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Dive into the research topics where Eugene Lerman is active.

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Featured researches published by Eugene Lerman.


Transactions of the American Mathematical Society | 1997

Hamiltonian torus actions on symplectic orbifolds and toric varieties

Eugene Lerman; Susan Tolman

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.


Journal of Geometry and Physics | 1988

On the Kostant multiplicity formula

Victor Guillemin; Eugene Lerman; Shlomo Sternberg

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.


Transactions of the American Mathematical Society | 2004

Homotopy groups of K-contact toric manifolds

Eugene Lerman

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.


Nonlinearity | 1998

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

Eugene Lerman; Stephanie Frank Singer

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


Advances in Mathematics | 2012

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

Eugene Lerman; Anton Malkin

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.


Journal of Geometry and Physics | 2004

Contact fiber bundles

Eugene Lerman

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.


Comptes Rendus De L Academie Des Sciences Serie I-mathematique | 1999

On relative normal modes

Eugene Lerman; Tadashi Tokieda

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.


International Mathematics Research Notices | 2001

The topological structure of contact and symplectic quotients

Eugene Lerman; Christopher Willett

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.


Letters in Mathematical Physics | 1988

How fat is a fat bundle

Eugene Lerman

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.


Symmetry Integrability and Geometry-methods and Applications | 2015

Dynamics on Networks of Manifolds

Lee DeVille; Eugene Lerman

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.

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Michèle Audin

University of Strasbourg

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Victor Guillemin

Massachusetts Institute of Technology

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D. Burns

University of Michigan

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David I. Spivak

Massachusetts Institute of Technology

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Susan Tolman

Massachusetts Institute of Technology

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