# 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) | Modified quantum difference Toda systems | Paul Terwilliger |

October 26 | Juliette Bruce | Covering Abelian Varieties and Effective Bertini | Local |

November 2 | Behrouz Taji (Notre Dame) | Remarks on the Kodaira dimension of base spaces of families of manifolds | Botong Wang |

November 9 | Rohit Nagpal (Michigan) | Finiteness properties of the Steinberg representation. | John WG |

November 16 | Wanlin Li | TBD | Local |

November 23 | Thanksgiving | No Seminar | |

November 30 | John Wiltshire-Gordon | TBD | Local |

December 7 | TBD | TBD | |

December 14 | TBD (this date is now open again!) | TBD |

## 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.

### Oleksandr Tsymbaliuk

**Modified quantum difference Toda systems**

The q-version of a Toda system associated with any Lie algebra was introduced independently by Etingof and Sevostyanov in 1999. In this talk, we shall discuss the generalization of this construction which naturally produces a family of 3^{rk(g)-1} similar integrable systems. One of the key ingredients in the proof is played by the fermionic formula for the J-factors (defined as pairing of two Whittaker vectors in Verma modules), due to Feigin-Feigin-Jimbo-Miwa-Mukhin. In types A and C, our construction admits an alternative presentation via local Lax matrices, similar to the classical construction of Faddeev-Takhtajan for the classical type A Toda system. Finally, we shall discuss the geometric interpretation of Whittaker vectors in type A.

This talk is based on the joint work with M. Finkelberg and R. Gonin.

### Juliette Bruce

**Covering Abelian Varieties and Effective Bertini**

I will discuss recent work showing that every abelian variety is covered by a Jacobian whose dimension is bounded. This is joint with Wanlin Li.

### Behrouz Taji

**Remarks on the Kodaira dimension of base spaces of families of manifolds**

A conjecture of Shafarevich and Viehweg predicted that a family of smooth projective manifolds with good minimal models have (log-)general type base spaces, if the family has maximal variation. Generalizing this problem, Kebekus and Kovács conjectured that the Kodaira dimension of base spaces of such manifolds should define an upper bound for the variation in the family, even if the variation is not maximal. My aim in this talk is to discuss a strategy to solve this problem.

### Rohit Nagpal

**Finiteness properties of the Steinberg representation**

We will show that the Steinberg modules for the general linear groups form a Koszul monoid in an appropriate symmetric monoidal category. Using this we will find bounds on the codimension-one cohomology of level-3 congruence subgroups. This Koszulness result can also be used to show Ash--Putman--Sam homological vanishing theorem for the Steinberg representations. This is a joint work with Jeremy Miller and Peter Patzt.

### John Wiltshire-Gordon

**Computing with FI-modules**

We explain what an FI-module is, giving examples in algebra and combinatorics, and show how to compute with an FI-module. We then demonstrate a new result about FI-modules that is joint work with Peter Patzt.

### Michael Brown

**Chern-Weil theory for matrix factorizations**

This is joint work with Mark Walker. Classical algebraic Chern-Weil theory provides a formula for the Chern character of a projective module P over a commutative ring in terms of a connection on P. I will discuss an analogous formula for the Chern character of a matrix factorization. Along the way, I will provide background on matrix factorizations, and also on classical Chern-Weil theory.