Commutative Algebra: Singularities in All Characteristics with Geometric Applications

University Of Utah, Salt Lake City UT

Investigators

Abstract

This research project explores questions in commutative algebra. Commutative algebra is the study of solutions of polynomial equations, which are ubiquitous throughout science. One way to study polynomial functions is to view them instead in a simpler clock-arithmetic-like setting (where for example, 6+7 = 1, in other words, 7 hours after 6 o'clock is 1 o'clock). Translating between these two number systems has been a fruitful strategy for centuries, with applications on both sides; for example, this sort of mathematics is essential in modern secure communications systems. This project aims to develop tools to help work directly in what is called a "mixed characteristic" setting, which sits between the classical numerical world and the clock-arithmetic-like setting. These tools will help unify the classical and clock arithmetic settings. The investigator plans to develop packages for open source software and to write a book on these topics. The project includes training of graduate students through involvement in the research. This project aims to develop a theory of singularities in mixed characteristic commutative algebra, with an eye towards geometric applications. This theory will unify and generalize the singularities coming from the minimal model program in algebraic geometry with the singularities coming out of tight closure theory in commutative algebra. Recent advances in commutative algebra utilizing the theory of perfectoid algebras and spaces has made now the right time to develop this theory. These tools should be able to replace Kodaira-type vanishing theorems in some applications where the latter do not apply. Inspired by these connections, the investigator will also study characteristic 0 and characteristic p > 0 commutative algebra and algebraic geometry. 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.