture of D. Anosov for 2-spheres proposed in 1974. In this lecture, I shall give a survey on recent progress on the problem of closed geodesics on spheres.

Mechanical is one of a number of situations where nonsmooth and discontinuous dynamics naturally arises. Other applications include electrical networks with diodes and hard resource limits in economics and ecology. But the mathematical theory to handle many such situations has been lacking, especially for mechanical impact. In this talk we will explore fundamental issues of existence, uniqueness and numerical approximation of solutions to nonsmooth problems which combine differential equations with ideas from constrained optimization.

Proper holomorphic maps are natural generalisations of biholomorphic maps and often arise in problems of complex analysis and geometry. In this talk I will concentrate on proper holomorphic maps between Reinhardt domains in complex space (a domain is called Reinhardt if it is invariant under rotations in each variable). One motivation for this study is a classical theorem due to H. Alexander (1977) that states that every proper holomorphic self-map of a ball in complex space is, in fact, biholomorphic. There have been many generalisations of this result, but the question of describing all Reinhardt domains for which every proper holomorphic self-map is a holomorphic automorphism has been open until recently even in dimension 2. In my talk I present a complete solution to this problem. It is a corollary of a more general result, which is a complete description of all proper holomorphic maps between Reinhardt domains in dimension 2. This work is joint with N. Kruzhilin (Steklov Mathematical Institute, Moscow).