# 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.

## Contents

## Algebra and Algebraic Geometry Mailing List

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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) | Conjecture D for matrix factorizations | 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.