Symbolic Powers and p-Derivations

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This is a project in commutative algebra, with connections to algebraic geometry, combinatorics, and arithmetic geometry. The research centers on the theme of symbolic powers, an active topic of research with connections to virtually all aspects of commutative algebra. This project aims to resolve two questions, on comparing the geometric notion of symbolic powers with the algebraic notion of natural powers, and on the application of the differential algebraic notion of p-derivations to commutative algebra. The investigator will direct undergraduate research experiences and organize a graduate workshop in commutative algebra.The unifying theme of this project is the study of symbolic powers of ideals. The research focuses primarily on two questions: the containment problem and applications of p-derivations to commutative algebra. While ideals in a polynomial ring correspond to the polynomials that vanish on a certain variety in affine space, their symbolic powers consist of the polynomials that vanish to a certain order on the given variety. This natural geometric notion has an algebraic description coming from the theory of primary decomposition, an ideal-theoretic version of the fundamental theorem of arithmetic. This is an old, rich, and ubiquitous topic, yet there is an abundance of questions about symbolic powers that are both easy to phrase and very difficult to solve. The containment problem attempts to compare symbolic powers, a natural geometric notion, with ordinary powers, a natural algebraic notion. This relates to other interesting questions, such as determining the minimal degrees of polynomials vanishing on a given variety. Another part of the project is to discover new connections between p-derivations and commutative algebra, with an eye towards a new singularity theory in mixed characteristic that combines p-derivations with differential operators in the spirit of the theory of F-singularities and its connections to D-modules.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.