Singularities and Multiplicities in Commutative Algebra

Purdue University, West Lafayette IN

Investigators

Abstract

The PI plans to study questions in commutative algebra. This is a field that deals with the study of algebraic varieties, that is, the solution set of a system of polynomial equations in several variables. For example, the parabola defined by the equation y=x^2 in the plane is smooth, meaning that locally, at any point, it looks like a line, while the curve defined by the equation y^2=x^3 is not smooth at the origin (it looks like a cusp near the origin). The singular or non-smooth points of an algebraic variety have very rich algebraic and geometric structures, and investigating their properties has many applications in other sciences and engineering. The projects that will be explored are the study of singular points of algebraic varieties (i.e., singularities), with a focus on measuring the singularities (e.g., assigning some invariant to compare them with each other) using the theory multiplier/test ideals and multiplicities. The PI plans to develop a mixed characteristic singularity theory that is parallel to singularities in the minimal model program and tight closure theory from equal characteristic. This is motivated by the recent breakthrough of Yves Andre that solved Hochster's long-standing direct summand conjecture and the existence of big Cohen-Macaulay algebras. Together with Karl Schwede, the PI utilizes Andre`'s techniques from perfectoid algebras and spaces to start building such a theory, introducing a perfectoid version of multiplier/test ideals. There will be applications on the study of symbolic powers in mixed characteristic, as well as the study of family of singularities when the characteristic varies. Another research direction is the Hilbert-Samuel multiplicities attached to singularities, with a focus on Lech's conjecture, Lech's inequality, and the Stuckrad--Vogel invariant. The PI has settled the dimension three equal characteristic case of the longstanding Lech's conjecture, and has recently settled the Stuckrad--Vogel conjecture (together with Klein, Pham, Smirnov and Yao). The proposed projects include attacking the higher dimensional case of Lech's conjecture by studying and improving the classical Lech's inequality. A related project is to investigate the relationship between multiplicity and colength for sufficiently deep 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.