Applying the Frobenius Morphism and Convexity to Study Singularities

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This project concerns the study of the set of solutions to certain systems of polynomials over the rational numbers. Polynomials are among the most basic and important functions in pure and applied mathematics, and come from iterating the basic functions of addition and multiplication on a set of variables. The set of solutions to polynomial equations can be interpreted geometrically, and many of the shapes appearing in nature such as spheres and ellipses, can be described using them. Polynomials and the geometric objects they describe are ubiquitous, both in pure and applied mathematics, as well as in statistics, engineering, and computer science. An important theme in this proposal is to study polynomial equations defined over the rational numbers via reduction to positive characteristic. Much of our approach involves the use of the Frobenius morphism, but also relies on basic notions from convex geometry (e.g., cones and polytopes). More precisely, we propose to 1) use Frobenius and convexity to study Bernstein-Sato polynomials; 2) use convexity and integration to investigate certain multiplicities in positive characteristic (e.g. F-signature and Hilbert-Kunz multiplicity); and 3) study the generalized Frobenius powers of an ideal, and relate them to test and multiplier ideals.