Winter 2014

January 9, 2014
A Prehistory of the Johnson Homomorphism — Nick Salter
In 1980, Dennis Johnson found an abelian quotient of the Torelli group which has since proven to be the centerpiece of its modern theory. In this talk I will provide some historical context by discussing earlier work of Sullivan and Chillingworth which Johnson credited with inspiring his own developments, and I will show how to interpret their theorems in the context of the Johnson homomorphism. No prior exposure to the Torelli group is assumed.
Talk notes.
January 16, 2014
Reconstruction of compact sets via noisy point clouds — Kate Turner
The scenario is that there is some unknown compact subset $A$ in Euclidean space (or some other Riemannian manifold with nowhere negative curvature) and what we have is a set of points that are a noisy approximation of it. We want to create a simplicial complex which is homotopic to $A$. We define a generalization of critical points of the distance function called $\mu$-critical points. We will find sufficient conditions in terms of $\mu$-critical points (and the amount of noise, etc) for the existence of a vector field whose flow gives a deformation retraction from the union of balls around the points in the point cloud down the the compact set itself – and hence proving the homotopy equivalence.
January 23, 2014
Uniquely ergodic IETs (are path-connected) — Sebastian Hensel
This talk will be mainly an introduction into interval exchange transformations, and (if time permits) at the end I will talk about a joint theorem with Jon Chaika (hint: the result may be contained in the title).
IETs are a very explicit model of a dynamical system which is related to abelian differentials on surfaces and foliations, and therefore they are interesting both to dynamicists and low-dimensional geometers.
One main appeal is that one can very explicitly work with them and really see by hand what is happening. I will explain Rauzy induction and show how this can be used to show all kinds of useful results.
January 30, 2014
Permutation groups and configuration spaces, or, How to use the cohomology of the pure braid group to count polynomials over finite fields — Jenny Wilson
In this talk I will describe some recent work by Church–Ellenberg–Farb, “Representation stability in cohomology and asymptotics for families of varieties over finite fields” (arXiv:1309.6038). I will overview some tools — twisted coefficients, transfer maps, Étale cohomology, and the Grothendieck-Lefschetz formula — which we will use to relate the cohomology of the pure braid group P_n to certain statistics for polynomials over finite fields. Results of Church–Ellenberg–Farb shows that these statistics stabilize in n in a particular sense.
Talk notes.
February 6, 2014
Stable Commutator Length and Quasimorphisms — Subhadip Chowdhury
In Topology, it is often important to be able to construct and
classify surfaces
of least complexity mapping to a given space, possibly subject to further
constraints and with boundary conditions. The relevant homological tool to
describe complexity in this context is *stable commutator length*. In this
talk, I will try to present the algebraic, geometric and functional
analytic characterization of scl and explain how it is related to
quasimorphisms and second bounded cohomology of groups. In addition, we
will also find out how Quasimorphisms arise from Hyperbolic geometry via
Euler class and in Symplectic geometry via Maslov class.

This will be a talk on some parts of my Topic presentation. Refreshments
will be provided to the attendees to show my gratitude 🙂

February 13, 2014
Surface diffeomorphisms with positive topological entropy — Clark Butler
Let f be a C^2 diffeomorphism of a closed surface which has positive topological entropy. Then f possesses invariant horseshoes on which an arbitrarily large proportion of the complexity of the dynamics is supported. This theorem, due to Katok, is one of the most striking applications of the theory of nonuniform hyperbolic dynamics. I will explain what horseshoes are, what topological entropy is, and the geometry behind the hypothesis and conclusion of the theorem. I will also give some idea of what goes into the proof, which invokes every single fancy concept from hyperbolic dynamics.
February 20, 2014
Your Favorite Solvable Groups are S-Arithmetic — Daniel Studenmund
What’s your favorite infinite solvable group? A lattice in SOL geometry? A solvable Baumslag-Solitar group? A lamplighter group? Or maybe something simple like Z or the Heisenberg group. In any case, your favorite solvable group is probably S-arithmetic.

In this talk I will define S-arithmeticity, exhibit each of the above as an S-arithmetic group, and discuss some rigidity results (for both solvable and semisimple groups).

February 27, 2014
Pontryagin classes of locally symmetric spaces — Bena Tshishiku
For a smooth manifold M, the Pontryagin classes p_i(M) are smooth invariants that live in the cohomology ring H^*(M). In this talk I will describe how these classes can be computed concretely for M a locally symmetric space (e.g. hyperbolic manifold) using the action of pi_1(M) on the visual boundary of the universal cover. This talk will include representation theory of Lie groups, Chern-Weil theory, cohomology of classifying spaces, and flat bundles. I will also mention Hirzebruch proportionality and connections to characteristic classes of surface bundles.
March 6, 2014
The wild world of aspherical manifolds and the Davis trick — Wouter van Limbeek
Aspherical manifolds are studied throughout topology (e.g. surfaces, many 3-manifolds, nonpositively curved manifolds, etc.), and based on these examples, you may conjecture that closed aspherical manifolds have all sorts of amazing topological properties. E.g. are their universal covers homeomorphic to R^n? Is every topological aspherical manifold smoothable? And since an aspherical manifold is determined up to homotopy by its fundamental group, you might as well conjecture amazing properties for such groups. E.g. are all such groups linear? As it turns out, counterexamples to all of these questions (and more) were found by a revolutionizing construction now known as the Davis trick. I will explain this construction and how to give the answers to (some of) the above questions.
March 13, 2014
What is lov? (baby, don’t hurt me) — Paul Apisa
Things that we will prove:

1. Conformal homeomorphisms all have topological entropy = 0.

2. Pseudo-Anosov surface homeomorphisms are entropy minimal in their homotopy class (we’ll compute the entropy!)

3. By Thurston’s classification of surface diffeomorphisms, after cutting along a system of nonisotopic stabilized curves any surface diffeomorphism can be broken into pseudo-Anosov and conformal pieces (after isotopy). This representative is entropy minimal and we’ll compute all of its ergodic measures.

Along the way, we’ll work out nice upper bounds for the entropy of quasi-conformal maps and wave in passing at the formula for the entropy of holomorphic maps between Kahler manifolds (which can be computed homologically).

Definitions of topological entropy and quasi-conformality will be reviewed and then illustrated with examples. The main ideas of proofs will be sketched through example. In sum, the things we discuss will be beautiful to contemplate.

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