Fellow 1992-93

Maarten J Bergvelt


Geometry of Integrable Systems

The fact that a physical system has symmetry implies (under quite general conditions) that there are conserved quantities. For instance, it does not matter (physically) where one plays billiards nor does it matter at what time one does. So we have here the symmetries of translation in space and in time and these are responsible for the conservation of linear momentum and of energy. 

A completely integrable system is, then, roughly speaking, a dynamical system with so many symmetries, i.e., so many conservation laws, that one can write down explicitly the solutions. A simple example is the harmonic oscillator (also known as the pendulum). A more complicated one is the Toda-lattice; this is a dynamical system consisting of a finite number of particles on a line that have exponential pairwise interaction with each other.

At the moment, the theory of integrable sytems is in rapid development, and it has been found that it has a very deep mathematical structure, making unexpected connections between, at first sight, unconnected parts of mathematics and physics. There are many groups all over the world finding exciting new results, but each speaking their own language; as a result, the connections between the various approaches are not always clear.

The two problems described in more detail in the research proposal form part of a longer-ranging project to give a unified framework for integrable systems (consisting mainly of representation theory of infinite dimensional groups and the geometry of infinite dimensional homogeneous spaces) and to study generalizations. The first problem deals with Toda lattice mentioned above; the aim is to explain the connection between generalizations of the Toda lattice and the geometry of finite and infinite dimensional Grassmannians. The second problem deals with the relation between the Hamiltonian structure of integrable systems and so-called W-algebras that have recently appeared in the context of string theory.