Eugene Lerman

Hamiltonian torus actions on symplectic orbifolds and toric varieties

In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus actions are classified by convex simple rational polytopes with a positive integer attached to each facet and that all such orbifolds are algebraic toric varieties.

On the Kostant multiplicity formula

Abstract The Kostant multiplicity formula is a recipe for computing the weight multiplicities of an irreducible representation of a compact semi-simple Lie group. We describe here a generalization of Kostants formula: Suppose τ is a Hamiltonian action of a compact Lie group on a compact symplectic manifold. For an appropriate «quantization», τ Q , of τ the weight multiplicaties of τ Q are given by a formula similar to Konstants. There is also an asymptotic version of this formula which gives a recipe for computing the Duistermaat Heckman polynomials associated with τ.

Homotopy groups of K-contact toric manifolds

Contact toric manifolds of Reeb type are a subclass of contact toric manifolds which have the property that they are classified by the images of the associated moment maps. We compute their first and second homotopy group terms of the images of the moment map. We also explain why they are K-contact.

Stability and persistence of relative equilibria at singular values of the moment map

We prove criteria for stability and propagation of relative equilibria in symmetric Hamiltonian systems at singular points of the momentum map. AMS classification scheme numbers: 58F05, 70H33

Hamiltonian group actions on symplectic Deligne–Mumford stacks and toric orbifolds

Abstract We develop differential and symplectic geometry of differentiable Deligne–Mumford stacks (orbifolds) including Hamiltonian group actions and symplectic reduction. As an application we construct new examples of symplectic toric DM stacks.

Contact fiber bundles

Abstract We define contact fiber bundles and investigate conditions for the existence of contact structures on the total space of such a bundle. The results are analogous to minimal coupling in symplectic geometry. The two applications are construction of K -contact manifolds generalizing Yamazaki’s fiber join construction and a cross-section theorem for contact moment maps.

On relative normal modes

Abstract We generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes near an equilibrium in a Hamiltonian system to a theorem on the existence of relative periodic orbits near a relative equilibrium in a Hamiltonian system with continuous symmetries. In particular, we prove that under appropriate hypotheses there exist relative periodic orbits near relative equilibria even when these relative equilibria are singular points of the corresponding moment map, i.e. when the reduced spaces are singular.

The topological structure of contact and symplectic quotients

We show that if a Lie group acts properly on a co-oriented contact manifold preserving the contact structure, then the contact quotient is topologically a stratified space (in the sense that a neighborhood of a point in the quotient is a product of a disk with a cone on a compact stratified space). As a corollary, we obtain that symplectic quotients for proper Hamiltonian actions are topologically stratified spaces in this strong sense thereby extending and simplifying previous work.

How fat is a fat bundle

Let P → M be a principal G-bundle with connection 1-form θ and curvature Θ. For a subset S of g* the given connection is S-fat (Weinstein, [5]) if for every μ in S the form μ ° Θ is nondegenerate on each horizontal subspace in TP.Let K be a compact group and K/H be its coadjoint orbit. The orthogonal projection t → h defines a connection on the principal H-bundle K → K/H. We show that this connection is fat off certain walls of Weyl chambers in h*. We then apply the result to the construction of symplectic fiber bundles over K/H. As an example, we show how higher-dimensional coadjoint orbits fiber symplectically over lower-dimensional orbits.

Dynamics on Networks of Manifolds

We propose a precise definition of a continuous time dynamical system made up of interacting open subsystems. The interconnections of subsystems are coded by directed graphs. We prove that the appropriate maps of graphs called graph brations give rise to maps of dynamical systems. Consequently surjective graph fibrations give rise to invariant subsystems and injective graph fibrations give rise to projections of dynamical systems.