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.

Title: Metric Based Approaches to Shape Comparison in Redistricting

Abstract:  In this talk we introduce the concept of a metric space, and motivate the use of metrics as a tool for shape comparison. By representing an object as a finite metric space, we can utilize families of metrics like Hausdorff and Gromov-Hausdorff distances to develop similarity measures between shapes. One particular focus is on applications to electoral redistricting, where the notion of shape is ubiquitous when investigating the political geography of a state. In particular, we will discuss how metrics were used in a recent effort to quantify the idea of ‘communities of interest’ in the recent redistricting cycle.

Prerequisites: Metric spaces; (hierarchical) clustering

Title: Rational points, symmetry, and derived categories

Abstract:  Together, we will explore the extent to which the derived category of coherent sheaves can be used to determine when a variety defined over an arbitrary field has points. In particular, we will give some examples of using the machinery of the derived category in an arithmetic setting, touching upon recent joint work with Matthew Ballard, Alex Duncan, and Patrick McFaddin. We will also discuss new related work with Aaron Bertram. 

Prerequisites: Groups, rings, linear algebra, permutations

Title: Relative information and the dual numbers

Abstract:  Relative information (Kullback-Leibler divergence) is a fundamental concept in statistics, machine learning and information theory. 

In the first half of the talk, I will define conditional relative information, list its axiomatic properties, and describe how it is used in machine learning. For example, the generalization error of a learning algorithm depends on the structure of algebraic geometric singularities of relative information.

In the second half of the talk, I will define the rig category InfoRig of random variables and their conditional maps, as well as the rig category R(e) of dual numbers. Relative information can then be constructed, up to a scalar multiple, via rig functors from InfoRig to R(e). If time permits, I may discuss how this construction relates to the information cohomology of Baudot, Bennequin and Vigneaux, and to the operad derivations of Bradley. 

Prerequisites: Random variables, probability distribution, conditional probability; category, object, morphism, functor; rig (semiring).

Title: Cost-sharing in parking games

Abstract:  We introduce parking games, which are coalitional cost-sharing games in characteristic function form derived from the total displacement of parking functions. Motivated by the "fair" distribution of parking costs, our main contribution is a polynomial-time algorithm to compute the Shapley value of these games. This is collaborative work with Jennifer Elder, Pamela E. Harris, and Jan Kretschmann, and can be found at arXiv:2309.12265.

Prerequisites: Some familiarity with permutations. However, all concepts (especially those from game theory) will be explained assuming no background.

Title: Combinatorics of colored multiset permutations

Abstract:  We will talk about descent polynomials of colored permutations on multisets and their properties (palindromicity, unimodality, real-rootedness). After defining each of these terms, we will construct a generating function associated to each descent polynomial, by generalizing a result of MacMahon. Using this connection, we will turn to polytopes and Ehrhart theory and employ techniques from there to characterize a large class of descent polynomials in terms of palindromicity and real-rootedness. The results in this talk are based on ongoing work with Liam Solus and Bin Han.

Prerequisites: Some familiarity with permutations is assumed. Other keywords that could be helpful: colored/signed permutations, multisets, polytopes, Ehrhart polynomials.