Topics in infinite dimensional algebra

具有大量参数的数学系统经常出现在数学和科学中。例如，在热力学中，人们可以用描述每个原子状态的参数来模拟具有大量原子的气体;或者在数学生物学中，人们可以考虑基因组的统计模型，其中每个碱基对都有参数。这样的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扮演着类似的角色。在过去的十年里，我们对这类系统的数学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。本计画将提供本科生、研究生及博士后之研究训练活动。此外，研究人员正在开发与本提案中的数学相关的临时资源，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）寡纯群的表示;（iv）无限维张量。前两个主题在过去的十年里一直是表征稳定性的流行主题，并有重要的应用（如阿廷猜想和斯蒂尔曼猜想的证明）。PI在这些领域的工作是以前工作的自然延续。对后两个主题的兴趣是最近的，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。该奖项反映了NSF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。

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.