Topics in infinite dimensional algebra

无限维代数主题

• 批准号: 2301871
• 负责人: Andrew Snowden
• 金额: $ 36.5万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2301871&HistoricalAwards=false
• 关键词: Topics; infinite; dimensional; algebra

Mathematical systems with very large numbers of parameters appear frequently within mathematics and science. For example, in thermodynamics one might model a gas with a large number of atoms, with parameters describing the state of each atom; or in mathematical biology, one might consider a statistical model of a genome, where there are parameters for each base pair. Such systems often exhibit a large amount of symmetry; for instance, in the gas system, the positions of the various atoms play a similar role to one another in the equations governing the system. The last decade has seen major progress in our understanding of the mathematics of such systems. This project aims to continue this progress on four specific fronts. This project will provide research training activities for undergraduates, graduate students, and post-docs. Additionally, the investigator is developing expository resources related to the mathematics in this proposal that will be available to the general public.The four areas of focus of this project are: (i) representations of categories; (ii) equivariant commutative algebra; (iii) representations of oligomorphic groups; and (iv) infinite dimensional tensors. The first two topics have been a prevalent theme in representation stability for the last decade, and have had important applications (such as proofs of the artinian conjecture and Stillman's conjecture). The PI's work in these areas is a natural continuation of previous work. Interest in the latter two topics is more recent, and based on exciting new connections to tensor categories, analytic number theory, and model theory. The PI's work in these areas will study these connections in detail.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.

具有大量参数的数学系统经常出现在数学和科学领域。例如，在热力学中，人们可以对具有大量原子的气体进行建模，并使用描述每个原子状态的参数；或者在数学生物学中，人们可能会考虑基因组的统计模型，其中每个碱基对都有参数。此类系统通常表现出大量的对称性。例如，在气体系统中，各种原子的位置在控制系统的方程中起着相似的作用。过去十年，我们对此类系统数学的理解取得了重大进展。该项目旨在在四个具体方面继续取得进展。该项目将为本科生、研究生和博士后提供研究培训活动。此外，研究人员正在开发与本提案中的数学相关的说明性资源，这些资源将提供给公众。该项目的四个重点领域是：(i) 类别的表示； (ii) 等变交换代数； (iii) 寡晶基团的表示； (iv) 无限维张量。前两个主题是过去十年表示稳定性中的一个普遍主题，并且具有重要的应用（例如阿蒂尼安猜想和斯蒂尔曼猜想的证明）。 PI 在这些领域的工作是先前工作的自然延续。对后两个主题的兴趣是最近才出现的，并且基于与张量范畴、解析数论和模型论的令人兴奋的新联系。 PI 在这些领域的工作将详细研究这些联系。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。