Topology/Geometry Seminar
3:20  4:40 p.m. Tuesday
124 HLL
Please contact
Steve Ferry,
Feng Luo,
Xiaochun
Rong,
Saul Schleimer, or
Chris Woodward
if you would like to speak or suggest a speaker.
Seminar Schedule  Fall, 2005
(Please scroll down for abstracts.)













































Schedule with abstracts







We consider the Quantum Teichmuller Space of the punctured surface introduced by ChekhovFockKashaev, and formalize it as a noncommutative deformation of the space of the algebraic functions on the Teichmuller space of the surface. We then investigate the representation theory of this algebraic object and define a quantum hyperbolic invariant for pseudoAnosov diffeomorphisms of the surface. 



This talk will be an introduction to the curve complex, especially its combinatorial structure, following the work of Masur and Minsky. We will discuss the geometry of subsurface projection maps from a completely elementary point of view. If there is time I will touch lightly on applications to hyperbolic threemanifolds, Teichmuller space, and the mapping class group. 



We will discuss how to use the Seidel element associated to a very particular family of hamiltonian circle actions on symplectic manifolds to compute their small quantum cohomology. 



(Joint work with Howard Masur.) This is a sequel to my previous talk: we will discuss the geometry of the disk complex and how it fits inside of the curve complex. I will sketch a proof of the classification of holes for the disk complex and give an application to Heegaard splittings. 



(Joint work with Francesco Costantino.)
It is known since 1954 that every 3manifold bounds a 4manifold.
Thus, for instance, every 3manifold has a surgery diagram. There are
many proofs of this fact, including several constructive ones, but
they do not bound the complexity of the 4manifold. (By "complexity"
of a manifold we mean the minimum number of simplices in a
triangulation.) Given a 3manifold M of complexity n, we show how to
construct a 4manifold bounded by M of complexity O(n^2). It is an
open question whether this quadratic bound can be replaced by a linear
bound.
The natural setting for this result is shadow surfaces, a representation of 3 and 4manifolds that generalizes many other representations of these manifolds. One consequence of our results is some intriguing connections between the complexity of a shadow representation and the hyperbolic volume of a 3manifold. Our results can also be phrased in terms of the singularities of smooth maps. In particular, the minimum number of "crossing singularities" of a map from a hyperbolic 3manifold to the plane is bounded below and above by the hyperbolic volume. 



(Joint work with Jason Behrstock and Cornelia Drutu.) Thickness is a condition on a metric space, invariant under quasiisometry, which when satisfied by a finitely generated group G with its word metric rules out strong relative hyperbolicity of G with respect to any finite collection of subgroups. We investigate thickness and give many examples from natural classes of groups, including mapping class groups of all finite type surfaces (except for the known low complexity cases where the mapping class group is virtually free) automorphism and outer automorphism groups of all finite rank free groups (except for ranks 1 and 2) and others. 



Cancelled 



(Joint work with Mladen Bestvina.) Let F be a finite rank free
group. Given an open sentence S(x) such as "\forall y \exists z (x^z
y^2 z^3 = 1 \wedge xyx^{1} y^{1} \not= 1)", we are interested in \{
a \in F \mid S(a) is true in F \}. Such sets are said to be
definable. The main question is: Which subsets of F are
definable? We will discuss a property that is shared by definable sets
and that is useful in showing that certain sets are not definable.
Another question is: Given a definable set, how many quantifiers are needed in its definition? We explore some invariance properties that can be used, for example, to explain an example of Razborov where two quantifiers are needed. Although the statements are logic theoretic in nature, the tools (pioneered by Sela) are geometric. We will explain how this works. 



(Joint work with B. Wilking.) We prove that the fundamental group of a manifold with an upper diameter and a lower Ricci curvature bound has a presentation with a universally bounded number of generators and relators. We also prove a conjecture of Gromov that a manifold which admits almost nonnegative Ricci curvature has a virtually nilpotent fundamental group. 



Graph manifolds are 3manifolds that have no hyperbolic pieces in their JSJ decomposition. They were classified almost 40 years ago by Waldhausen. We will describe how a less focussed view of the classification yields sharper views of the properties of these manifolds. 



We defined two new complete Kahler metrics: the Ricci metric and the perturbed Ricci metric, on the moduli space of hyperbolic Riemann surfaces. We derived curvature formulae of these new metrics and showed that they have good curvature properties and asymptotic behavior. These new metrics are used to anchor the KahlerEinstein metric. As corollaries, we showed that all the complete canonical metrics on the Teichmuller space and moduli space are equivalent. Furthermore, the KahlerEinstein metric has strongly bounded geometry. Also, the logarithmic cotangent bundle of the DeligneMumford moduli is strictly stable with respect to the canonical polarization. We also proved the goodness of the WeilPetersson metric and the new metrics which imply that the Chern classes of the log cotangent bundle can be computed by the Chern forms of these metrics as currents. 



The length spectrum of a manifold does not behave well when the manifold is deformed. Manifolds which converge smoothly can have suddenly appearing geodesic loops, and manifolds which converge in the GromovHausdorff sense can have geodesics disappear. Special subsets of the length spectrum don't have these difficulties. One subset, the Covering Spectrum, developed with Guofang Wei of UCSB, behaves continuously with respect to the GH convergence of manifolds and still captures the lengths of smoothly closed geodesics representing generators of the fundamental group. Another collection of subsets, the 1/k geodesics, cover the complete length spectrum and never disappear under GH convergence, although naturally they may suddenly appear. 



I would like to present the following results:
(1) Let $p$ be any point on a closed Riemannian manifold $M^n$ of dimension $n$. Then there exists a geodesic loop based at that point of length $\leq 2nd$, where $d$ is the diameter of $M^n$. (2) Let $M^n$ be a closed Riemannian manifold of dimension $n$. Then the length of a shortest geodesic net on $M^n$ is $\leq (n+1)d$, where $d$ is the diameter of $M^n$ and $\leq (n+1)(n+2) FillRad M^n$, where $FillRad M^n$ is the Filling Radius of $M^n$. I will also talk about some estimates for the length of a shortest closed geodesic and for the smallest area of a minimal surface. 



In this talk, we will study the singularities of the Ricci flow on manifolds with positive curvature operator. We first refine Hamilton's dimension reduction theorem. Then we prove a convergence result of the dilation limit. Finally we will talk about the limiting case: the shrinking Ricci soliton, we show that under a addtional condition, all the compact gradient shrinking Ricci solitons with positive curvature operator must be Einstein, hence of constant curvature. 