Singularities in Positive and Mixed Characteristic Commutative Algebra

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Commutative algebra studies Noetherian commutative rings, examples of which are sets of polynomials and the ordinary integers. It is used in a wide variety of applied settings, ranging from error-correcting codes in computer science and genomics to control theory and modeling in engineering. Other fields of mathematics that use commutative algebra, and commutative rings in particular, are algebraic and arithmetic geometry, as well as complex analysis, topology, and representation theory. Execution of the projects in this grant will both advance theoretical understanding in commutative algebra and shed light on important problems from some of these related fields. In addition, the principal investigator is dedicated to promoting mathematics education, developing future generations of researchers, and assisting in building a strong STEM workforce in the US. Towards these goals, the PI will supervise, train, and mentor doctoral students and postdoctoral fellows throughout the course of the grant. The PI will also facilitate a number of seminars, workshops, and reading courses for undergraduate and graduate students.The PI specializes in the study of singularities in positive and mixed characteristic commutative algebra, and will address a number of questions arising naturally from recent developments in the theory of Frobenius splittings, tight closure, the homological conjectures, and absolute integral closure. The PI aims to exploit the connection between singularities defined via the Frobenius map in positive characteristic and those arising in complex algebraic geometry in order to carry out research projects in three new directions. First, the PI will investigate properties of the dual F-signature limit in characteristic p 0, showing its existence and deriving new properties with applications to F-rationality. Second, also in positive characteristic, the PI will further the understanding of F-singularities outside of the F-finite setting by exploring quotients of excellent regular rings and generalizing adjunction statements. Lastly, leveraging breakthrough results on the absolute integral closure in mixed characteristic, the PI will attack open questions regarding the splinter condition and exhibit new geometric applications.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.