Uniform phenomena in commutative algebra

This project aims to bring to light hidden similarities between objects that are studied in the fields of commutative algebra, algebraic geometry and arithmetic geometry. These are some of the oldest fields that have been continuously studied in mathematics, and they have broad ranging applications such as in cryptography and in physical, social and biological sciences. At a fundamental level, these fields aim to understand the behavior of shapes that can be described using polynomial equations. The behavior of the shapes are, in turn, influenced by algebraic properties of the functions on them. The broad goal of this project is to establish results that exhibit unexpected uniform behavior among these algebraic objects. A particular focus is on using techniques that arise from modular, or “clock”, arithmetic. This project also includes outreach to the community, opportunities for students, and conference organization.

The discovery of unexpected uniform behavior in Noetherian rings has been a central theme in commutative algebra over the past fifty years. Among all Noetherian rings, there is an important class consisting of those considered to be the most geometric, known as excellent rings. Several uniformity results such as uniform Artin-Rees and uniform Briançon-Skoda remain open for the class of excellent rings in general, and in the mixed characteristic setting even for finite type algebras over a discrete valuation ring. The principal investigator will use techniques from prime characteristic commutative algebra, non-Archimedean functional analysis and recent advances in mixed characteristic singularity theory to tackle these open problems for new classes of excellent rings, especially those arising in an arithmetic setting. With the recent prolific use of arithmetic techniques in commutative algebra and algebraic geometry, the study of such rings is becoming increasingly important. In addition, these rings have also helped clarify how some fundamental notions of singularities defined via the Frobenius map behave outside the settings of finite type algebras over a field, and rings with finite Frobenius.

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.

Award Number: 2502333
Principal Investigator: Rankeya Datta
Funds Obligated: $116,251
State: MO