Support for K-theory Conferences; 2003-2006

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Abstract Proposal 0303519: "K-theory Conferences" Eric M. Friedlander and Daniel R. GraysonThis grant will contribute to the support of four research conferences in algebraic K-theory over a 3-year period. One conference, a 5-day conference, is to take place in Canada at the Centre de Recherches Mathematiques on the campus of the Universite de Montreal. The majority of the funding for that conference is expected to come from Canadian sources. The other three conferences will be the 10th, 11th, and 12th in the series of annual "Great Lakes K-theory" weekend conferences occurring at American universities. These conferences will focus on exhilarating progress and expected new developments in settling long-open conjectures. In August, 2002, Voevodsky was awarded the Fields Medal at the ICM in Beijing in recognition for his fundamental contributions, which have yielded important results modulo 2: the Milnor conjecture and the Bloch-Kato conjecture. The proof of the Milnor conjectures tells us something remarkable and special about fields: that the Galois cohomology ring modulo 2 is presented by explicit generators in degree 1 and explicit relations in degree 2. Work of Rost and Voevodsky already in hand is expected to lead to the proof of the Bloch-Kato conjecture at odd primes, which by work of Suslin and Voevodsky will imply the Beilinson conjecture relating motivic cohomology to etale cohomology, which in turn is expected to lead to a proof of the Quillen-Lichtenbaum conjecture using work of Grayson, Friedlander, and Suslin or of Bloch, Lichtenbaum, Friedlander, and Suslin, that relates algebraic K-theory for varieties to motivic cohomology. The Quillen-Lichtenbaum conjecture has been the driving force for research in K-theory since the early 1970s, and has been the focus of the interplay between related areas of geometry, number theory and topology. A consequence will be an explicit computation of the K-groups of the ring of integers, covering the p-primary parts for prime numbers p for which the Vandiver conjecture is known. The Vandiver conjecture is an old conjecture from number theory, has been checked by computations for all prime numbers smaller than 12 million, and recent theoretical progress using algebraic K-theory has been made on it by Kurihara and Soule. The conferences to be supported by this grant will contribute to the dissemination of dramatic new developments in the subject as well as introduce this important mathematical subject to the next generation of American mathematicians. Because of K-theory's influence in much of modern mathematics, we expect that these conferences will make a significant contribution to the American mathematical community's efforts to maintain its world leadership in fundamentalmathematics.K-theory is a relatively new field of mathematics which has grown and prospered in the past 40 years. One now finds that K-theory plays an important role in mathematical physics (e.g., various conformal field theories), classical actions of groups on vector spaces, number theory, and especially number theory. K-theory is a way of examining features of systems of polynomial equations by considering the possible ways to associate flat planes (or spaces of any dimension) to each solution. As a concrete example, imagine that the solutions are the points on the surface of the earth, and for each point consider the plane containing that point and the horizon. In general, these planes or spaces may twist and turn as one moves from one point to another, nearby or far away, so a proper understanding of the possibilities requires the use of topology, the study of gradual change. Motivic cohomology is another way to glean information about solutions of equations that uses homotopy theory and topology in a different way. One examines the possible ways to augment the original system of equations by new ones that have one or more free parameters. The conferences supported by this grant will continue a strong tradition of delivering the highest quality research talks to the mathematical community, including graduate students and junior mathematicians in postdoctoral positions. These conferences should encourage a new generation of younger American mathematicians to participate in various research programs concerning algebraic K-theory. With many issues still unsettled, the field is ripe for further exciting developments.