# CMU Algebra and Combinatorics Seminar

## Fall 2018

### Meeting Times

Fridays, 11:15am–12:15pm, in Pearce 226.

### Abstracts

Speaker: Botong Wang
Title: The log-concave conjectures of graphs and matroids
Abstract: The chromatic polynomial is an important invariant in graph theory introduced by Birkhoff. A generalization of chromatic polynomial is the characteristic polynomial in matroid theory. It was conjectured by Rota, Heron and Welsh in the 70’s that the coefficients of the characteristic polynomials are log-concave. I will talk about a beautiful proof of this conjecture by Karim Adiprisito, June Huh and Eric Katz. I will also discuss a log-concavity result about the number of independent sets, which is joint work with June Huh and Benjamin Schroter. The key idea to proof the log-concavity properties is to use the Hodge-Riemann relation in algebraic geometry.

Speaker: Paramasamy Karuppuchamy
Title: Schubert varieties and toric varieties
Abstract: Degeneration of Schubert varieties to toric varieties is completed in the paper "Toric degeneration of Schubert varieties" by Philippe Caldero. In an attempt to find a geometric proof of this result we realized that certain Schubert varieties are already toric varieties: A Schubert variety $X_w$ is a toric variety if and only if $w$ is a product of distinct simple reflections. Part of this result can be found (not explicitly mentioned) in Deodhar's article "On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cell." The author was unaware of this fact while writing his article. In $A_n$ type, Masuda and Lee have different approach to get this result in their recent paper "Generic torus orbit closure in Schubert varieties." In this talk we give an overview of this topic.

Speaker: Olivia Dumitrescu
Title: Interplay between ribbon graphs and CohFT
Abstract: I will review an axiomatic formulation of a 2D TQFT whose formalism is based on the edge-contraction operations on graphs drawn on a Riemann surface (cellular graphs). I will describe a new result, that ribbon graphs provide both cohomological field theory and a visual explanation of Frobenius-Hopf duality, that plays a crucial role in Givental-Teleman's classification theorem of CohFTs. No prerequisite is assumed. This is based on a work in progress with Motohico Mulase.

Speaker: Jordan Watts
Title: Diffeological Groups
Abstract: Lie groups, groups equipped with a smooth manifold structure, are a broadly-studied and import class of groups with a natural connection to Lie algebras. They arise naturally in representation theory, classical mechanics, and other fields. However, there are plenty of groups that do not admit a smooth manifold structure, but we still wish to treat them as though they did… The concept of diffeological group remedies this by equipped these groups, such as the irrational torus and diffeomorphism groups, with a smooth structure that is not necessarily that of a manifold. We can also easily make the natural connection to Lie algebras of these groups. This talk will be designed for graduate students, will discuss Lie groups, diffeological groups, their Lie algebras, and have some helpful examples throughout.

Speaker: Dmitry Zakharov
Title: Maps to trees and loci in the moduli space of tropical curves
Abstract: Tropical geometry is a large and growing field of mathematics that aims to find combinatorial, piecewise-linear analogues of various algebraic and geometric objects. One particularly well-developed correspondence is the one between algebraic curves and metric graphs, which are also called tropical curves. There exist tropical analogues of many constructions and results for algebraic curves, such as meromorphic functions, divisors, linear equivalence, the Riemann–Roch theorem, Jacobians, and moduli spaces. Such objects can be studied using purely combinatorial methods, and results about them can then be used to understand their algebraic analogues. A classical problem in algebraic geometry is the study of loci in the moduli space of algebraic curves consisting of curves admitting linear systems of a particular type. A major difference between tropical and algebraic curves is that the former usually have a much larger collection of principal divisors than the latter. For this reason, the loci of tropical curves admitting specific linear systems have unexpectedly large dimension in moduli. I will talk about an approach to this problem in which, instead of looking at tropical curves with linear systems, we look at tropical curves admitting maps to trees of a particular type.

Speaker: Takumi Murayama
Title: Frobenius-Seshadri constants and characterizations of projective space
Abstract: Mori and Mukai conjectured that projective spaces can be characterized by their intersection theory. While their conjecture has been proved over the complex numbers by Cho, Miyaoka, and Shepherd-Barron, the conjecture is still open in positive characteristic. We will describe how Seshadri constants and a positive-characteristic analogue of these constants, called Frobenius-Seshadri constants, can be used to give a partial answer to Mori and Mukai's conjecture in positive characteristic. Frobenius-Seshadri constants were introduced by Mustata-Schwede and the presenter, and are obtained in a way similar to how Hilbert-Kunz multiplicity is obtained from Hilbert-Samuel multiplicity.