CAREER: New Frontiers for Frobenius, Singularity Theory, Differential Operators, and Local Cohomology

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The field of commutative algebra provides a framework in which to study polynomial equations and their set of solutions. Given the prominent role of such equations in our society (e.g., in pure and applied math, computer science and technology, engineering, physics, biology, and chemistry), the applications of commutative algebra are broad and impactful. For example, commutative algebra is fundamental to cryptography, which many of us rely on daily, but also to genomics. This project advances the field of commutative algebra, and also trains junior scientists to develop skills relevant to a variety of scientific careers. This project also includes three initiatives focused on education, outreach and scientific leadership. These initiatives support research collaboration among women algebraists, implement an REU training program serving students from groups that are underrepresented in the STEM fields, and create new computer algebra software for research and education.This project aims to further our understanding of commutative rings and algebraic varieties using prime characteristic methods, differential operators, and local cohomology. In prime characteristic, one goal is to use the Frobenius map to study hypersurfaces by providing a better understanding of test ideals and Frobenius jumping exponents. The project also seeks effective algorithms for computing the Bernstein- Sato polynomial over the complex numbers. Another goal of the project is to investigate differential operators, and modules over them, in non-regular settings, and then apply this new theory (for example, to study multiplier ideals). Finally, the project proposes to investigate new geometric and topological properties determined by local cohomology, with an emphasis on connectedness properties of the irreducible components of a ring's spectrum.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.