Support for K-theory Conferences; 2003-2006

摘要建议0303519：“K理论会议”Eric M.Friedlander和Daniel R.Grayson这笔赠款将在三年内资助四个代数K理论研究会议。其中一个为期5天的会议将在加拿大蒙特利尔大学校园内的数学研究中心举行。该会议的大部分资金预计将来自加拿大。其他三个会议将是在美国大学举行的年度“五大湖K理论”周末会议系列中的第10、11和12次。这些会议将聚焦于在解决长期悬而未决的猜测方面取得的令人振奋的进展和预期的新发展。2002年8月，沃沃茨基在北京被授予菲尔兹奖，以表彰他的基本贡献，这些贡献产生了模2的重要结果：米尔纳猜想和布洛赫-加藤猜想。Milnor猜想的证明告诉我们关于域的一些值得注意和特殊的事情：Galois上同调模2是由1次显式生成元和2次显式关系表示的。Rost和Voevodsky已经在手的工作有望导致在奇素数处证明Bloch-Kato猜想，Suslin和Voevodsky的工作将暗示Beilinson猜想将基元上同调与上同调联系起来，这反过来又有望导致Quillen-Lichtenbaum猜想的证明使用Grayson，Friedlander和Suslin or Bloch，Lichtenbaum，Friedlander和Suslin的工作来证明Quillen-Lichtenbaum猜想。自20世纪70年代初以来，Quillen-Lichtenbaum猜想一直是K-理论研究的驱动力，并一直是几何、数论和拓扑学相关领域相互作用的焦点。一个结果将是对整数环的K-群的显式计算，包括已知Vandiver猜想的素数p的p-初等部分。Vandiver猜想是数论中的一个古老猜想，对所有小于1200万的素数的计算都得到了验证，栗原和Soule利用代数K理论对其进行了最新的理论研究。由这笔赠款支持的会议将有助于传播这一学科的戏剧性新发展，并将这一重要的数学学科介绍给下一代美国数学家。由于K理论在许多现代数学中的影响，我们预计这些会议将为美国数学界保持其在基础数学领域的世界领先地位做出重大贡献。K理论是一个相对较新的数学领域，在过去的40年里得到了发展和繁荣。人们现在发现，K-理论在数学物理(例如，各种保形场论)、群在向量空间上的经典作用、数论，特别是数论中扮演着重要的角色。K-理论是一种通过考虑将平面(或任何维度的空间)与每个解相关联的可能方法来检查多项式方程组的特征的方法。作为一个具体的例子，假设解是地球表面上的点，对于每个点，考虑包含该点和地平线的平面。一般来说，当一个人从一个点移动到另一个点时，这些平面或空间可能会扭曲和转动，无论是近距离还是远距离，因此正确理解这种可能性需要使用拓扑学，即研究渐变。Motivic上同调是另一种收集有关方程解的信息的方法，它以不同的方式使用同伦理论和拓扑学。人们研究了用具有一个或多个自由参数的新的方程系统来扩充原始方程系统的可能方法。由这笔赠款支持的会议将继续向数学界提供最高质量的研究演讲的强大传统，包括研究生和博士后职位的初级数学家。这些会议应该会鼓励新一代年轻的美国数学家参与有关代数K理论的各种研究计划。由于许多问题仍然悬而未决，这一领域已经成熟，可以进行更令人兴奋的发展。

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.