It seems hard to believe, but mathematicians were not interested in integration problems on infinite-dimensional nonlinear structures up to 70s of our century. At least the author is not aware of any publication concerning this theme, although as early as 1967 L. Gross mentioned that the analysis on infinite dimensional manifolds is a field of research with rather rich opportunities in his classical work [2. This prediction was brilliantly confirmed afterwards, but we shall return to this later on. In those days the integration theory in infinite dimensional linear spaces was essentially developed in the heuristic works of RP. Feynman [1], I. M. Gelfand, A. M. Yaglom [1]). The articles of J. Eells [1], J. Eells and K. D. Elworthy [1], H. -H. Kuo [1], V. Goodman [1], where the contraction of a Gaussian measure on a hypersurface, in particular, was built and the divergence theorem (the Gauss-Ostrogradskii formula) was proved, appeared only in the beginning of the 70s. In this case a Gaussian specificity was essential and it was even pointed out in a later monograph of H. -H. Kuo [3] that the surface measure for the non-Gaussian case construction problem is not simple and has not yet been solved. A. V. Skorokhod [1] and the author [6,10] offered different approaches to such a construction. Some other approaches were offered later by Yu. L. Daletskii and B. D. Maryanin [1], O. G. Smolyanov [6], N. V.

Infinite dimensional manifolds, Lie groups and algebras arise naturally in many areas of mathematics and physics. Having been used mainly as a tool for the study of finite dimensional objects, the emphasis has changed and they are now frequently studied for their own independent interest. On the one hand this is a collection of closely related articles on infinite dimensional Kähler manifolds and associated group actions which grew out of a DMV-Seminar on the same subject. On the other hand it covers significantly more ground than was possible during the seminar in Oberwolfach and is in a certain sense intended as a systematic approach which ranges from the foundations of the subject to recent developments. It should be accessible to doctoral students and as well researchers coming from a wide range of areas. The initial chapters are devoted to a rather selfcontained introduction to group actions on complex and symplectic manifolds and to Borel-Weil theory in finite dimensions. These are followed by a treatment of the basics of infinite dimensional Lie groups, their actions and their representations. Finally, a number of more specialized and advanced topics are discussed, e.g., Borel-Weil theory for loop groups, aspects of the Virasoro algebra, (gauge) group actions and determinant bundles, and second quantization and the geometry of the infinite dimensional Grassmann manifold.

Measure and Integration Theory on Infinite-Dimensional Spaces

This book lays the foundations of differential calculus in infinite dimensions and discusses those applications in infinite dimensional differential geometry and global analysis not involving Sobolev completions and fixed point theory. The approach is simple: a mapping is called smooth if it maps smooth curves to smooth curves. Up to Fr‚chet spaces, this notion of smoothness coincides with all known reasonable concepts. In the same spirit, calculus of holomorphic mappings (including Hartogs' theorem and holomorphic uniform boundedness theorems) and calculus of real analytic mappings are developed. Existence of smooth partitions of unity, the foundations of manifold theory in infinite dimensions, the relation between tangent vectors and derivations, and differential forms are discussed thoroughly. Special emphasis is given to the notion of regular infinite dimensional Lie groups. Many applications of this theory are included: manifolds of smooth mappings, groups of diffeomorphisms, geodesics on spaces of Riemannian metrics, direct limit manifolds, perturbation theory of operators, and differentiability questions of infinite dimensional representations.

The book discusses the following topics in stochastic analysis: 1. Stochastic analysis related to Lie groups: stochastic analysis of loop spaces and infinite dimensional manifolds has been developed rapidly after the fundamental works of Gross and Malliavin. (Lectures by Driver, Gross, Mitoma, and Sengupta.)

This is the third version of a book on differential manifolds. The first version appeared in 1962, and was written at the very beginning of a period of great expansion of the subject. At the time, I found no satisfactory book for the foundations of the subject, for multiple reasons. I expanded the book in 1971, and I expand it still further today. Specifically, I have added three chapters on Riemannian and pseudo Riemannian geometry, that is, covariant derivatives, curvature, and some applications up to the Hopf-Rinow and Hadamard-Cartan theorems, as well as some calculus of variations and applications to volume forms. I have rewritten the sections on sprays, and I have given more examples of the use of Stokes' theorem. I have also given many more references to the literature, all of this to broaden the perspective of the book, which I hope can be used among things for a general course leading into many directions. The present book still meets the old needs, but fulfills new ones. At the most basic level, the book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. In differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differentiable maps in them (immersions, embeddings, isomorphisms, etc.).

This book contains the proceedings of the special session in honor of Leonard Gross held at the annual Joint Mathematics Meetings in New Orleans (LA). The speakers were specialists in a variety of fields, and many were Professor Gross's former Ph.D. students and their descendants. Papers in this volume present results from several areas of mathematics. They illustrate applications of powerful ideas that originated in Gross's work and permeate diverse fields. Topics include stochastic partial differential equations, white noise analysis, Brownian motion, Segal-Bargmann analysis, heat kernels, and some applications. The volume should be useful to graduate students and researchers. It provides perspective on current activity and on central ideas and techniques in the topics covered.