Schedule
Sep 20th Saturday:
9:00-9:30: Breakfast and Registration
9:30-10:20: Talk 1-Jake Levinson (Lecture I)
10:20-10:40: Break
10:40-11:30: Talk 2-Rohini Ramadas (Lecture I)
11:30-1:00: Lunch
1:00-2:00: Talk 3-Rob Silversmith
2:00-2:30: Coffee Break
2:30-3:20: Talk 4-Jake Levinson (Lecture II)
3:20-3:40: Break
3:40-4:30: Talk 5-Rohini Ramadas (Lecture II)
4:30-5:45: Poster Session
Sep 21st Sunday:
8:30-9:00: Breakfast
9:00-9:50: Talk 6-Jake Levinson (Lecture III)
9:50-10:10: Break
10:10-11:10: Talk 7-Haggai Liu
11:10-11:40: Coffee Break
11:40-12:40: Talk 8-David Jensen
Abstracts
Talk-1 Jake Levinson (Lecture I)
Title: Equations for M_{0,n}-bar
Abstract: Explicit equations for moduli spaces, such as those known for toric varieties and Grassmannians, are somewhat hard to come by. I will discuss an embedding of M_{0, n}-bar in a product of projective spaces, built out of what are called the psi classes, for which explicit equations were conjectured by Monin and Rana in 2017. Then, I'll discuss the proof of the conjecture (with M. Gillespie and S. Griffin), which combines tree colorings and determinants.
Talks-2 & 5 Rohini Ramadas (Lecture I & II)
Title: Tropical moduli spaces outside algebraic geometry.
Abstract: I will introduce moduli spaces of tropical curves, and connections to non-algebro-geometric objects such as: Teichmuller space and mapping class groups, free groups and Outer Space, and handlebodies.
Talk-3 Rob Silversmith
Title: Stable curves and chromatic polynomials
Abstract: To any finite simple graph G, one can naturally associate a sequence of intersection numbers on moduli spaces of stable curves. I'll give a surprising formula for these numbers in terms of the chromatic polynomial of G, and explain how this formula emerges from the classical theory of hyperplane arrangements. I’ll also discuss the generalization of this story to directed graphs, which leads to some cool speculations and combinatorial open problems. Joint with Bernhard Reinke.
Talks-4 & 6 Jake Levinson (Lecture II & III)
Title: Psi classes, tropical geometry and enumeration
Abstract: I will examine psi classes from the perspective of tropical and enumerative geometry. These classes arise naturally from geometric considerations, and are moreover known to have rich combinatorial structure. In the first part, I will discuss some forthcoming work with Griffin-Ramadas-Silversmith on tropicalizing the psi classes, along with the corresponding hypersurfaces and maps to projective space. We were motivated by certain calculations in cohomology, and this work accordingly led us to a new tropical technique for computing limits of cycles. In the second part, I will explore some of the rich combinatorics (particularly of trees and graphs) that arises in studying intersection products involving psi classes.
Talk-7 (Haggai Liu)
Title: Moduli Spaces of Weighted Stable Curves and their Fundamental Groups
Abstract: A certain equivariant fundamental group of the moduli space $\overline{M_{0,n+1}}(\mathbb{R})$ of real $(n+1)$-marked stable curves of genus $0$ in known as cactus groups $J_n$ and have applications both in geometry and the representation theory of Lie algebras. We compute the corresponding equivariant fundamental groups of the Hassett space of weighted real stable curves $\overline{M_{0,\mathcal{A}}}(\mathbb{R})$ with a weight vector $\mathcal{A} = (1/a, \ldots, 1/a, 1)$ that is symmetric on the first $n$ marked points, which we call a \emph{weighted cactus group} $J_n^a$. We show that $J_n^a$ is obtained from the usual cactus presentation by introducing braid relations. Our proof is by decomposing $\overline{M_{0,\mathcal{A}}}(\mathbb{R})$ as a polytopal complex, generalizing a similar known decomposition into cubes for $\overline{M_{0,n+1}}(\mathbb{R})$ by Davis, Jan., and Scott. Our decomposition for the weighted space consists of products of permutahedra indexed by weighted stable trees.
Talk-8 (David Jensen)
Title: Recent Developments in Brill-Noether Theory
Abstract: The central question in Brill-Noether theory is: given a curve C, describe all maps from C to projective space. A series of results in the 1980's answers this question when C is sufficiently general. But what if C is special? If C admits one unexpected map to projective space, what does this imply about the existence and behavior of other unexpected maps from C to projective space? In this talk we will survey recent results in this direction.