Michèle Audin

Torus Actions on Symplectic Manifolds

Introductory preface.- How I have (re-)written this book.- Acknowledgements.- What I have written in this book.- I. Smooth Lie group actions on manifolds.- I.1. Generalities.- I.2. Equivariant tubular neighborhoods and orbit types decomposition.- I.3. Examples: S 1-actions on manifolds of dimension 2 and 3.- I.4. Appendix: Lie groups, Lie algebras, homogeneous spaces.- Exercises.- II. Symplectic manifolds.- II.1What is a symplectic manifold?.- II.2. Calibrated almost complex structures.- II.3. Hamiltonian vector fields and Poisson brackets.- Exercises.- III. Symplectic and Hamiltonian group actions.- III.1. Hamiltonian group actions.- III.2. Properties of momentum mappings.- III.3. Torus actions and integrable systems.- Exercises.- IV. Morse theory for Hamiltonians.- IV.1. Critical points of almost periodic Hamiltonians.- IV.2. Morse functions (in the sense of Bott).- IV.3. Connectedness of the fibers of the momentum mapping.- IV.4. Application to convexity theorems.- IV.5. Appendix: compact symplectic SU(2)-manifolds of dimension 4.- Exercises.- V. Moduli spaces of flat connections.- V.1. The moduli space of fiat connections.- V.2. A Poisson structure on the moduli space of flat connections.- V.3. Construction of commuting functions on M.- V.4. Appendix: connections on principal bundles.- Exercises.- VI. Equivariant cohomology and the Duistermaat-Heckman theorem.- VI.1. Milnor joins, Borel construction and equivariant cohomology.- VI.2. Hamiltonian actions and the Duistermaat-Heckman theorem.- VI.3. Localization at fixed points and the Duistermaat-Heckman formula.- VI.4. Appendix: some algebraic topology.- VI.5. Appendix: various notions of Euler classes.- Exercises.- VII. Toric manifolds.- VII.1. Fans and toric varieties.- VII.2. Symplectic reduction and convex polyhedra.- VII.3. Cohomology of X ?.- VII.4. Complex toric surfaces.- Exercises.- VIII. Hamiltonian circle actions on manifolds of dimension 4.- VIII.1. Symplectic S 1-actions, generalities.- VIII.2. Periodic Hamiltonians on 4-dimensional manifolds.- Exercises.

Symplectic rigidity: Lagrangian submanifolds

This chapter is supposed to be a summary of what is known today about Lagrangian embeddings. We emphasise the difference between flexibility results, such as the h-principle of Gromov applied here to Lagrangian immersions (and also to the construction of examples of Lagrangian embeddings) and rigidity theorems, based on existence theorems for pseudo-holomorphic curves.

Lectures on gauge theory and integrable systems

In these notes, I will describe the moduli space of flat connections on a principal bundle over a surface and its Poisson structure. I will then give examples of integrable systems on these spaces, following ideas of Goldman, Jeffrey and Weitsman, Fock and Roslyi, and Alekseev.

Symplectic geometry in Frobenius manifolds and quantum cohomology

The multiplication of vector fields on a Frobenius manifold M defines a Lagrangian submanifold of T∗M. In this paper, we give a proof of this “folklore” fact based on the formalism of Higgs pairs and we explain how it can be applied in the quantum cohomology situation (following Givental and Kim (1995)).

Symplectic and almost complex manifolds

The aim of this chapter is to introduce the basic problems and (soft!) techniques in symplectic geometry by presenting examples—more exactly series of examples— of almost complex and symplectic manifolds: it is obviously easier to understand the classification of symplectic ruled surfaces if you have already heard of Hirzebruch surfaces for instance.

Remembering Sofya Kovalevskaya

Sofyas chronology.- Sofyas names.- Stories.- The thesis of Sofya, the Cauchy-Kovalevskaya theorem.- The Solid.- A letter to Mittag-Leffler.- Stockholm.- A letter to Vollmar.- The Bordin prize and Sofyas reputation.- The women of Men of mathematics.- I remember Sofya, by George, Gosta, Julia and all the rest.- I too remember Sofya.- Bibliography.

Vladimir Igorevich Arnold and the Invention of Symplectic Topology

In 1965, with a Comptes rendus note of Vladimir Arnold, a new discipline, symplectic topology, was born. In 1986, its (remarkable) first steps were reported by Vladimir Arnold himself. In the meantime…

From Floer to Morse

We prove that in the case of a nondegenerate Hamiltonian that does not depend on time and is sufficiently small, when we are able to define the Morse complex and the Floer complex, the two coincide.

Two notions of integrability

We investigate the relation between two notions of integrability, to have enough first integrals on the one hand, and to have meromorphic solutions on the other, that are present in Kowalevskayas famous memoire on the rigid body problem. We concentrate on the examples of the rigid body and of the system of Henon-Heiles.

Introduction Applications of pseudo-holomorphic curves to symplectic topology

This chapter is an introduction to the book. First we will describe some problems in symplectic geometry, or more exactly topology, and the way to solve them using pseudo-holomorphic curves techniques. Then we describe very roughly the contents of the book. For the basic results in geometry, the reader can consult chapters I, II or III.