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The first edition of this book presented simple proofs of the Atiyah-Singer Index Theorem for Dirac operators on compact Riemannian manifolds and its generalizations (due to the authors and J.-M. Bismut), using an explicit geometric construction of the heat kernel of a generalized Dirac operator; the new edition makes this popular book available to students and researchers in an attractive softcover. The first four chapters could be used as the text for a graduate course on the applications of linear elliptic operators in differential geometry and the only prerequisites are a familiarity with basic differential geometry. The next four chapters discuss the equivariant index theorem, and include a useful introduction to equivariant differential forms. The last two chapters give a proof, in the spirit of the book, of Bismut's Local Family Index Theorem for Dirac operators.

The purpose of the present memoir is two-fold. First, the authors obtain a non-abelian localization theorem when M is any even dimensional compact manifold : following an idea of E. Witten, the authors deform an elliptic symbol associated to a Clifford bundle on M with a vector field associated to a moment map. Second, the authors use this general approach to reprove the $[Q,R] = 0$ theorem of Meinrenken-Sjamaar in the Hamiltonian case and obtain mild generalizations to almost complex manifolds. This non-abelian localization theorem can be used to obtain a geometric description of the multiplicities of the index of general $spin^c$ Dirac operators.

• A collection of research-oriented monographs, reports, notes arising from lectures or seminars • Quickly published concurrent with research • Easily accessible through international distribution facilities • Reasonably priced • Reporting research developments combining original results with an expository treatment of the particular subject area • A contribution to the international scientific community: for colleagues and for graduate students who are seeking current information and directions in their graduate and post-graduate work

This volume is devoted to the theme of Noncommutative Harmonic Analysis and consists of articles in honor of Jacques Carmona, whose scientific interests range through all aspects of Lie group representations. The topics encompass the theory of representations of reductive Lie groups, and especially the determination of the unitary dual, the problem of geometric realizations of representations, harmonic analysis on reductive symmetric spaces, the study of automorphic forms, and results in harmonic analysis that apply to the Langlands program. General Lie groups are also discussed, particularly from the orbit method perspective, which has been a constant source of inspiration for both the theory of reductive Lie groups and for general Lie groups. Also covered is Kontsevich quantization, which has appeared in recent years as a powerful tool. Contributors: V. Baldoni-Silva; D. Barbasch; P. Bieliavsky; N. Bopp; A. Bouaziz; P. Delorme; P. Harinck; A. Hersant; M.S. Khalgui; A.W. Knapp; B. Kostant; J. Kuttler; M. Libine; J.D. Lorch; L.A. Mantini; S.D. Miller; J.D. Novak; M.-N. Panichi; M. Pevzner; W. Rossmann; H. Rubenthaler; W. Schmid; P. Torasso; C. Torossian; E.P. van den Ban; M. Vergne; and N.R. Wallach

The AMS-IMS-SIAM Joint Summer Research Conference “Integer Points in Polyhedra—Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics” was held in Snowbird, Utah in June 2006. This proceedings volume contains research and survey articles originating from the conference. The volume is a cross section of recent advances connected to lattice-point questions. Similar to the talks given at the conference, topics range from commutative algebra to optimization, from discrete geometry to statistics, from mirror symmetry to geometry of numbers. The book is suitable for researchers and graduate students interested in combinatorial aspects of the above fields.

For more than five decades Bertram Kostant has been one of the major architects of modern Lie theory. Virtually all his papers are pioneering with deep consequences, many giving rise to whole new fields of activities. His interests span a tremendous range of Lie theory, from differential geometry to representation theory, abstract algebra, and mathematical physics. Some specific topics cover algebraic groups and invariant theory, the geometry of homogeneous spaces, representation theory, geometric quantization and symplectic geometry, Lie algebra cohomology, Hamiltonian mechanics, modular forms, Whittaker theory, Toda lattice, and much more. It is striking to note that Lie theory (and symmetry in general) now occupies an ever increasing larger role in mathematics than it did in the fifties. This is the first volume (1955-1966) of a five-volume set of Bertram Kostant's collected papers. A distinguished feature of this first volume is Kostant's commentaries and summaries of his papers in his own words.

All the papers in this volume are research papers presenting new results. Most of the results concern semi-simple Lie groups and non-Riemannian symmetric spaces: unitarisation, discrete series characters, multiplicities, orbital integrals. Some, however, also apply to related fields such as Dirac operators and characters in the general case.