Fall 2023

Title: Why we love spanning trees

Abstract: Spanning trees are to graphs as bases are to vector spaces.  And that's just the start of the story of why we love spanning trees.  I'll talk about some of the surprising and deep facts from the last 70 years or so about trees and randomness, and give some fresh applications (including to voting rights!).

Prerequisites: Graphs, basic probability

Title: Homological algebra on toric varieties

Abstract:  When studying subvarieties of projective space, homological algebra over the standard graded polynomial ring provides several useful tools (free resolutions, syzygies, Castelnuovo-Mumford regularity, etc.) which capture nuanced geometric information. One might hope that there are analogous tools over multigraded polynomial rings, which provide similar geometric information for subvarieties of other toric varieties. I will discuss recent work developing such tools, as well as some of the subtleties that arise when moving to toric varieties beyond projective space. This is joint work with Lauren Cranton Heller and Mahrud Sayrafi.

Prerequisites: I will assume some familiarity with polynomial rings, ideals, as well as linear algebra.

Title: Moduli spaces of curves

Abstract:  I'll introduce the concept of moduli spaces in algebraic geometry with the example of the moduli space of circles. This is an example of a moduli space of "embedded curves." However, as I'll explain, the associated "abstract curves" are all the same. I'll finish by talking about moduli spaces of abstract curves and share some recent results, which are joint work with Samir Canning.

Prerequisites: The only essentials are complex numbers, polynomials, and a willingness to visualize! Near the end, I'll mention elliptic curves and rings.

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