# Algebra and Algebraic Geometry Seminar Spring 2019

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

Here is the schedule for the previous semester, for 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).

## Spring 2019 Schedule

date | speaker | title | host(s) |
---|---|---|---|

January 25 | Daniel Smolkin (Utah) | Symbolic Powers in Rings of Positive Characteristic | Daniel |

February 1 | Juliette Bruce | Asymptotic Syzgies for Products of Projective Spaces | Local |

February 8 | Isabel Vogt (MIT) | Low degree points on curves | Wanlin and Juliette |

February 15 | Pavlo Pylyavskyy (U. Minn) | TBD | Paul Terwilliger |

February 22 | Michael Brown (Wisconsin) | Chern-Weil theory for matrix factorizations | Local |

March 1 | Shamgar Gurevich (Wisconsin) | Harmonic Analysis on GLn over finite fields, and Random Walks | Local |

March 8 | Jay Kopper (UIC) | TBD | Daniel |

March 15 | TBD | TBD | TBD |

March 22 | No Meeting | Spring Break | TBD |

March 29 | Chris Eur (UC Berkeley) | TBD | Daniel |

April 5 | TBD | TBD | TBD |

April 12 | Eric Canton (Michigan) | TBD | Michael |

April 19 | Eloísa Grifo (Michigan) | TBD | TBD |

April 26 | TBD | TBD | TBD |

May 3 | TBD | TBD | TBD |

## Abstracts

### Daniel Smolkin

**Symbolic Powers in Rings of Positive Characteristic**

The n-th power of an ideal is easy to compute, though difficult to describe geometrically. In contrast, symbolic powers of ideals are difficult to compute while having a natural geometric description. In this talk, I will describe how to compare ordinary and symbolic powers of ideals using the techniques of positive-characteristic commutative algebra, especially in toric rings and Hibi rings. This is based on joint work with Javier Carvajal-Rojas, Janet Page, and Kevin Tucker. Graduate students are encouraged to attend!

### Juliette Bruce

**Title: Asymptotic Syzygies for Products of Projective Spaces**

I will discuss results describing the asymptotic syzygies of products of projective space, in the vein of the explicit methods of Ein, Erman, and Lazarsfeld’s non-vanishing results on projective space.

### Isabel Vogt

**Title: Low degree points on curves**

In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to e at least 4. We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This is joint work with Geoffrey Smith.

### Shamgar Gurevich

**Harmonic Analysis on GLn over finite fields, and Random Walks**

There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}:

$$ trace(\rho(g))/dim(\rho), $$

for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).