Support for K-theory Conferences; 2003-2006

支持K理论会议；

• 批准号: 0303519
• 负责人: Daniel Grayson
• 金额: $ 3.1万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0303519&HistoricalAwards=false
• 关键词: Support; theory; Conferences; 2003; 2006

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.

摘要提案0303519：“K理论会议”埃里克M。Friedlander和丹尼尔R.格雷森这笔赠款将有助于支持四个研究会议在代数K理论在3年的时间。 一个会议，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资金将来自加拿大。 其他三个会议将是第10次，第11次和第12次在一系列的年度“五大湖K理论”周末会议发生在美国大学。 这些会议将集中讨论在解决长期未决问题方面令人振奋的进展和预期的新发展。 2002年8月，Voevodsky被授予菲尔兹奖在ICM在北京，以表彰他的基本贡献，其中产生了重要的结果模2：米尔诺猜想和布洛赫-加藤猜想。 Milnor定理的证明告诉我们一些关于域的显著和特殊的东西：模2的伽罗瓦上同调环由1度的显式生成元和2度的显式关系表示。 Rost和Voevodsky的工作已经在手上，预计将导致在奇素数上证明Bloch-Kato猜想，通过Suslin和Voevodsky的工作，这将意味着Beilinson猜想将motivic上同调与etale上同调联系起来，这反过来预计将导致使用Grayson，Friedlander和Suslin或Bloch，Lichtenbaum，Friedlander和Suslin的工作证明Quillen-Lichtenbaum猜想，将簇的代数K理论与动机上同调联系起来。 奎伦-利希滕鲍姆猜想自20世纪70年代初以来一直是K理论研究的驱动力，并且一直是几何、数论和拓扑学相关领域之间相互作用的焦点。 一个结果将是一个明确的计算的K-群的整数环，涵盖了p-主要部分的素数p的范德维尔猜想是已知的。 Vandiver猜想是数论中的一个古老猜想，已经通过对所有小于1200万的素数的计算进行了验证，最近Kurihara和Soule使用代数K理论在理论上取得了进展。 会议将支持这笔赠款将有助于传播戏剧性的新发展的主题，以及介绍这一重要的数学学科的下一代美国数学家。 由于K-理论的影响，在许多现代数学，我们预计，这些会议将作出重大贡献，美国数学界的努力，以保持其世界领导地位fundamentalmathematics. K-理论是一个相对较新的领域，数学已经成长和繁荣，在过去的40年。 人们现在发现K理论在数学物理中起着重要作用（例如，各种共形场论），向量空间上的群的经典作用，数论，尤其是数论。 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式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，这些平面或空间可能会随着一个人从一个点移动到另一个点而扭曲和转向，附近或远处，因此正确理解这些可能性需要使用拓扑学，研究逐渐变化。 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学。 人们研究了用具有一个或多个自由参数的新方程组来扩充原始方程组的可能方法。 由这笔赠款支持的会议将继续提供最高质量的研究会谈，以数学界，包括研究生和初级数学家在博士后职位的强大传统。 这些会议应鼓励新一代的年轻美国数学家参加各种研究计划有关代数K理论。由于许多问题尚未解决，该领域已经成熟，可以进一步取得令人兴奋的发展。