Stable Cohomology: Foundations and Applications

稳定上同调：基础和应用

• 批准号: 1804126
• 负责人: Mark Walker
• 金额: $ 3.5万
• 依托单位: University of Nebraska-Lincoln
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-03-01 至 2019-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1804126&HistoricalAwards=false
• 关键词: Stable; Cohomology; Foundations; Applications

This award supports participation in a 5-day workshop to be held May 28-June 1, 2018 in Snowbird, Utah. The workshop will bring together world experts and early career researchers to explore some fundamental questions in algebra and representation theory, subjects that are important in many scientific and mathematical settings. The workshop will lead to a better understanding of stable cohomology, its foundations, and its applications. The workshop will thus have substantial impact on research in the many fields to which stable cohomology can be applied. At the same time, the workshop is an opportunity for early career mathematicians to learn from experts in a collaborative environment. Further details are on the conference website: http://www.math.ttu.edu/~lchriste/snowbird2018.htmlStable cohomology goes back to Tate, who in 1952 announced a new cohomology theory for finite groups, with a view towards applications in number theory. It deviates only slightly from classical (co)homology of finite groups, but the difference is decisive: it essentially combines group homology and group cohomology into one theory. Tate cohomology has since evolved along several different tracks, and there are applications in several branches of mathematics, including local algebra, singularity theory, representation theory, and homological and categorical algebra. Evolutions include generalizations to other groups and rings, as well as a Hochschild analogue for some associative algebras. Today one has several theoretical frameworks, depending on context, for treating stable cohomology. They expose different facets of the theory, and some properties are as evident in one framework as they are obscure in another. The purpose of the workshop is to bring together experts on homological algebra and mathematicians who apply stable cohomology in their research to address such broad questions as: Is there a unifying theoretical framework for stable cohomology in which all its properties emerge naturally? What kind of information, exactly, is encoded in stable cohomology?This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该奖项支持参加2018年5月28日至6月1日在犹他州雪鸟举行的为期5天的研讨会。研讨会将汇集世界专家和早期职业研究人员，探讨代数和表示理论中的一些基本问题，这些问题在许多科学和数学环境中都很重要。这次研讨会将有助于更好地理解稳定上同学及其基础和应用。因此，讲习班将对可以应用稳定上同论的许多领域的研究产生重大影响。与此同时，研讨会为早期职业数学家提供了一个在协作环境中向专家学习的机会。进一步的细节可以在会议网站上看到：http://www.math.ttu.edu/~lchriste/snowbird2018.htmlStable上同调可以追溯到Tate，他在1952年宣布了一个新的有限群上同调理论，以期在数论中的应用。它与有限群的经典（co）同调只有轻微的差别，但差别是决定性的：它实质上把群同调和群上同调合二为一。自那以后，Tate上同调沿着几个不同的轨道发展，在数学的几个分支中有应用，包括局部代数、奇点理论、表示理论、同调代数和范畴代数。进化包括对其他群和环的推广，以及对一些结合代数的Hochschild模拟。今天，根据不同的背景，人们有几种理论框架来处理稳定上同。它们揭示了理论的不同方面，有些性质在一个框架中很明显，而在另一个框架中却很模糊。研讨会的目的是将同调代数的专家和在研究中应用稳定上同调的数学家聚集在一起，以解决以下广泛的问题：是否存在一个稳定上同调的统一理论框架，使其所有性质自然出现？确切地说，什么样的信息是用稳定上同调编码的？该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。