Algebraic Geometry/Commutative Algebra Seminar, 2026–2027

To volunteer to give a talk, or for anything else regarding the seminar, contact Claudiu Raicu, Ritvik Ramkumar, or Eric Riedl.

Fall Schedule

The seminar will meet on Thursdays, 3:30–4:30pm in 258 Hurley, unless otherwise noted. Related events are also listed below.

Abstracts

Sep. 3, 2026

Speaker

Fuxiang Yang (Notre Dame)

Title

Recent progress on the Eisenbud-Huneke-Ulrich conjecture

Abstract

Over a polynomial ring, an ideal I is said to be linearly presented if its first syzygy module is generated by linear relations. More generally, the resolution of I is linear for p steps if this linearity continues through the first p steps. In this talk, we explore some fundamental properties of ideals with partially linear minimal free resolution, such as asymptotic behavior of powers, Castelnuovo-Mumford regularity, and lower bounds on the number of minimal generators. In particular, we prove the Eisenbud-Ulrich conjecture on powers of linearly presented ideals and establish a linear effective bound toward the more general Eisenbud-Huneke-Ulrich conjecture.

Sep. 10, 2026

Speaker

Teresa Yu (Notre Dame)

Title

Module theory in infinite-dimensional equivariant commutative algebra

Abstract

A foundational result in equivariant commutative algebra is Cohen's theorem that the infinite variable polynomial ring R=C[x1, x2, x3,…] is Noetherian up to the action of the infinite symmetric group. This result has led to uniformity and finiteness theorems for finite-dimensional structures in algebraic geometry, statistics, and algebraic topology, and more generally motivates the study of the equivariant commutative algebra of R. In this talk, we will survey the module-theoretic aspects of this field, with particular emphasis on the infinite GL and symmetric group settings. Along the way, we will see how FI-modules, originally introduced in the context of representation stability, provide a useful bridge between finite-dimensional representation theory and infinite-dimensional equivariant algebra.

Sep. 17, 2026

Speaker

Nikola Kuzmanovski (Notre Dame)

Title

A structural property of generic initial ideals

Abstract

For over a century mathematicians have studied how to transfer information between a homogeneous ideal and its initial ideal. In this talk I will take this topic in the context of regular sequences. I will present an asymptotic structural property of generic initial ideals. This single phenomenon yields results on Hilbert functions, persistence, hyperplane restriction, graded Betti numbers, combinatorial shadow minimization, and Lefschetz properties.

Sep. 24, 2026

Speaker

Amadou Bah (Notre Dame)

Title

Uniform generic vanishing theorems and equidistribution on G^d_m

Abstract

As an application of Deligne's work on the Weil conjectures, Katz (1980) showed that Gauss sums are equidistributed with respect to the Haar measure on the unit circle. One does so by interpreting them as traces of Frobenius acting on a local system on G_m and applying the Grothendieck-Lefschetz trace formula (as outlined by Deligne). Katz subsequently extended this strategy to perverse sheaves M on G_m and deduced equidistribution for other exponential sums (Kloosterman sums, Evans sums, Rudnick sums). The key step in this business is bounding the set of characters χ of G_m for which the cohomology of the twisted perverse sheaf M_χ fails to vanish. In a joint work with K.V. Shuddhodan, we establish a uniform bound of the analogous set of characters for perverse sheaves on G_m^d (d-dimensional split torus) and prove a "horizontal" equidistribution theorem in this setting, extending Katz's result in d=1. I will motivate this work and explain some of our results.

Oct. 1, 2026

Speaker

Jonghyun Lee (Michigan)

Title

Bernstein-Sato polynomials and log resolutions by orbifolds

Abstract

The Bernstein-Sato polynomial is an important invariant of singularities, intimately related to the monodromy of the Milnor fiber, the log canonical threshold, and rational and Du Bois singularities. A classical method introduced by Kashiwara and refined by Lichtin provides estimates for the roots of the Bernstein-Sato polynomial in terms of a log resolution. In this talk, we extend their method to log resolutions by orbifolds, which has applications to the singularities of weighted homogeneous and Khovanskii non-degenerate complete intersections.

Oct. 8, 2026

Speaker

Shuddhodan Kadattur Vasudevan (Notre Dame)

Title

Vanishing in the limit

Abstract

An affine variety of dimension $d$ has no mod $\ell$ cohomology above degree $d$, whereas a projective variety of positive dimension has cohomology in degree $2d$ (the class of a point!). In characteristic zero, Scholze showed that for subvarieties of projective space the cohomology above the dimension vanishes in the limit along the “mock Frobenius” tower, and Esnault proved this in every characteristic different from $\ell$. Bhatt-Schnell-Scholze asked whether the same holds for subvarieties of an abelian variety $A$ in characteristic $p\neq\ell$, along the tower of multiplication by $\ell$. We prove that it does. This is joint work with Ashutosh Roy Choudhury.

Nov. 12, 2026

Speaker

Manav Batavia (Purdue)

Title

A sharp upper bound on cohomological dimension in unramified 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.