Winter 2015

January 8, 2015
Arithmetic statistics and the topology of discriminant complements — Sean Howe
How many squarefree polynomials of degree d are there with coefficients in a finite field F_q? What’s the average number of linear factors of such a beast? How about irreducible quadratics? By applying the theory of representation stability to configuration spaces and using the Grothendieck-Lefschetz formula, Church, Ellenberg, and Farb are able to answer these questions and many more about the arithmetic statistics of polynomials.
What if we want to ask similar questions about multivariable polynomials? For example, how many smooth hypersurfaces of degree d in P^n are there over a finite field F_q?
Or, how about smooth curves? For example, how many smooth genus g curves are there over a finite field F_q?
In this talk, I will explain a way to think about representation stability in these contexts, and some of the difficulties that arise in trying to carry out these computations. The emphasis will be on how various results (some recent, some old, none of them mine!) on stable cohomology naturally fit into this framework.
January 15, 2015
Methods in group actions on surfaces — Nick Salter
Let M be a smooth manifold. What do the (finitely-generated) subgroups of Diff(M) look like? For a fixed group Gamma, can we classify all homomorphisms Gamma \to Diff(M)? Does anything special happen when Gamma is a higher-rank lattice? What about when Gamma is the mapping class group of M? In this talk, I will discuss some results addressing the above questions when M is a surface. This is an especially nice setting because of the interplay with the rich theory of the mapping class groups of surfaces.
January 22, 2015
Compute cohomology by counting — Weiyan Chen
In the late 40’s, Andre Weil observes that computing cohomology will help to count objects over finite fields. In this talk, we are going to switch the server and the master in Weil’s brilliant observation (number theory will serve topology this time!) and consider how counting can help us compute cohomology. I will illustrate this idea by presenting examples of computations, one of which is to compute the twisted cohomology of the configuration space (which we know to stabilize by a paper of Church-Ellenberg-Farb. But what do they stabilize to?).
January 29, 2015
The Teichmüller space (with the Teichmüller metric) is negatively
curved — Ian Frankel
In 1974, Masur disproved a longstanding conjecture by showing
Teichmüller space (with the Teichmüller metric) violates every definition
of a non-positively curved metric space. About 40 years, later, along with
two alumni of this department, he showed that statistically, the
Teichmüller metric exhibits negative curvature properties. We will explain
this with the help of several interesting tools, including Minsky’s
product regions theorem, random walks, and volume estimates for balls in
Teichmüller space.
February 5, 2015
Building 3-manifolds with mapping classes (the old way) — Sebastian Hensel
This talk will be about nice, old results that (in my opinion) everyone should know.
We’ll start with some basics of Heegaard splittings and show (via mapping class groups) that every closed 3-manifold is obtained from S^3 by surgery. Depending on the time we’ll then explore some very interesting connections between the Torelli group and homology 3-spheres.
February 10, 2015
Euler class and its application — Lei Chen
Euler class in Homeo(S^1) is actually a class of bounded cohomology. In Ghys’ paper, he proved a very beautiful theorem that the pullback of this class actually determines something called semiconjugacy. Using this, we can prove that all the homomorphism of PSL(2,Z) to Homeo(S^1) is up to conjucagy the same.(if it’s not trivial) Then by using some properties of Homeo(S^1), we know that all the homomorphism of Homeo(S^1) to itself is a conjugacy.
February 19, 2015
Conformal rigidity for boundaries of negatively curved manifolds — Clark Butler
Given a closed negatively curved Riemannian manifold M, what are sufficient conditions for the existence of an isometry M —> N, where N is a hyperbolic manifold? Rigidity theory has provided numerous examples of conditions which produce such an isometry, many of which have interesting geometric, dynamical, and probabilistic interpretations. I will survey a subset of these results and then focus on one that has been of recent interest for myself: if the boundary of the universal cover of M carries a C^{1} structure with respect to which the fundamental group of M acts uniformly quasiconformally, then M is isometric to a hyperbolic manifold. The proof demonstrates the interesting interplay between geometry and dynamics which is common to these results.
February 26, 2015
What Kobayashi hyperbolicity can do for you! — Paul Apisa
Here’s a great theorem: “holomorphic maps between hyperbolic Riemann surfaces are distance non-increasing” (this is the Schwarz-Pick theorem). Corollary 1: the great Picard theorem. Corollary 2: Hol(X, Y) is finite for X and Y compact hyperbolic Riemann surfaces. Corollary 3: The only holomorphic maps from C to a hyperbolic Riemann surface are constant.
In this talk we’ll define Kobayashi hyperbolicity using the Schwarz-Pick theorem as our guide and we will show that for compact complex manifolds X being Kobayashi hyperbolic is equivalent to the property that Hol(C, X) only contains constant functions. We’ll motion at a vast generalization of the great Picard theorem for Kobayashi hyperbolic manifolds. Then we’ll show that moduli space with the Teichmuller metric is Kobayashi hyperbolic and apply everything we talked about previously to study holomorphic families of Riemann surfaces. Finally we’ll have an application to a result of McMullen on rigidity of Teichmuller curves.
I want this talk to be panoramic and impressionistic. Pictures will be drawn. Complete proofs will not be given. Lots of proofs will be sketched.
March 5, 2015
How Lipschitz is a nullhomotopy? — Fedya Manin
Let X and Y be CW complexes with Y simply-connected.  In
1999 Gromov conjectured that two homotopic L-Lipschitz maps from X to
Y are homotopic through CL-Lipschitz maps, with C depending on X and
Y.  In 2011, Ferry and Weinberger remarked that a more meaningful
question is whether the homotopy itself is CL-Lipschitz.  It turns out
that the answer to this question in general is no, but even the case
of S^3 -> S^2 is still unclear.  On the other hand, Gromov’s
conjecture may well be correct.  In my talk, I will focus on
establishing upper and lower bounds in some interesting special cases.
This is ongoing joint work with Chambers, Dotterrer, and Weinberger.
March 12, 2015
Visualizing the unit ball for the Teichmuller metric — Ronen Mukamel
The Teichmuller metric is a Finsler metric on the moduli space of Riemann surfaces whose study has celebrated applications to the classification of mapping classes, the geometry of three manifolds and dynamics of polygonal billiard tables.  In this talk, I will describe an algorithm for computing the Teichmuller norm for Riemann surfaces presented as algebraic curves.