Algebra and Algebraic Geometry Seminar Fall 2018
The seminar meets on Fridays at 2:25 pm in room B235.
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Fall 2018 Schedule
|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)||Conjecture D for matrix factorizations||Michael and Daniel|
|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 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|
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.
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.
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.