Raoul Bott

Developed from a first-year graduate course in algebraic topology, this text is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory. The materials are structured around four core areas: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes. By using the de Rham theory of differential forms as a prototype of cohomology, the machineries of algebraic topology are made easier to assimilate. With its stress on concreteness, motivation, and readability, this book is equally suitable for self-study and as a one-semester course in topology.

K-theory.

These are the proceedings of the joint seminar by M.I.T. and Harvard on the current Developments in mathematics for the year 1997. Established in 1995, this seminar has been continued on the third weekend of November every year. The organizing committee for the seminar consisted of distinguished mathematicians from the mathematics departments of both Raoul Bott, Arthur Jaffe, and Shing-Tung Yau from Harvard, and David Jerison, George Lustig and Isadore Singer from M.I.T.. We trust that these proceedings will be of interest to many mathematicians, and will inspire future developments and research pursuits in mathematics. Papers Alain Connes, College de France, "Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function"Lawrence C. Evans, U.C. Berkeley, "Partial Differential Equations and Monge-Kantorovich Mass Transfer"Peter Sarnak, Princeton, "Quantum Chaos, Symmetry and Zeta Functions"W. Soergel, "Character Formulas for Tilting Modules over Quantum Groups at Roots of One"U. Frisch, Observatoire de Nice, "Is There Finite-Time Blow-Up in 3-D Euler Flow?"

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These are the terse notes for a graduate seminar which I conducted at Harvard during the Fall of 1963. By and large my audience was acquainted with the standard material in bundle theory and algebraic topology and I therefore set out directly to develop the theory of characteristic classes in both the standard cohomology theory and K-theory. Since 1963 great strides have been made in the study of K(X), notably by Adams in a series of papers in Topology. Several more modern accounts of the subject are available. In particular the notes of Atiyah, IINotes on K-theoryll not only start more elementarily, but also carry the reader further in many respects. On the other hand, those notes deal only with K-theory and not with the characteristic vii 406 viii classes in the standard cohomology. The main novelty of these lectures is really the systematic use of induced representation theory and the resulting formulae for the KO-theory of sphere bundles. Also my point of view toward the J -invariant, e(E) is slightly different from that of Adams. I frankly like my groups H Z+; KO(X)) and there is some indication that the recent work of Sullivan will bring them into their own.