Spring 2018 Room 705 Hill Center, Thursdays 11 AM - 12 PM

Deformations of the moduli space of polygons
January, Thursday 25 - Marco Castronovo

We construct a Kähler moduli space parametrizing polygons in R^3 with given side lengths. A path in the positive cone of lengths gives a deformation of the moduli of polygons, which changes topology at some critical times. Along a canonical path, this deformation matches the Kähler-Ricci flow. Specialising to moduli spaces of pentagons, we explain why they are generically toric del Pezzo surfaces and describe natural Lagrangian tori in them. Some open Gromov-Witten invariants of these tori are encoded by a Laurent polynomial called disk potential, and we show how to compute it in practice. Finally, we highlight some open questions regarding the disk potential and their connection with Kontsevich's homological mirror symmetry.

Geodesics in H^3 and Bianchi groups
February, Thursday 8 - Katie McKeon

We will explain the action of SL(2,C) on the upper half-plane model of hyperbolic three-space H^3 via quaternions, and 3D analogues of the modular surface, Bianchi manifolds. Our goal is to study the behavior of closed geodesics in the manifold using symbolic dynamics.

The geometry of spacetimes
February, Thursday 15 - Annegret Burtscher

In Einstein's general theory of relativity, gravity is described as a geometric property of space and time. More precisely, these spacetimes are four-dimensional Lorentzian manifolds, and via the Einstein equations their curvature is related to the energy and momentum of whatever matter is present. We are particularly interested in stellar models, which can be described by a perfect fluid. According to the (mostly resolved) “fluid ball conjecture” non-rotating stellar models are automatically also spherically symmetric and are thus modeled by the Tolman-Oppenheimer-Volkoff equation. With all these restrictions in place, one still has to solve a system of singular and highly nonlinear ordinary differential equations. We will discuss what is known about the geometry of these solutions, and how the asymptotic behavior relates to properties of the fluid and other solutions in general relativity.

Motivic integration and the Grothendieck ring of varieties
February, Thursday 22 - Vernon Chan

Motivic integration is an extension of p-adic integration which takes values in the Grothendieck ring instead of real numbers. In this talk I will explain the basic idea of motivic integration and construct the motivic counting measure. I will also discuss the closely related Grothendieck ring of varieties and its relation to universal additive invariants, e.g. Hodge polynomials. If time allows, I will discuss the motivic integral on the formal arc space of a smooth variety Y and an effective divisor D, and mention some important results.

Integrable systems and S^1 symmetries
March, Thursday 8 - Joseph Palmer

Roughly speaking, an integrable system is a physical system with the maximal number of independent symmetries. The results of Atiyah, Guillemin-Sternberg, and Delzant from the 1980s classify so-called toric integrable systems, which are 2n-dimensional systems which admit a Hamiltonian n-torus action, and in 2011 this classification was extended in dimension 4 by Pelayo-Vu Ngoc to a class of systems known as semitoric. Semitoric systems are four dimensional integrable systems which admit a circle action instead of a 2-torus action, and have been the subject of a large amount of recent work. In this talk we give an introduction to integrable systems, with special focus on toric systems, and discuss new developments and conjectures related to semitoric systems.

Hyperbolic surfaces, winding numbers, automorphic forms
March, Thursday 22 - Claire Burrin

I will present ongoing work on a topological and geometric invariant computing the winding of closed geodesics on hyperbolic surfaces. If time permits, this will include a discussion of Selberg's trace formula, and how it can be used to deduce distribution results about these winding numbers.

Graph formulas of classes on moduli spaces of curves
March, Thursday 29 - Nicola Tarasca

This talk will be an introduction to the intersection theory of moduli spaces of curves. I will focus on the use of graph formulas to describe intersections of tautological classes, and discuss applications to the study of effective classes.

Canonical measures on metric graphs and Kazhdan's theorem
April, Thursday 5 - Chenxi Wu

A theorem by Kazhdan says that the limit of canonical (Arakelov) metric converges to the hyperbolic metric when passing to increasingly high finite regular covers that converge to the universal cover. We proved an analogy for it to the case of graphs as well as manifolds and metrized simplicial complexes, and which allows any infinite regular cover to be the limit. This is joint work with Farbod Shokrieh and Harry Baik.

An introduction to complete quadrics
April, Thursday 19 - Chengxi Wang

This talk will be an introduction to complete quadrics which preceded the study of the symmetric objects of complete collineations. We will give the description of a complete quadrics in two different points of view. Also, I will explain to you the structure of the moduli space of complete quadrics.