Angular Momentum, Convex Polyhedra and Algebraic Geometry

The three families of classical groups of linear transformations (complex, orthogonal, symplectic) give rise to the three great branches of differential geometry (complex analytic, Riemannian and symplectic). Complex analytic geometry derives most of its interest from complex algebraic geometry, while symplectic geometry provides the general framework for Hamiltonian mechanics. These three classical groups "intersect" in the unitary group and the three branches of differential geometry correspondingly "intersect" in Kahler geometry, which includes the study of algebraic varieties in projective space. This is the basic reason why Hodge was successful in applying Riemannian methods to algebraic geometry in his theory of harmonic forms. In the past few years it has been realised that some of the ideas from symplectic geometry can also be applied to algebraic geometry. The key notion is that of angular momentum and the main technical result is a convexity theorem [1] [6] which asserts that the simultaneous values of commuting angular momenta form a convex polyhedron. My aim in this talk is to illustrate some of these new ideas and I will begin in the next two Sections by describing two results connecting algebra with convex polyhedra. These appear quite unrelated and neither is in a geometric form. Nevertheless I will show in subsequent sections how they both fit elegantly into the symplectic framework, which is explained in Section 4. In Section 7 I will discuss briefly the very important application to geometric invariant theory. Finally in Section 8 I will explain the "exact integration formulae" of Duistermaat and Heckman [5]. This is of interest not so much in algebraic geometry but as a prototype of infinite-dimensional counterparts which arise as models in theoretical physics.