# Combinatorial AG Seminar

Welcome to the homepage for the Combinatorial Algebraic Geometry Seminar!

This seminar is co-organized by Cameron Wright and Andrew Tawfeek at the University of Washington.

## Meetings:

We meet weekly on Fridays in Thomson Hall (THO) 325 from 10:30AM to 11:50AM.

Talks often are 50-minutes in length, with the remaining 30-minutes used for discussion.

# Winter 2022 Talks

## Friday, January 20th

### Title: Algebraic Curves, Dual Graphs, and Jacobians Thereof

Speaker: Cameron Wright

Abstract: Algebraic curves are a central class of objects in algebraic geometry. Smooth curves in particular are a well-understood class of curves, and possess a robust moduli theory. On the other hand, curves with (nodal) singularities are somewhat less well-behaved; still, these curves feature prominently in modern algebraic geometry, in particular in the context of the Deligne-Mumford compactification of the moduli space of smooth curves. As such, it is of great interest to understand curves with these mild singularities. Over the past century, it became clear to algebraic geometers that much could be gained from studying combinatorial objects derived from these curves. In this talk, we survey this combinatorial approach and, if time permits, compare the construction of Jacobians in both the algebraic and combinatorial settings.

## Friday, January 27th

### Title: Tropical Ceresa cycles

Speaker: Caelan Ritter

Abstract: The Ceresa cycle C - C^{-} of a smooth algebraic curve C is a tautological algebraic cycle contained in the Jacobian J(C). It is homologically trivial, but Ceresa showed that if C is very general of genus at least 3, then it is not algebraically trivial. It is in some sense the simplest algebraic cycle satisfying these properties, leading to applications for the étale fundamental group, intersection theory, and the theory of heights. We will discuss the extent to which the Ceresa cycle and the proof of non-triviality carry over to the tropical setting. Along the way, we will introduce important tools in the study of rational polyhedral spaces, namely, tropical cycles and homology.

## Friday, February 3rd

### Title: The Virtual Euler Characteristic for Binary Matroids

Speaker: Andrew Tawfeek

Abstract: In the graph chain complex work of Kontsevich, he computed the graphic orbifold Euler characteristic and showed it may be fascinatingly expressed through Bernoulli numbers. Inspired by this, Madeline Brandt, Juliette Bruce, and Daniel Corey in arXiv: 2301.10108 define a virtual Euler characteristic for any finite set of isomorphism classes of matroids of rank r, then prove this similarly expression is possible over the finite field F_2 (i.e. binary matroids). Additionally, they apply their methods to craft recursive formulas for subsets of the Grassmannian in the Grothendieck ring of varieties. In the talk, we briefly overview the combinatorial and algebro-geometric aspects of matroids and the Grassmannian before then delving into a discussion of their methods. We conclude the talk with various future directions one could take their work as well.

## Friday, February 10th

### Title: TBA

Speaker: Harry Richman

Abstract: TBA

## Friday, February 17th

### Title: TBA

Speaker: TBA

Abstract: TBA

## Friday, February 24th

### Title: TBA

Speaker: TBA

Abstract: TBA

## Friday, March 3rd

### Title: TBA

Speaker: TBA

Abstract: TBA

## Friday, March 10th

### Title: TBA

Speaker: TBA

Abstract: TBA

# Speaking at the seminar:

We are more than delighted to have people voulenteer to speak at the seminar! Virtually any topic that falls within the scope of combinatorics, algebra, and/or geometry is welcome -- particularly those that cover all three.

Talks are 50-minutes in length and the remaining 30-minutes are used for a discussion between the speaker and audience.

## To volunteer to speak:

Please email either

Cameron Wright: wrightc8@uw.edu

Andrew Tawfeek: atawfeek@uw.edu

with the following information:

Talk title

Abstract

Preferred date

Please refer to dates marked "TBA" in schedule for the quarter.

Your website

Include this only if you are comfortable with this homepage linking to your website.

Anything else you wish for us to take into account!