Frobenius Splitting in Algebraic Geometry, Commutative Algebra, and Representation Theory

代数几何、交换代数和表示论中的弗罗贝尼乌斯分裂

• 批准号: 0968646
• 负责人: Mircea Mustata
• 金额: $ 2.5万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-03-01 至 2012-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0968646&HistoricalAwards=false
• 关键词: Frobenius; Splitting; Algebraic; Geometry; Commutative

The property of a variety or ring being F-split (under mild conditions, equivalently F-pure) is an extremely powerful condition. Perhaps most famously, in the 1980s, these techniques were applied in the study of Schubert varieties, and continue to be actively used in the study of algebraic groups. On the other hand, in the 1970s, these techniques were used to prove fundamental results about rings of invariants by reductive groups. These methods anticipated the fundamental ideas behind the tight closure theory. In the 1990s, it was discovered that there is a precise dictionary between some of the notions coming from the minimal model program, and invariants defined by variants of Frobenius splitting and tight closure theory (a correspondence that is still not fully understood). Some of these methods are also related to the study of vector bundles in characteristic p, another active area of research which has had numerous applications.This conference that will take place at the University of Michigan, Ann Arbor, May 17th--May 22nd, 2010. The organizing committee consists of:M. Blickle (Universitat Duisburg-Essen), M. Brion (Universite de Grenoble), F. Enescu (Georgia State University), S. Kumar (University of North Carolina at Chapel Hill), M. Mustata (University of Michigan), K.Schwede (University of Michigan). The conference will focus on Frobenius splitting and related notions, methods, and applications to the following important areas of mathematics: the representation theory of algebraic groups, commutative algebra, and higher dimensional algebraic geometry. The conference will bring researchers together and stimulate communication between the various groups (communication which previously has been somewhat limited). It is expected that this conference will impact the mathematical community in a number of ways. Firstly, by exposing researchers to new potential applications of their own work and also to different points of view, the meeting will inspire new communication, collaboration and research. The participants of the conference will have different backgrounds, and thus many of the talks will necessarily be focused at a non-expert audience. Therefore, secondly, the talks given will be suitable for young mathematicians, especially graduate students and junior faculty. Finally, we also expect to attract other established researchers interested in learning about these techniques.

簇或环被 F 分裂（在温和条件下，相当于 F 纯）的性质是一个极其强大的条件。 也许最著名的是，在 20 世纪 80 年代，这些技术被应用于舒伯特簇的研究，并继续积极地用于代数群的研究。 另一方面，在 20 世纪 70 年代，这些技术被用来证明还原群关于不变量环的基本结果。 这些方法预见了紧闭理论背后的基本思想。 在 20 世纪 90 年代，人们发现来自最小模型程序的一些概念与 Frobenius 分裂和紧闭理论的变体定义的不变量之间存在精确的字典（这种对应关系尚未完全理解）。 其中一些方法还与特征 p 中向量束的研究有关，这是另一个活跃的研究领域，已拥有众多应用。本次会议将于 5 月 17 日至 5 月 22 日在密歇根大学安娜堡分校举行， 2010.组委会组成：M. Blickle（杜伊斯堡-埃森大学）、M. Brion（格勒诺布尔大学）、F. Enescu（乔治亚州立大学）、S. Kumar（北卡罗来纳大学教堂山分校）、M. Mustata（密歇根大学）、K.施韦德（密歇根大学）。会议将重点讨论弗罗贝尼乌斯分裂以及相关概念、方法和在以下重要数学领域的应用：代数群的表示论、交换代数和高维代数几何。该会议将把研究人员聚集在一起，并促进各个团体之间的交流（以前的交流有些有限）。 预计这次会议将以多种方式影响数学界。 首先，通过让研究人员接触到自己工作的新的潜在应用以及不同的观点，会议将激发新的沟通、合作和研究。 会议的参与者将有不同的背景，因此许多演讲必然会针对非专家观众。 因此，其次，演讲内容适合年轻数学家，特别是研究生和初级教师。 最后，我们还希望吸引其他有兴趣了解这些技术的知名研究人员。