Measuring singularities in commutative algebra

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This project involves questions in the theory of commutative algebra. This is a field that deals with the local properties of algebraic varieties, i.e., the solution set of a system of polynomial equations in several variables. For example, the solution set of a single polynomial equation in two variables can be realized as a curve in the plane (e.g., a parabola y=x^2). The singular or non-smooth points of an algebraic variety have rich algebraic and geometric structures, and detailing their properties is crucial in many investigations. For example, the parabola is nonsingular (meaning that locally it looks like a line), while the curve defined by y^2=x^3 is nonsingular except at the origin, where it locally looks like a cusp. The projects that will be explored are focused on the singular points of algebraic varieties (i.e., singularities), with a focus on measuring singularities using various algebraic techniques. The PI will involve his graduate students and post-docs in this research project.The PI will construct a mixed characteristic singularity theory in collaboration with other experts in this area. One focus is to develop a mixed characteristic version of test ideals. Projects include studying their behaviors under localization and completion, and their connections to multiplier ideals from birational geometry. Another focus is on perfectoid signature and perfectoid Hilbert-Kunz multiplicity, which are defined using the perfectoidization functor of Bhatt-Scholze and are inspired by F-signature and Hilbert-Kunz multiplicity from positive characteristic. Projects include understanding the behavior of these numerical invariants under localization and in families. The PI will also continue the study on Hilbert-Samuel multiplicities, with a focus on the longstanding Lech's conjecture. The proposed projects include attacking the three dimensional mixed characteristic case, and exploring the existence of lim Ulrich sequences beyond the graded case. The PI will also investigate the existence of lim Cohen-Macaulay sequences and their variations to attack the longstanding Serre's conjecture on intersection multiplicities.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.