Purdue CA Seminar

Spring 2026

Wednesday, January 21

Speaker: Reinhold Huebl (Resensburg)
Title: Equivalence of curve singulariies and δ-invariants
Abstract: This talk is about joint work with I. Swanson. A longstanding question in algebraic geometry is the classification of reduced and irreducible local complete one-dimensional domains R over an algebraically closed field k. It is known that such a ring is completely determined once it is known up to a “sufficiently high” power of its maximal ideal, where this sufficiently high power depends on the singularity degree δ of the ring.

In this talk we show that two parametrizations of curves parametrize isomorphic curve singularities if they agree up to degree 2δ + 2, strengthening a result of Greuel and Pfister, and that two curve singularities are already isomorphic if they are isomorphic mod m^{2δ + 2}, strengthening a result of Hironaka.

Wednesday, February 4

Speaker: Manav Batavia (Purdue)
Title: A sharp upper bound on cohomological dimension in mixed characteristic
Abstract: Given an ideal I in a regular local ring A, the cohomological dimension of I in A is the index of the highest non-vanishing local cohomology of A supported at I. Determining effective upper bounds on the cohomological dimension in terms of topological invariants of Spec(A/I) is a central problem in commutative algebra: foundational results include the Hartshorne--Lichtenbaum Vanishing Theorem and the Second Vanishing Theorem.

In equal characteristic, Faltings established in 1980 a general bound on the cohomological dimension of an ideal in terms of its “big height”. In this talk, we extend Faltings’ results to the unramified mixed characteristic setting and show that the resulting bound is sharp.

Wednesday, February 11

Speaker: Kangjin Han (DGIST, Korea)
Title: TBA
Abstract: TBA

Wednesday, February 25

Speaker: C-Y. Jean Chan (Central Michigan University)
Title: On A^1-Milnor numbers
Abstract: TBA

Wednesday, March 4

Speaker: Siva Somasundram (Purdue)
Title: TBA
Abstract: TBA

Thursday, March 12 (3 pm); note special date and time.

Speaker: Marta Benozzo (Paris-Saclay)
Title: TBA
Abstract: TBA

Wednesday, April 1

Speaker: Sasha Pevzner (Northeastern)
Title: TBA
Abstract: TBA

Wednesday, April 8

Speaker: Dipendranath Mahato (Tulane)
Title: TBA
Abstract: TBA

Wednesday, April 29

Speaker: Vaibhav Pandey (Purdue)
Title: TBA
Abstract: TBA

Wednesday, September 3

Speaker: Aryaman Maithani (Utah)
Title: Polynomial invariants of GL₂: Conjugation over finite fields
Abstract: Consider the conjugation action of GL₂(K) on the polynomial ring K[X₂ₓ₂]. When K is an infinite field, the ring of invariants is a polynomial ring generated by the trace and the determinant. We describe the ring of invariants when K is a finite field, and show that it is a hypersurface.

Wednesday, September 10

Speaker: Vignesh Jagathese (UIC)
Title: Quasi-F-Purity and F-pure Thresholds
Abstract: A weakening of Frobenius splitting, Quasi-F-Splittings have proven to be a vital invariant in the study of varieties in positive characteristic, with numerous applications to birational geometry and singularities. In this talk I'll provide an overview of Quasi-F-Splittings and introduce a local analogue, Quasi-F-Purity, and discuss its various permanence properties. I will also discuss how quasi-F-pure hypersurfaces are "as close to being F-pure as possible" by computing the F-pure Threshold of an arbitrary quasi-F-pure hypersurface. This talk includes joint work with JJ Garzella.

Wednesday, September 24

Speaker: Manav Batavia (Purdue)
Title: The arithmetic rank of residual intersections of a complete intersection ideal
Abstract: The arithmetic rank of a variety is the minimal number of equations needed to define it set-theoretically, i.e., the smallest number of polynomials generating the defining ideal upto radical. Computing this invariant is notoriously difficult: the minimal generators up to radical often bear little relation to the given ideal generators and can vary unpredictably across characteristics.
Residual intersections provide a natural extension of the classical notion of algebraic links. We establish a general upper bound for the arithmetic rank of any residual intersection of a complete intersection ideal in an arbitrary Noetherian ring, and we show that this bound is sharp under specific characteristic assumptions. This work is joint with Kesavan Mohana Sundaram, Taylor Murray, and Vaibhav Pandey.

Wednesday, October 1

Speaker: Reinold Huebl (Regensburg)
Title: Parametrization of algebroid Curve Singularities
Abstract: This talk is based on joint work with C. Huneke, S. Maitra, and V. Mukundan.
A reduced and irreducible one-dimensional local complete domain R over an algebraically closed field k can be parametrized in many different ways as a subring of its integral closure R = k[[t]]. In this talk we will show that there is a unique sequence of positive integers a₁ < ... < a_n such that any set x₁, ..., x_n in R with valuations v_R(x_i) = a_i defines a minimal parametrization of R. We will then use this result to study the torsion of differential forms and to prove a theorem about modifications of parametrizations of R.

Wednesday, October 8

Speaker: Kangjin Han (DGIST, Korea)
Title: On the ideals of cyclic Gaussian graphical models
Abstract: In this talk, we introduce a conjecture due to Sturmfels and Uhler concerning generation of the prime ideal of the variety associated to the Gaussian graphical model of any cycle graph and explain how to prove it using commutative algebra and combinatorics. Our methods are also applicable to a large class of ideals with radical initial ideals. This work is done jointly with A. Conner and M. Michalek.

Wednesday, October 15

Speaker: Joel Castillo Rey (BCAM, Basque)
Title: The Strong Watanabe–Yoshida conjecture for complete intersections.
Abstract: The Watanabe–Yoshida conjecture states that the Hilbert–Kunz multiplicity attains its minimal value across singularities exactly at quadric hypersurfaces. We present an affirmative answer for complete intersections in every positive characteristic, improving a theorem by Enescu and Shimomoto.

Wednesday, October 22

Speaker: Dan Bath (KU Leuven)
Title: The strong monodromy conjecture and certain projective plane curves
Abstract: I will discuss ongoing work with Wim Veys on the "next open case" of the Strong Monodromy Conjecture and how to solve the D-module part of the story.

Wednesday, November 19

Speaker: Ryan Watson (UNL)
Title: Cohomological Support Varieties Under Local Homomorphisms
Abstract: I will discuss cohomological support varieties and recent work on how they behave when restricting along a local homomorphism.

Wednesday, December 3

Speaker: C-Y. Jean Chan (Central Michigan University)
Title: Fibered Sums of Affine Semigroups and Algebras
Abstract: Let S, S₁ and S₂ be affine semigroups with S embedded in S₁ and S₂ respectively. We introduce the fibered sum of two such affine semigroup algebras. This is joint work with I-Chiau Huang and Jung-Chen Liu.