Support for K-theory Conferences; 2003-2006

0303519：“k -理论会议”Eric M. Friedlander和Daniel R. grayson这项资助将用于在3年内支持四个代数k理论研究会议。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。预计该会议的大部分经费将来自加拿大。另外三场会议将是在美国大学举行的“五大湖k理论”年度周末会议系列的第10、11和12场会议。这些会议将集中讨论在解决长期悬而未决的猜想方面令人振奋的进展和预期的新发展。2002年8月，沃沃斯基被授予菲尔兹奖，以表彰他的基础性贡献，他在模2中产生了重要的结果：米尔诺猜想和布洛赫-加托猜想。米尔诺猜想的证明告诉我们关于场的一些非凡的和特殊的东西：伽罗瓦上同环模2是由1次显式生成子和2次显式关系表示的。Rost和Voevodsky已经完成的工作有望导致奇素数下的Bloch- kato猜想的证明，而Suslin和Voevodsky的工作将暗示与动机上同调有关的Beilinson猜想，而Beilinson猜想反过来有望导致使用Grayson， Friedlander和Suslin的工作证明Quillen-Lichtenbaum猜想，或者Bloch， Lichtenbaum， Friedlander和Suslin的工作证明Quillen-Lichtenbaum猜想，将变量的代数k理论与动机上同调联系起来。自20世纪70年代初以来，Quillen-Lichtenbaum猜想一直是k理论研究的推动力，并且一直是几何，数论和拓扑学相关领域相互作用的焦点。一个结果将是整数环的k群的显式计算，包括已知Vandiver猜想的素数p的p个主要部分。Vandiver猜想是数论中的一个老猜想，已经通过对小于1200万的所有素数的计算进行了验证，Kurihara和Soule最近利用代数k理论在此基础上取得了理论进展。由该基金支持的会议将有助于传播该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，我们期望这些会议将对美国数学界保持其在基础数学领域的世界领导地位的努力做出重大贡献。k理论是一个相对较新的数学领域，在过去的40年里发展和繁荣。人们现在发现k理论在数学物理（例如，各种共形场论），群在向量空间上的经典作用，数论，特别是数论中起着重要作用。k理论是一种通过考虑将平面（或任何维度的空间）与每个解联系起来的可能方法来检验多项式方程系统特征的方法。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，或近或远，所以对这些可能性的正确理解需要使用拓扑学，即渐变的研究。动机上同调是另一种收集方程解信息的方法，它以不同的方式使用同伦理论和拓扑。一种是研究用具有一个或多个自由参数的新方程组来扩充原方程组的可能方法。该基金支持的会议将继续向数学界（包括研究生和博士后职位的初级数学家）提供最高质量的研究演讲的强大传统。这些会议应该鼓励新一代年轻的美国数学家参与代数k理论的各种研究项目。由于许多问题仍未解决，该领域进一步令人兴奋的发展已经成熟。

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.