Syzygies and Mirror Symmetry Virtual Seminar

Spring 2026

Wednesdays 3:00-4:00 pm CT

Organizers:

Matthew Ballard, Christine Berkesch, Michael Brown, Daniel Erman, David Favero, Andrew Hanlon, Sheel Ganatra

Background: This seminar was launched in Spring 2024 with the goal of creating new connections among researchers working on the circle of ideas related to multigraded syzygies, homological mirror symmetry, derived categories of toric varieties, and more. The seminar is now part of the FRGMS: Multigraded Commutative Algebra, Toric Varieties, and Homological Mirror Symmetry.

If you are interested in speaking in our seminar, please fill out this form (you may also want to email one of the organizers directly).

Anyone is welcome to attend this seminar, so please feel free to share this site with anyone you think might be interested.

Logistics: You can sign up for the mailing list here. The zoom link is here or you can use

Meeting ID: 953 1307 7795 and Passcode: RingsHave1

The format of the semester changes each semester, varying between expository talks, research talks, reading seminars, and so on. In Spring 2026 we are focusing on slightly shorter research talks with a longer Q&A time.

Jan 21 - Pankaj Singh: Classifying torsors of tori with Brauer groups

Using Mackey functors, we develop a general theory for classifying torsors under algebraic tori in terms of Brauer groups of finite field extensions of the base field. This extends Blunk’s description of the tori associated with del Pezzo surfaces of degree 6 to all retract rational tori, which is essentially the largest class for which such a description is possible. I will also discuss ongoing work on the classification of forms of centrally symmetric smooth Fano varieties.

Jan 28 - Dan Kaplan: Kuznetsov Components for GLSMs

Gauged linear sigma models (GLSMs) were introduced in physics by Witten, as a framework for the Calabi-Yau / Landau-Ginzburg correspondence. Independently, Kuznetsov defined the residual (or Kuznetsov) component of certain Fano varieties as the left orthogonal to an exceptional collection.

In joint work with Favero and Kelly, we define a Kuznetsov component for GLSMs using the determinant character. We use this definition to give a uniform description of the Kuznetsov component of complete intersections of ample divisors in a smooth projective toric variety. As a second application, we show that the non-trivial complete intersection of ample divisors in a Fano variety is a Fano visitor, meaning its derived category embeds in the derived category of some Fano variety.

Feb 4 - Juliette Bruce: Seshadri Regions and the Asymptotic Shape of Multigraded Regularity

I will discuss how the asymptotic Castelnuovo-Mumford regularity for powers of ideals sheaves on a smooth projective toric variety is closely related to a convex region, callted the Seshadri region, that measures the positivity of along subvarieties.

Feb 11- Jeff Hicks: Toric degenerations and tropical geometry

One of the ways that we can access the toolkit of toric geometry is to "degenerate" our favorite space into one glued together from toric varieties. For example, the Hesse pencil of elliptic curves in the projective plane degenerates an elliptic curve into the union of three projective lines. The reverse process—extracting data about the original space from the simpler central fiber—is a deformation theory problem. The "zeroth-order" obstruction to this deformation theory is encoded in tropical geometry. In this talk, we'll look at the simplest example of such a degeneration and discuss how the deformation theory problem can be understood via symplectic geometry and cellular resolutions.

Feb 18- John Nolan: Geometric Construction of Quiver Tensor Products

(Based on joint work with Daigo Ito.) Beilinson showed that the derived category of coherent sheaves on projective space is equivalent to the derived category of representations of a certain quiver with relations. There is a natural "quiver tensor product" on the latter category, given by vertexwise tensor products of quiver representations. I will present a geometric "B-model" description of this quiver tensor product as an "extended convolution" product of coherent sheaves on projective space. I will also discuss the generalization of these results to toric varieties of Bondal-Ruan type and some related open problems connected to Cox categories.

Feb 25- Michael Perlman

Mar 4- Caitlin Davis

Mar 11- No seminar

Mar 18- Sebastian Haney: TBA

Mar 25- Sasha Pevzner: Syzygies of the transfer map and polynomials with common roots

In modular invariant theory for finite groups, the image of the transfer map is an ideal of the ring of invariants which carries rich structure and information on the group action. In the case of the symmetric group over a finite field, the image of the transfer is an elimination ideal for a finite collection of polynomials with generic coefficients. We will discuss how this leads to a certain stability in the syzygies of this ideal, as well as conjectures on a determinantal presentation of the ideal. This is based on joint work with Harm Derksen.