Seminar Announcement
Date: Wednesday, 5 February 2025
Time: 3:30 PM
Venue: Seminar Hall
Recent developments in numerical invariants in prime characteristic commutative algebra
Alapan Mukhopadhyay
Ecole Polytechnique Federale de Lausanne (EPFL).
05-02-25

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Abstract

Hilbert-Kunz multiplicity, F -signature and F -threshold are some numerical invariants that quantify the severity of the singularity of a prime characteristic local ring. In the graded setup, the theory of Hilbert-Kunz multiplicity has recently witnessed two generalizations: Hilbert-Kunz density function and Frobenius-Poincar´e function. These two facilitate use of tools from projective geometry in the graded setting. Hilbert-Kunz density function has been recently used to answer several questions regarding the Hilbert-Kunz multiplicity. In this talk, we will discuss an extension of the theory of Hilbert-Kunz density function to any prime characteristic local ring. We will handle the additional challenges posed by the lack of grading by developing a theory of h-functions that we introduce. This h-function treats Hilbert-Kunz multiplicity, Hilbert-Samuel multiplicity and F -threshold on an equal footing. We will show some applications of the h-function highlighting this feature.