Algebraic and Geometric Properties of Multiplicities and the Frobenius Endomorphism

This project lies in the field of Commutative Algebra, which provides the local language of Algebraic Geometry, offering a powerful framework for studying local solutions to systems of polynomial equations, as well as more general systems that can be approximated by such equations. This project includes opportunities for students, and outreach to local and academic communities. The field of Commutative Algebra is foundational not only to pure mathematics but also to a wide range of applied disciplines, including computer science, physics, and engineering. It is closely intertwined with other areas of mathematics, such as Number Theory, Complex Analysis, Topology, Geometry, and Representation Theory. The Principal Investigator specializes in singularity theory, a branch of Commutative Algebra concerned with the behavior of algebraic objects that fail to be smooth. Unlike smooth structures, where the tools of differential calculus apply directly, singularities require alternative methods. One such method involves Rees valuations, which generalize the concept of tangent spaces and provide an algebraic mechanism for understanding local behavior near singular points. These valuations correspond to ideal blowups, a geometric construction central to singularity theory. This project will focus on the role of Rees valuations in understanding singularities, especially in settings of positive characteristic, an arithmetic context where classical tools of differentials often break down. The Principal Investigator will investigate how concepts like multiplicity, valuation theory, and S2-ification (a type of algebraic approximation) interact in this setting. The work aims to address fundamental conjectures in the field, including the Weak Implies Strong Conjecture and the LC Conjecture, which are central to understanding the behavior of singularities in positive characteristic. A key component of the project is the study of symbolic powers of ideals, particularly in singular rings. The Principal Investigator will develop and apply new techniques to advance the Uniform Symbolic Topology Property, which connects the algebraic properties of ideals with the underlying geometry of the space they define. Ultimately, the goal is to extend known theorems from smooth settings to singular ones, thereby deepening our understanding of singularities and their applications across mathematics and the sciences. 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.