# Difference between revisions of "Algebra and Algebraic Geometry Seminar Fall 2018"

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|October 19 | |October 19 | ||

|Oleksandr Tsymbaliuk (Yale) | |Oleksandr Tsymbaliuk (Yale) | ||

− | | | + | |Modified quantum difference Toda systems |

|Paul Terwilliger | |Paul Terwilliger | ||

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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. | 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=== | ===Juliette Bruce=== |

## Revision as of 21:30, 15 October 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

- Please join the AGS Mailing List to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).

## 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) | TBD | Botong Wang |

November 9 | Rohit Nagpal (Michigan) | TBD | John WG |

November 16 | Wanlin Li | TBD | Local |

November 23 | Thanksgiving | No Seminar | |

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

December 7 | Michael Brown | TBD | Local |

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.