These are expanded notes from a set of lectures given at the school “Actions Hamiltoniennes: leurs invariants et classification” at Luminy in April 2009. The topics center around the theorem of Kempf and Ness [58], which describes the equivalence between the notion of quotient in geometric invariant theory introduced by Mumford in the 1960’s [80], and the notion of symplectic quotient introduce...

The classical Pfaff-Darboux Theorem, which provides local ‘normal forms’ for 1-forms on manifolds, has applications in the theory of certain economic models [3]. However, the normal forms needed in these models come with an additional requirement of convexity, which is not provided by the classical proofs of the Pfaff-Darboux Theorem. (The appropriate notion of ‘convexity’ is a feature of the e...

Delzant’s theorem for symplectic toric manifolds says that there is a one-to-one correspondence between certain convex polytopes in Rn and symplectic toric 2n-manifolds, realized by the image of the moment map. I review proofs of this theorem and the convexity theorem of Atiyah-Guillemin-Sternberg on which it relies. Then, I describe Honda’s results on the local structure of near-symplectic 4-m...

Caffarelli-Friedman [7] proved a constant rank theorem for convex solutions of semilinear elliptic equations in R2, a similar result was also discovered by Yau [28] at the same time. The result in [7] was generalized to R by Korevaar-Lewis [27] shortly after. This type of constant rank theorem is called microscopic convexity principle. It is a powerful tool in the study of geometric properties ...

The convexity theorem of Atiyah and Guillemin-Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar-Lerman proved that the Marsden-Weinstein reduction of a connected Hamitonian G-manifold is a stratified symplectic space. Suppose 1 → A → G → T → 1 is an exact sequence of compa...

Tverberg’s theorem asserts that every (k − 1)(d + 1) + 1 points in R can be partitioned into k parts, so that the convex hulls of the parts have a common intersection. Calder and Eckhoff asked whether there is a purely combinatorial deduction of Tverberg’s theorem from the special case k = 2. We dash the hopes of a purely combinatorial deduction, but show that the case k = 2 does imply that eve...

A classical result of Schur and Horn [Sc, Ho] states that the set of diagonal elements of all n x n Hermitian matrices with fixed eigenvalues is a convex set in IRn. Kostant [Kt] has generalized this result to the case of any semisimple Lie group. This is often referred to as the linear convexity theorem of Kostant: picking up the diagonal of a Hermitian matrix is a linear operation. This resul...