Stable Cohomology: Foundations and Applications

该奖项支持参加将于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。该研讨会将汇集世界各地的专家和早期职业研究人员，探讨代数和表示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。该研讨会将导致更好地理解稳定上同调，其基础和应用。 因此，研讨会将有实质性的影响，在许多领域的研究，稳定的上同调可以应用。与此同时，研讨会是一个机会，为早期职业数学家学习专家在协作环境。进一步的细节是在会议的网站：http://www.math.ttu.edu/~lchriste/snowbird2018.htmlStable上同调可以追溯到泰特，谁在1952年宣布了一个新的上同调理论的有限群，以期对应用数论。它只稍微偏离有限群的经典（上）同调，但差异是决定性的：它本质上是将群同调和群上同调结合到一个理论中。泰特上同调已经沿着几个不同的轨道发展，并且在数学的几个分支中有应用，包括局部代数，奇点理论，表示论，同调代数和范畴代数。演化包括对其他群和环的推广，以及一些结合代数的Hochschild模拟。今天，人们有几个理论框架，根据上下文，处理稳定上同调。它们揭示了理论的不同方面，并且一些性质在一个框架中是明显的，而在另一个框架中是模糊的。研讨会的目的是汇集专家对同调代数和数学家谁适用于稳定的上同调在他们的研究，以解决这样一个广泛的问题：是否有一个统一的理论框架稳定的上同调，其中所有的属性自然出现？什么样的信息，确切地说，是编码在稳定的上同调？该奖项反映了NSF的法定使命，并被认为是值得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的支持。

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.