Studies in Commutative Algebra and Algebraic Geometry

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Algebraic geometry studies solutions of families of polynomial equations. One can study the solution set geometrically, using higher dimensional analogues of the graph of an equation, or algebraically, by investigating the behavior of functions on the geometric solution set that form what is called a commutative ring. This provides a valuable dual perspective. The project will yield a deeper understanding of central, fundamental notions in this area, several of which have been used recently to make progress on long-standing conjectures. The results will give both quantitative and qualitative information about the nature of the solution sets of equations and are expected to lead to significant progress on a number of long-standing questions. The project is multi-faceted and will provide many opportunities for collaboration with graduate students and postdoctoral faculty that will foster their growth as researchers. Some portions of the project will be suitable for training undergraduates to do research.The project will explore applications of the strength of a polynomial. This notion of strength was recently introduced by the PI, in joint work with a collaborator, and proved to be a critical element in their proof of Stillman's conjecture. The project will use polynomial strength to answer several remaining questions, for example, about obtaining bounds for primary decomposition--independent of the number of variables--that are as sharp as possible. Another direction is to use ideas from perfectoid geometry to construct a tight closure theory, valid in all characteristics, that has both persistence and a satisfactory theory of test elements. Perfectoid techniques have already led to a great deal of progress in this area. The PI will also continue to work on the theory of lim Cohen-Macaulay modules, aimed at resolving a long-standing conjecture about the behavior of intersection multiplicities. Other directions include the study of: finiteness of minimal primes of local cohomology modules; filtration theorems for local cohomology that can be used to investigate strongly F-regular rings; a long-standing conjecture of Eisenbud, Green, and Harris on the behavior of Hilbert functions of ideals in polynomial rings; and the uniform comparison of ordinary and symbolic powers of ideals.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.