Friday, January 20th, 2023

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, 2023

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, 2023

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, 2023

Title: Weierstrass Points on Algebraic Curves and Tropical Curves

Speaker: Harry Richman

Abstract: An algebraic curve of high genus can be a difficult landscape to navigate. In such a situation, we would like a set of well-placed landmarks as a guide. This is what Weierstrass points provide for us. We will cover the definition of Weierstrass points, and overview results that quantify in what sense they are "well-placed". We will also mention some applications in algebraic geometry and tropical geometry, in particular:

• When does an algebraic / tropical curve have finitely many automorphisms?

• When is the moduli space of algebraic / tropical curves "general type"?

Friday, February 24th, 2023

Title: Compactifications of Jacobians of Nodal Curves

Speaker: Cameron Wright

Abstract: Jacobians of smooth curves are well-studied classical objects which have been studied since the work of Riemann. In the time since then, there have been many generalizations of this theory, including in particular constructions of Jacobians for curves with nodal singularities. These Jacobians are typically not projective and, as such, much work has been invested to compactify these objects. In this talk we study a family of compactifications of Jacobians of nodal curves due to Oda and Seshadri. These constructions rely on combinatorial and convex-geometric machinery which is interesting in its own right. Here, we survey both the algebro-geometric and the convex-geometric constructions needed to appreciate Oda-Seshadri's results.

Friday, March 3rd, 2023

Title: Tropical Vector Bundles and their Chern Classes

Speaker: Andrew Tawfeek

Abstract: We introduce the notion of a tropical vector bundle (of rank n) over a tropical cycle X, which is equivalently interpretable as a polyhedral complex living "above" X or GL_n-torsor. Primarily relying on the more-accessible combinatorial definition, we go on to look at morphisms, rational sections, and pull-backs of these bundles. After some examples, we will discuss how one can take local intersection products and construct Chern classes -- concluding the talk with a classification of all vector bundles on a (tropical) elliptic curve up to isomorphism, which will (surprisingly!) coincide with Atiyah's 1957 result in classical algebraic geometry.

Friday, March 10th, 2023

Title: Pseudo-Divisors and Polyhedral Connections

Speaker: Natasha Crepeau

Abstract: Divisors on graphs foster a rich connection between algebraic geometry and combinatorics, especially connections between the moduli theory of algebraic curves to graph theory and polyhedral geometry. In this talk we will walk through results presented in A universal tropical Jacobian over M_g^trop by Abreu, Andria, Pacini, and Taboada: we'll start with defining relevant divisors on tropical curves and graphs, introduce some stability conditions and special polytopes, and conclude with a construction of a universal tropical Jacobian.