CAREER: Differential Operators and p-Derivations in Commutative Algebra

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Understanding the solutions of polynomial equations is a fundamental problem with applications throughout the basic sciences. Over the most familiar number systems—real numbers and complex numbers—one can think of these solutions sets geometrically via Cartesian coordinates. It is important for various reasons to also consider polynomial equations over more exotic number systems. For example, computer data and arithmetic are built on a system of two numbers (zero and one). The main research goal of this project is to develop new tools to study the small-scale behavior of systems of polynomial equations, both over familiar number systems (of characteristic zero) and over more exotic number systems (of positive or mixed characteristic). Specifically, the PI aims to extend aspects of the theory of differential operators for smooth varieties of characteristic zero, such as holonomicity and Bernstein-Sato theory, to singular varieties in characteristic zero and positive characteristic. The PI will also apply Buium and Joyal's notion of p-derivation to the study of singularities in mixed characteristic. The main educational goals of this project are to increase participation in mathematics by high school students and members of the growing Hispanic community in Nebraska, to foster international academic relationships among graduate students, particularly between the United States and Mexico, to create connections between traditionally different areas of mathematics themed around emerging problems, and to train graduate students, both directly and more broadly through dissemination of pedagogical materials. These will be pursued in a range of activities, including the formation of a high school math circle, cross-topical graduate workshops, and visits from senior and junior mathematicians from Latin America.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.