Geometry and Topology Seminar

Friday January 9, 14:00, room MS.05

Jean Raimbault (Toulouse)

Homology of hyperbolic three--manifolds and analytical torsion

Abstract: In this talk I'll try to survey recent work on the integral homologies of hyperbolic three--manifolds of finite volume, essentially in the two following directions :

• Given a manifold $M$, and a sequence of finite covers $M_n$, when can we establish exponential growth of the order of the torsion subgroup of the first homology $H_1(M_n)$ in the degree of the cover $M_n \to M$?
• For a sequence of arithmetic (in a somewhat restricted sense) hyperbolic three--manifolds (which do not have to be covers of a fixed manifold, or even to be commensurable with each other), we should always have exponential growth of torsion homology (with respect to the volume).

I will begin by giving a quick account of the conjectural picture surrounding the two items above. Then I will describe analytic tools (analytic torsion and the Cheeger--Müller Theorem) that can be applied to prove analogues of it for the homology with twisted coefficients, first in the case of compact manifolds (this is due to Bergeron--Venkatesh), and then in the general case of manifolds with cusps.

Thursday January 22, 15:00, room MS.04

Arnaud Chéritat (Toulouse/Bordeaux/CNRS)

Abstract: Smale proved in the 1960's that one can turn the sphere inside out in euclidean 3-space, via a sequence of immersions. However, he did not give an explicit description of the necessary regular homotopy. Since then many people have given explicit sphere eversions. I will give a brief history and will then describe a, possibly new, way to turn the sphere inside out.

Thursday January 29, 15:00, room MS.04

Mark Bell (Warwick)

Conjugacy done quickly

Abstract: We will look at the conjugacy problem for the mapping class group of a surface. In particular, we will discuss an effective solution using train tracks and the action of the mapping class group on the space of measured laminations.

Thursday February 12, 15:00, room MS.04

David Epstein (Warwick)

Multi-dimensional splines, Tarski's quantifier elimination, and manifold learning

Abstract: Suppose $Z \subset \RR^n$ is finite: for example, the data points recorded from a scientific experiment. When the underlying natural laws are well behaved the points of $Z$ are sampled, with some noise, from a smoothly embedded, connected, compact manifold $M$ of dimension $m$ much less than $n$. We wish to recover an approximation of the unknown manifold $M$, knowing only $Z$.

No procedure will work on all examples. Nonetheless, we may try the following.

1. Estimate $m$ and estimate the size of the noise.
2. Carefully choose a sparse subset $L \subset Z$.
3. Construct a linearly embedded simplical complex $K \subset \RR^n$, with vertices $K^0 = L$, such that $K$ is a combinatorial $m$-manifold.
4. Consider all countinuous piecewise polynomials $f \from K \to \RR^n$ of degree at most $d$. The condition that $f(K)$ should be a $C^1$ submanifold of $\RR^n$ gives rise to constraints on the coefficients of $f$. The constraints are computed via the "cylindrical algebraic decomposition" algorithm, arising from software implementations of Tarski's theorem.
5. Define a projection $\pi \from Z \to K$.
6. Minimize $\sum_{z \in Z} ||z - f(\pi(z))||^2$ subject to the constraints found in Step 4. This solves for the coefficients of $f$, restricted to each simplex, and thus determines both $f$ and the approximating manifold $f(K)$.

Thursday February 19, 15:00, room MS.04

Brian Bowditch (Warwick)

Large scale geometry of Teichmüller space and mapping class groups

Abstract: We describe a number of results regarding the coarse rank and quasi-isometric rigidity of various spaces associated to a compact surface. Of particular interest are the mapping class group (viewed as a geometric object) as well as Teichmüller space in the Teichmüller or Weil-Peterson metrics. Some of this reproves known results by several authors, though other results appear to be new. The general theme is that these spaces all admit a certain ``coarse median'' structure, which allows much of this material to be put in a unified framework. A basic idea was originally inspired by constructions of Behrstock and Minsky.

Thursday February 26, 15:00, room MS.04

Martin Möller (Frankfurt)

Teichmüller curves in genus three and unlikely intersections

Abstract: The group $\SL_2(\RR)$ acts on the moduli space of flat surfaces and, after the solution of the Ratner-type question by Eskin-Mirzakhani-Mohammadi, the classification problem for orbit closures is the main open question. For almost a decade no new examples of one-dimensional orbit closures, that is, Teichmüller curves, have been found.

In this talk we present various techniques to rule out the existence of Teichmüller curves. Besides geometric arguments, we use one number-theoretic result: we extend the theory of unlikely intersections from the multiplicative group $G_m$ to products with $G_a$ factors.

Thursday March 5, 15:00, room MS.04

Peter Haïssinsky (Toulouse)

Quasi-isometric rigidity of the class of convex-cocompact Kleinian groups

Abstract: The talk will be devoted to discussing background and ingredients for the proof of the following theorem: a finitely generated group quasi-isometric to a convex-cocompact Kleinian group contains a finite index subgroup isomorphic to a convex-cocompact Kleinian group.

Thursday March 12, 15:00, room MS.04

TBA (TBA)

TBA

Abstract: TBA