# Colloquium

The Colloquium is held on Mondays at 4pm in Deady 208.

### Fall Quarter, 2018

- September 24, No Colloquium
- October 1,
**Justin Sawon** (University of North Carolina)

Moduli of vector bundles
**Abstract**: A vector bundle on an algebraic variety associates a vector space to each point of the variety. A moduli space of vector bundles is a space parametrizing all vector bundles of a particular kind. A simple example is the Jacobian of a curve: the Jacobian is the moduli space of all line bundles of degree zero on the curve.

Moduli space of vector bundles exhibit very rich and interesting properties, often reflecting the geometry of the underlying variety. For instance, moduli space of vector bundles on holomorphic symplectic surfaces (such as K3 and abelian surfaces) are higher-dimensional holomorphic symplectic manifolds.

In this talk I will review the development of the theory of moduli spaces of vector bundles, culminating in recent advances involving not just vector bundles, but so-called `stable complexes’ of vector bundles.

- October 15,
**Richard Hain** (Duke)

Completions of Mapping Class Groups
**Abstract**: Mapping class groups are groups of orientation preserving diffeomorphisms of a compact surface. They are a natural generalization of the modular group SL_2(Z), which is the mapping class group of a torus. These mysterious groups play an important role in topology and algebraic geometry, and an increasingly important role in number theory. Even though there is a vast literature on mapping class groups that begins with the seminal work of Dehn and Nielsen in the 1930s, there is a large (and increasing) number of important open

questions about them.

One useful tool for studying discrete groups,such as mapping class groups, and also for applying results about them to questions in algebraic and arithmetic geometry, is “relative (unipotent) completion”. In this talk I will define relative completion, give explicit presentations of the relative completions of mapping class groups in genus 3 or more. I will also explain relative completion of the modular group SL_2(Z) and how it is related to modular forms and their periods. I will give some applications of relative completion of mapping class groups to algebraic and arithmetic geometry, and to topology and, if time, mention some open problems.

- October 29,
**Mario Bonk** (UCLA)

The Quasiconformal Geometry of Fractals
**Abstract**: Many questions in analysis and geometry lead to problems of quasiconformal geometry on non-smooth or fractal spaces. For example, there is a close relation of this subject to the problem of characterizing fundamental groups of hyperbolic 3-orbifolds or to Thurston’s characterization of rational functions with finite post-critical set.

In recent years, the classical theory of quasiconformal maps between Euclidean spaces has been successfully extended to more general settings and powerful tools have become available. Fractal 2-spheres or Sierpin ́ski carpets are typical spaces for which this deeper understanding of their quasiconformal geometry is particularly relevant and interesting. In my talk I will give a survey on some recent developments in this area.

- November 9,
**Dan Christensen** (Ontario)

An introduction to homotopy type theory

**Abstract**: Type theory is a formal system that was originally intended to describe set-like objects, and which is well-suited to formalizing proofs so that they can be verified by a computer. Recently, it was realized that type theory is intrinsically homotopical, and can be used to reason about spaces and other homotopical categories.

Voevodsky introduced an axiom that he calls Univalence, which says roughly that homotopy equivalence and equality agree. This axiom holds for spaces, and makes the theory truly homotopical. This talk will start with an introduction to type theory, will introduce Univalence and give examples of its consequences, and will briefly discuss some recent work on developing the theory of localization in homotopy type theory.

**Special Time** : 4pm in 208 Deady

- November 26,
**Tom Roby** (UConn)

Dynamical Algebraic Combinatorics: Actions, Orbits, and Averages

**Abstract**: Dynamical Algebraic Combinatorics explores actions on sets of discrete combinatorial objects, many of which can be built up by small local changes, e.g., Schützenberger’s promotion and evacuation, or the rowmotion map on order ideals. There are strong connections to the combinatorics of representation theory and with Coxeter groups. Birational liftings of these actions are related to the Y-systems of statistical mechanics, thereby to cluster algebras, in ways that are still relatively unexplored.

The term “homomesy” (coined by Jim Propp and the speaker) describes the following widespread phenomenon: Given a group action on a set of combinatorial objects, a statistic on these objects is called “homomesic” if its average value is the same over all orbits. Along with its intrinsic interest as a kind of “hidden invariant”, homomesy can be used to prove certain properties of the action, e.g., facts about the orbit sizes. Proofs of homomesy often involve developing tools that further our understanding of the underlying dynamics, e.g., by finding an equivariant bijection.

This talk will be a introduction to these ideas, giving a number of examples of such actions and pointing out connections other areas.

- December 3,
**Maciej Zworski** (Berkeley)

Microlocal methods in chaotic dynamics
**Abstract**: Microlocal analysis exploits mathematical manifestations of the classical/quantum (particle/wave) correspondence and has been a successful tool in spectral theory and partial differential equations. Recently, microlocal methods have been applied to the study of classical dynamical problems, in particular of chaotic (Anosov, Axiom A) flows. I will survey results obtained with Dyatlov and present some more recent results of, among others, Guillarmou, Dang–Riviere, Shen, Bonthonneau–Weich.

### Winter Quarter, 2019

- February 25,
**Dan Margalit** (Georgia Tech)

- March 4,
**Craig Westerland** (Minnesota)

### Spring Quarter, 2019

**Previous years:** 2017 2016 2015 2013 2012 2011 2010