Algebra and Algebraic Geometry Seminar Fall 2018

The seminar meets on Fridays at 2:25 pm in room B235.

Here is the schedule for the previous semester, the next semester, and for this semester.

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Fall 2018 Schedule

date	speaker	title	host(s)
September 7	Daniel Erman	Big Polynomial Rings	Local
September 14	Akhil Mathew (U Chicago)	Kaledin's noncommutative degeneration theorem and topological Hochschild homology	Andrei
September 21	Andrei Caldararu	Categorical Gromov-Witten invariants beyond genus 1	Local
September 28	Mark Walker (Nebraska)	TBD	Michael and Daniel
October 5
October 12	Jose Rodriguez (Wisconsin)	TBD	Local
October 19	Oleksandr Tsymbaliuk (Yale)	TBD	Paul Terwilliger
October 26	Juliette Bruce	TBD	Local
November 2	Behrouz Taji (Notre Dame)	TBD	Botong Wang
November 9	Saved	TBD	Local
November 16	Wanlin Li	TBD	Local
November 23	Thanksgiving	No Seminar
November 30	Eloísa Grifo (Michigan)	TBD	Daniel
December 7	Michael Brown	TBD	Local
December 14	John Wiltshire-Gordon	TBD	Local

Abstracts

Akhil Mathew

Title: Kaledin's noncommutative degeneration theorem and topological Hochschild homology

For a smooth proper variety over a field of characteristic zero, the Hodge-to-de Rham spectral sequence (relating the cohomology of differential forms to de Rham cohomology) is well-known to degenerate, via Hodge theory. A "noncommutative" version of this theorem has been proved by Kaledin for smooth proper dg categories over a field of characteristic zero, based on the technique of reduction mod p. I will describe a short proof of this theorem using the theory of topological Hochschild homology, which provides a canonical one-parameter deformation of Hochschild homology in characteristic p.

Andrei Caldararu

Categorical Gromov-Witten invariants beyond genus 1

In a seminal work from 2005 Kevin Costello defined numerical invariants associated to a Calabi-Yau A-infinity category. These invariants are supposed to generalize the classical Gromov-Witten invariants (counting curves in a target symplectic manifold) when the category is taken to be the Fukaya category. In my talk I shall describe some of the ideas involved in Costello's approach and recent progress (with Junwu Tu) on extending computations of these invariants past genus 1.

Mark Walker

Conjecture D for matrix factorizations

Matrix factorizations form a dg category whose associated homotopy category is equivalent to the stable category of maximum Cohen-Macaulay modules over a hypersurface ring. In the isolated singularity case, the dg category of matrix factorizations is "smooth" and "proper" --- non-commutative analogues of the same-named properties of algebraic varieties. In general, for any smooth and proper dg category, there exist non-commutative analogues of Grothendieck's Standard Conjectures for cycles on smooth and projective varieties. In particular, the non-commutative version of Standard Conjecture D predicts that numerical equivalence and homological equivalence coincide for such a dg category. Recently, Michael Brown and I have proven the non-commutative analogue of Conjecture D for the category of matrix factorizations of an isolated singularity over a field of characteristic 0. In this talk, I will describe our theorem in more detail and give a sense of its proof.