Commutative Algebra: F-Regularity in Algebraic Geometry and Non-Commutative Algebra

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This project investigates polynomials and the geometric shapes they define, called algebraic varieties. The problems belong to the field of mathematics called algebraic geometry. Algebraic geometry has many applications throughout industry and national security, including to error-correcting codes, machine learning, cryptography, computer aided design, and 3D printing. Some of these applications are concerned with studying the sets where polynomials are zero module certain prime numbers, the particular focus of this project. This project is basic research into understanding the singularities of algebraic varieties defined over finite fields, including criteria for understanding how far the varieties are from being smooth, and techniques for circumventing the failure of smoothness in some cases. The project includes research designed to be student-ready to provide training for the next generation of mathematicians.In more precise terms, the project investigates strong F-regularity in several contexts in non-commutative algebra, combinatorics, and birational algebraic geometry. It investigates whether or not, for a strongly F-regular local ring R of prime characteristic, the algebra A of R-linear endomorphisms of F*R, where F is the Frobenius map, is a non-commutative resolution of singularities in the sense of Van den Bergh. In addition, a theory of derived functors of differential operators will be investigated, with an eye toward using it to show that rings of differential operators over fields of characteristic zero can be reduced to characteristic p under some conditions. Finally, building on recent work in valuation rings, the project aspires to show that certain section rings of varieties are always finitely generated, an important step in the minimal model program in characteristic p.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.