The Seminar usually takes place on Wednesdays at 2:30 PM in Room 617 on the sixth floor of Wachman Hall.
Andrew Cooper, NC State
Given a space \(X\), the configuration space \(F(X,n)\) is the space of possible ways to place \(n\) points on \(X\), so that no two occupy the same position. But what if we allow some of the points to coincide?
The natural way to encode the allowed coincidences is as a simplicial complex \(S\). I will describe how the configuration space \(M(S,X)\) obtained in this way gives rise to polynomial and homological invariants of \(S\), how those invariants are related to the cohomology ring \(H^*(X)\), and what this has to do with the topology of spaces of maps into \(X\).
I will also mention some potential applications of this structure to problems arising from international relations and economics.
This is joint work with Vin de Silva, Radmila Sazdanovic, and Robert J Carroll.
Andrew Yarmola, Princeton University
Abstract: At the interface of discrete conformal geometry and the study of Riemann surfaces lies the Koebe-Andreev-Thurston theorem. Given a triangulation of a surface \(S\), this theorem produces a unique hyperbolic structure on \(S\) and a geometric circle packing whose dual is the given triangulation. In this talk, we explore an extension of this theorem to the space of complex projective structures - the family of maximal \(CP^1\)-atlases on \(S\) up to Möbius equivalence. Our goal is to understand the space of all circle packings on complex projective structures with a fixed dual triangulation. As it turns out, this space is no longer a unique point and evidence suggests that it is homeomorphic to Teichmüller space via uniformization - a conjecture by Kojima, Mizushima, and Tan. In joint work with Jean-Marc Schlenker, we show that this projection is proper, giving partial support for the conjectured result. Our proof relies on geometric arguments in hyperbolic ends and allows us to work with the more general notion of Delaunay circle patterns, which may be of separate interest. I will give an introductory overview of the definitions and results and demonstrate some software used to motivate the conjecture. If time permits, I will discuss additional ongoing work with Wayne Lam.
Michael Landry, Yale University
Let \(M\) be a closed hyperbolic 3-manifold which fibers over \(S^1\), and let \(F\) be a fibered face of the unit ball of the Thurston norm on \(H^1(M;R)\). By results of Fried, there is a nice flow on \(M\) naturally associated to \(F\). We study surfaces which are almost transverse to \(F\) and give a new characterization of the set of homology directions of \(F\) using Agol’s veering triangulation of an auxiliary cusped 3-manifold.