This book constitutes the proceedings of the 2000 Howard conference on "Infinite Dimensional Lie Groups in Geometry and Representation Theory." It presents some important recent developments in this area. It opens with a topological characterization of regular groups, treats among other topics the integrability problem of various infinite dimensional Lie algebras, presents substantial contributions to important subjects in modern geometry, and concludes with interesting applications to representation theory. The book should be a new source of inspiration for advanced graduate students and established researchers in the field of geometry and its applications to mathematical physics.

This book offers a detailed presentation of results needed to prove the Morse Homology Theorem using classical techniques from algebraic topology and homotopy theory. The text presents results that were formerly scattered in the mathematical literature, in a single reference with complete and detailed proofs. The core material includes CW-complexes, Morse theory, hyperbolic dynamical systems (the Lamba-Lemma, the Stable/Unstable Manifold Theorem), transversality theory, the Morse-Smale-Witten boundary operator, and Conley index theory.

In the 60's, the work of Anderson, Chernavski, Kirby and Edwards showed that the group of homeomorphisms of a smooth manifold which are isotopic to the identity is a simple group. This led Smale to conjecture that the group Diff'" (M)o of cr diffeomorphisms, r ~ 1, of a smooth manifold M, with compact supports, and isotopic to the identity through compactly supported isotopies, is a simple group as well. In this monograph, we give a fairly detailed proof that DifF(M)o is a simple group. This theorem was proved by Herman in the case M is the torus rn in 1971, as a consequence of the Nash-Moser-Sergeraert implicit function theorem. Thurston showed in 1974 how Herman's result on rn implies the general theorem for any smooth manifold M. The key idea was to vision an isotopy in Diff'"(M) as a foliation on M x [0, 1]. In fact he discovered a deep connection between the local homology of the group of diffeomorphisms and the homology of the Haefliger classifying space for foliations. Thurston's paper [180] contains just a brief sketch of the proof. The details have been worked out by Mather [120], [124], [125], and the author [12]. This circle of ideas that we call the "Thurston tricks" is discussed in chapter 2. It explains how in certain groups of diffeomorphisms, perfectness leads to simplicity. In connection with these ideas, we discuss Epstein's theory [52], which we apply to contact diffeomorphisms in chapter 6.

The book introduces the basic notions in Symplectic and Contact Geometry at the level of the second year graduate student. It also contains many exercises, some of which are solved only in the last chapter.We begin with the linear theory, then give the definition of symplectic manifolds and some basic examples, review advanced calculus, discuss Hamiltonian systems, tour rapidly group and the basics of contact geometry, and solve problems in chapter 8. The material just described can be used as a one semester course on Symplectic and Contact Geometry.The book contains also more advanced material, suitable to advanced graduate students and researchers.

In the 60's, the work of Anderson, Chernavski, Kirby and Edwards showed that the group of homeomorphisms of a smooth manifold which are isotopic to the identity is a simple group. This led Smale to conjecture that the group Diff'" (M)o of cr diffeomorphisms, r ~ 1, of a smooth manifold M, with compact supports, and isotopic to the identity through compactly supported isotopies, is a simple group as well. In this monograph, we give a fairly detailed proof that DifF(M)o is a simple group. This theorem was proved by Herman in the case M is the torus rn in 1971, as a consequence of the Nash-Moser-Sergeraert implicit function theorem. Thurston showed in 1974 how Herman's result on rn implies the general theorem for any smooth manifold M. The key idea was to vision an isotopy in Diff'"(M) as a foliation on M x [0, 1]. In fact he discovered a deep connection between the local homology of the group of diffeomorphisms and the homology of the Haefliger classifying space for foliations. Thurston's paper [180] contains just a brief sketch of the proof. The details have been worked out by Mather [120], [124], [125], and the author [12]. This circle of ideas that we call the "Thurston tricks" is discussed in chapter 2. It explains how in certain groups of diffeomorphisms, perfectness leads to simplicity. In connection with these ideas, we discuss Epstein's theory [52], which we apply to contact diffeomorphisms in chapter 6.

This book offers a detailed presentation of results needed to prove the Morse Homology Theorem using classical techniques from algebraic topology and homotopy theory. The text presents results that were formerly scattered in the mathematical literature, in a single reference with complete and detailed proofs. The core material includes CW-complexes, Morse theory, hyperbolic dynamical systems (the Lamba-Lemma, the Stable/Unstable Manifold Theorem), transversality theory, the Morse-Smale-Witten boundary operator, and Conley index theory.