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.

一个簇或环的性质是F-分裂的（在温和条件下，等价于F-纯的）是一个非常强大的条件。 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，这些技巧被用来证明约化群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由弗罗贝纽斯分裂和紧闭理论的变体定义的不变量（一种尚未完全理解的对应关系）之间存在一个精确的字典。 这些方法中的一些也与特征p中的向量丛的研究有关，这是另一个活跃的研究领域，已经有了许多应用。这个会议将于2010年5月17日-5月22日在密歇根大学安阿伯举行。 组委会由以下人员组成：M. Blickle（Universitat Duisburg-Essen），M. Brion（Universite de Grenoble），F. Enescu（格鲁吉亚州立大学），S.库马尔（北卡罗来纳州查佩尔山大学），M。Mustata（密歇根大学），K.Schwede（密歇根大学）。会议将重点讨论Frobenius分裂和相关的概念，方法和应用到以下重要的数学领域：代数群的表示理论，交换代数和高维代数几何。这次会议将把研究人员聚集在一起，促进各个群体之间的交流（以前的交流是有限的）。 预计这次会议将影响数学界在许多方面。 首先，通过让研究人员了解他们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，因此许多会谈将必然集中在非专家观众。 因此，第二，讲座将适合年轻的数学家，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学习这些技术的知名研究人员。