Topics in infinite dimensional algebra

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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.