Speaker: Dr.Kriti Goel

Title of the talk: Local theory of prime characteristic singularities - an algebraic viewpoint

Date ,Time & Venue: 15 July 2025, 11:00 AM, Seminar Hall, Dept. of Mathematics

Abstract: In this talk, we present some tools to study singularities of a ring in the prime characteristic case. These include the study of closure operations on ideals and associated Hilbert functions. Singularities defined in terms of the behavior of the Frobenius endomorphism on the ring are called 'F-singularities'. We study tight closure, a closure operation on ideals introduced by M. Hochster and C. Huneke in 1986, and the Hilbert-Kunz function of ideals and their blowup algebras. Hilbert-Kunz functions were introduced by E. Kunz in 1969 in his work characterizing regular local rings in the prime characteristic setting. In the process, we introduce the concept of tight Hilbert polynomials, whose coefficients help characterize F-rational singularities, and explore the behavior of Hilbert-Kunz functions in various classes of rings, including face rings of simplicial complexes.

Finally, we consider a finite group acting on a polynomial ring over a field. We study the connection between the algebraic properties of the invariant ring and the properties of the group action in the prime characteristic setting; the focus being the local cohomology modules of the invariant ring. 

About the Speaker: Dr. Kriti Goel completed her Ph.D. from IIT Bombay. Her research area is Commutative Algebra and its interactions with Algebraic Geometry and Combinatorics.  She is joining the University of Missouri, USA, as a Postdoctoral Fellow in August 2025, after completing a postdoc stint at the Basque Center for Applied Mathematics (BCAM), Spain (February 2023 - February 2025). Before moving to Spain, she was a Fulbright-Nehru Postdoctoral Research Fellow at the University of Utah, USA (January 2022 - January 2023) and Early Career Fellow at IIT Gandhinagar, India (October 2020 - October 2021).