We prove a Hitchin-Kobayashi correspondence for affine vortices generalizing a result of Jaffe-Taubes [19] for the action of the circle on the complex line. Namely, suppose a compact connected Lie group K acts on a Kähler manifold X with proper moment map so that stable=semistable for the action of the complexified Lie group G and X is equivariantly convex at infinity. Then, for some sufficient...

Foreword These notes cover a short course on symplectic toric manifolds, delivered in six lectures at the summer school on Symplectic Geometry of Integrable Hamiltonian Systems, mostly for graduate students, held at the Centre de Recerca Matemàtica in Barcelona in July of 2001. The goal of this course is to provide a fast elementary introduction to toric manifolds (i.e., smooth toric varieties)...

We consider compact Kähler manifolds acted on by a connected compact Lie group K of isometries in a Hamiltonian fashion. We prove that the squared moment map ||μ|| is constant if and only if the manifold is biholomorphically and K-equivariantly isometric to a product of a flag manifold and a compact Kähler manifold which is acted on trivially by K. The authors do not know whether the compactnes...

This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties. In particular, we explain the relation between linear precision and a particular linear projection we call the algebraic moment map.

By moduli we mean the set of points up to the equivalence of lying in the same G orbit. More generally, we are interested in understanding W up to G-equivalence; here G does not preserve W . We study this question when W is smooth and there exists a reductive subgroup H of G which measures the G-action along W . The notion of measuring the G-action is by means of the moment map and we say that ...

Tolman proposed that complex animal behavior is mediated by the cognitive map, an integrative learning system that allows animals to reconfigure previous experience in order to compute predictions about the future. The discovery of place cells in the rodent hippocampus immediately suggested a plausible neural mechanism to fulfill the 'map' component of Tolman's theory. Recent work examining hip...

We consider the moment map m : PVn 7→ iu(n) for the action of GL(n) on Vn = Λ(C) ⊗ C. The critical points of the functional Fn = ||m|| : PVn 7→ R are studied, in order to understand the stratification of Ln ⊂ PVn defined by the negative gradient flow of Fn, where Ln is the algebraic subset of all Lie algebras. We obtain a description of the critical points which lie in Ln in terms of those whic...

The purpose of this paper is to discuss the relationship between commutative and non-commutative integrability of Hamiltonian systems and to construct new examples of integrable geodesic flows on Riemannian manifolds. In particular, we prove that the geodesic flow of the biinvariant metric on any bi-quotient of a compact Lie group is integrable in the non-commutative sense by means of polynomia...

We prove a criterion for stability of relative equilibria in symmetric Hamiltonian systems at singular points of the momentum map. This generalizes a theorem of G.W. Patrick. The method of the proof is also useful in studying the bifurcation of relative equilibria.