# Fall 2013

 October 3, 2013 Hermitian symmetric domains and moduli of Hodge Structures — Sean Howe We will explore the connection between Hermitian symmetric domains and moduli of Hodge Structures. For example, the upper half plane is a connected component of the parameter space for complex structures on R^2 (via sending a point $\tau$ to the complex structure J with i-eigenspace spanned by [$\tau$, 1]), and it is also a Hermitian symmetric domain with the standard metric. If instead of the complex structure we consider the equivalent Hodge structure, then we can view the upper half plane as a parameter space for a certain type of Hodge structure closely related to elliptic curves. In general, parameter spaces for Hodge Structures that “look like” they come from algebraic varieties will have Hermitian symmetric domains as their connected components; this is the result we will try to understand. In the talk I will define most of the notions above. Talk notes.
 October 10, 2013 Iterated integrals — Fedya Manin Much of what I think about revolves around the question: how can we understand rational homotopy groups geometrically? The simplest example of the ideas at stake is the Hopf invariant, which distinguishes classes in $\pi_{4n-1}(S^{2n})$. I will start by defining the Hopf invariant in various ways, including using differential forms. For a more general recipe, we use iterated integrals, a way of constructing differential forms on loop space which are dual to rational homotopy classes. Since loop spaces are “formal”, that is, their rational (co)homology is generated by their rational homotopy, any rational homotopy invariant of simply connected spaces can be constructed this way. Lest one think this is too general to be of any use, we will prove a lower bound for Lipschitz constants of rational homotopy classes. This is largely work of Kuo-Tsai Chen and Richard Hain with handwaving by Sullivan and Gromov.
 October 17, 2013 Simple closed curves on surfaces — Katie Mann As a counterpart to Jenya’s talk on non-simple closed curves on surfaces, I’ll tell you a very different story about simple closed curves. Recall that a hyperbolic structure on a surface is defined by a discrete, faithful representation of its fundamental group to PSL(2,R). Bowditch asked the question “if you have a representation of a surface group to PSL(2,R) such that the image of every element representing a simple closed curve is hyperbolic in PSL(2,R), is the representation necessarily discrete and faithful?” I’ll present some background and a possible approach to this problem using a construction of Calegari that relates the Gromov norm, stable commutator length, and simple closed curves. The talk should be accessible even if you have no idea what I mean by “Gromov norm”.
 October 24, 2013 GL(2,R) orbit closures of translation surfaces — Alex Wright A translation surface can be thought of as a polygon in R^2 with parallel side identifications (up to cut and paste). For example, a regular octagon with opposite sides identified gives a genus 2 translation surface, which is flat everywhere except at one “singularity”. The standard linear action of GL(2,R) on the plane induces an action of GL(2,R) on the moduli space of all translation surfaces. Over the past three decades it has been discovered that understanding the closure of the orbit of a translation surface is necessary for understanding the geometry and dynamics present on the translation surface. We will survey recent progress on the problem of classifying orbit closures, as well as hopes for the future.
 October 31, 2013 The Auslander Conjecture* — Wouter van Limbeek In 1900, Hilbert asked (among many other things) if there are only finitely many n-manifolds locally isometrically modeled on Euclidean n-space. About 10 years later, Bieberbach proved that all such manifolds have virtually abelian fundamental group, and used this to solve Hilbert’s problem. Inspired by this, we could study compact manifolds that have an affine structure (instead of a Euclidean structure) and ask what their fundamental groups can be. The Auslander conjecture states that the fundamental group of these manifolds are virtually solvable. I will discuss positive results towards this conjecture, as well as some open questions. * Not conjectured by Auslander, but proven** by him. However, asked*** by Milnor. **: Auslander’s proof contains a flaw. ***: For complete affine manifolds, not necessarily compact ones. Noncompact counterexamples have been found by Margulis.
 November 7, 2013 The mapping class group and the moduli space of Riemann surfaces — Weiyan Chan The mapping class groups are natural generalizations of SL(2,Z) to surfaces with higher genera. Remarkably, many properties of SL(2,Z) that seem to depend on linear algebra extend to topological analogs on MCG (for example, we can classify individual element in MCG in the same way we classify hyperbolic isometries). In this talk, I will try to answer the following three questions. (1) Why should we care about MCG? (2) What are the similarity and the difference that occur when we pass from genus 1 to higher genus? (3) How can surfaces bundle, representation theory, and Teichmuller space help us understand MCG, and possibly vice versa? If time permits, I will discuss a representation theoretic version of the Riemann-Hurwitz formula. This will be a practice talk for my topic presentation. To express my gratitude for those who come, ice cream will be served.
 November 14, 2013 The topology of hyperplane complements — Jenny Wilson This will be an expository talk on some old and new results on hyperplane complements. For a finite real reflection group W, let A denote the set of its (complexified) reflecting hyperplanes, and M their complement in C^n. I will survey results by Fox–Neuwirth, Arnold, Brieskorn, Deligne, Orlik–Solomon, and Lehrer–Solomon on the topology of M, addressing questions such as: What is the fundamental group of M, and when is M a K(pi, 1) space? What is the cohomology of M, and how is it encoded by the combinatorics of the hyperplane arrangement? What is H*(M) as a representation of W? Recent progress has been made on a conjecture of Lehrer and Solomon related to this last question. Talk notes.
 November 21, 2013 Train tracks for Out(F_n) — Sebastian Hensel Most of us know that automorphisms of free groups can be realised as homotopy equivalences of graphs. But, not all such realisations are created equal. In this talk, I’ll describe the nicest class of representatives, called train track maps. I will give examples of these, try to explain why they are better than others, and even show where the trains come from (and go to). In a single talk I will not be able to cover everything these ideas were used for, but I will highlight some famous results, and other random ones that I find personally interesting.
 December 5, 2013 Rotor-routing, sandpile groups, ribbon graphs and spanning trees — Tom Church The sandpile group Pic_0(G) of a finite graph G is a discrete analogue of the Jacobian of a Riemann surface. Pic_0(G) is always a finite group, and its order coincides with the number #T(G) of spanning trees of G, so you could look for an identification of Pic_0(G) with T(G). The rotor-routing process on ribbon graphs is a discrete dynamical system that provides one such identification (sort of). I’ll explain the rotor-routing process, use it to describe the sandpile group and why Pic_0(G) is an appropriate name for it, and draw a bunch of spanning trees on the blackboard. Joint work with Melody Chan and Joshua Grochow. (FI-modules and representation stability will not appear in this talk.)

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