Studies in Commutative Algebra and Algebraic Geometry

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The Principal Investigator will work on several questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of many polynomial equations in many variables. Understanding this problem is of fundamental importance in many sciences, in engineering, and in other disciplines as well. It is often difficult to determine whether to expect any solution, finitely many solutions, or infinitely many: in the last case one wants to know how many degrees of freedom one has in describing the solutions. One can study the solutions geometrically, or algebraically, by investigating certain functions on the solution space that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry that is very valuable. The problems that will be studied are long standing and of fundamental, central importance. The results obtained will be disseminated by journal and book publications, lectures at conferences and workshops, and via the internet. There is a strong educational component. The principal investigator will continue to mentor graduate students, postdocs, high school students through several programs that he has established over the years with a particular attention to attracting students that underrepresented in the mathematical sciences.The Principal Investigator will continue exploring several long standing questions in the theory of Noetherian rings. Tigran Ananyan and the PI have proved Stillman's conjecture in all characteristics, but the bounds on projective dimension and on other properties, such as numbers of generators of ideals associated with primary decompositions, can certainly be improved, and this problem will be studied. The PI will also study whether there is an analogue of tight closure in mixed characteristic that has the certain standard properties: both the usual colon-capturing and Dietz's version of this property, the property that all ideals of regular rings are closed, persistence, and a test element theory. It is the final two that appear to be most difficult. It is expected that new methods from perfectoid geometry will be among the tools needed. Jointly with Sema Gunturkun, the PI is studying the long standing Eisenbud-Green-Harris conjecture on quadratic forms, which predicts minimum values for the Hilbert functions of certain ideals. Jointly with Bhargav Bhatt and Linquan Ma, the PI will continue to study the existence and properties of lim Cohen-Macaulay sequences of modules. This work, as well as some other asymptotic approaches, may resolve the long open question of whether Serre intersection multiplicities are positive, in general, in the case of mixed characteristic regular rings. Another direction for research includes the finiteness of support and other properties of local cohomology, such as the faithfulness of the highest non-vanishing local cohomology module over a local domain with support in a specified ideal. The PI has a continuing project with Jack Jeffries exploring the latter problem. A number of these problems are intended for collaboration with graduate students and postdoctoral faculty.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.