The PI plans to 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 proposed for study are long standing and of fundamental, central importance. The results obtained will be disseminated by journal and book publication, lectures at conferences and workshops, and via the internet. There is a strong educational component. The principal investigator has had thirty-nine Ph.D. students including fourteen women (and has five other Ph.D. students currently, including four women, one of whom is African-American), and has served as mentor to fourteen junior faculty members, of whom six were women. This level of activity will continue. The PI will integrate undergraduate students into his research.
The PI will investigate several long standing questions in the theory Noetherian rings. One is to prove Stillman's conjecture bounding projective dimension, which Tigran Ananyan and Hochster have done in characteristic not 2, 3 for degree at most four. Another is to continue the development of tight closure theory: Neil Epstein and Hochster have developed a new version with many of the properties of the original theory that gives a smaller closure, is defined in both characteristic p and equal characteristic 0, and commutes with localization. This theory raises new questions while offering insight into existing ones. Jointly with Bhargav Bhatt, Hochster has given a new proof of the positivity of Serre intersection multiplicities over regular rings in positive characteristic using an idea that has promise for solving the more than fifty year old problem of settling the general case in mixed characteristic. Other directions include the relatively recent theory of quasilength and content of local cohomology, finiteness of support of local cohomology, Lech's conjecture and generalizations, a new approach to the long standing direct summand conjecture, and the geometry of certain algebraic sets associated with matrices. A number of these problems are intended for collaboration with graduate students and postdoctoral faculty.
