Geometry and Topology Seminar

Thursday October 2, 16:00, room B3.01

Stefan Friedl (Warwick)

Symplectic 4-manifolds and fibered 3-manifolds

Abstract: If N is a fibered 3-manifold, then Thurston showed in 1976 that S^1xN is symplectic. In this talk we will show the converse, i.e. if N is a 3-manifold such that S^1xN is symplectic, then N is fibered.

Thursday October 9, 16:00, room B3.01

Corinna Ulcigrai (Bristol)

Cutting sequences for linear flows in the octagon

Abstract: Consider a linear trajectory in a regular polygon with opposite sides identified; a cutting sequence is the sequence of sides which are hit.

The cutting sequences that arise in a square, known as Sturmian sequences, were characterized in terms of a renormalization operation in a paper by C. Series and are connected with the continued fraction expansion of the slope.

In a joint work with John Smillie, we give a similar characterization of cutting sequences in the case of the regular octagon and more in general regular 2n-gons. We exploit the renormalization dynamics in the space of translation surfaces and the affine diffeomorphisms in the Veech group of the corresponding lattice surfaces.

Thursday October 16, 16:00, room B3.01

Ser Peow Tan (Singapore)

On the SL(2,C) character variety of a free group on two generators

Abstract: The SL(2,C) character variety of a free group π on two generators is identified with the three dimensional complex space C^3. The outer automorphism group Out(π) acts on the character variety via polynomial automorphisms. This action is somewhat mysterious and not well understood.

We will give a brief introduction of how this can be studied via the trace maps on the so called Markoff tree, sketch some relations with real and complex dynamics, hyperbolic geometry and Kleinian groups, number theory and mathematical physics (which the speakers of the mini-workshop the following week will talk on), and mention some open problems.

Thursday October 23, 16:00, room B3.01

Workshop: Dynamics of the Markov tree

No seminar

Abstract: None

Thursday October 30, 16:00, room B3.01

Hideki Miyachi (Osaka)

Teichmuller rays and the Gardiner-Masur boundary of Teichmuller space

Abstract: I will talk about a property of the Gardiner-Masur boundary of Teichmuller space. Indeed, any point of the Gardiner-Masur boundary is the projective class of a positive function on the homotopy classes of non-trivial and non-peripheral simple closed curves on the surface, like the Thurston boundary. I will show that such positive functions can extend on the space of measured foliations. If time allows, I will discuss the behaviours of Teichmuller rays near the boundary.

Thursday November 6, 16:00, room B1.16
(Note room change.)

Yasushi Yamashita (Nara)

Computer experiments on the growth function of two-bridge link groups

Abstract: Computation of growth series of finitely generated groups is an interesting challenge. In this talk, we consider two-bridge link groups. First, we present some calculations of growth series using the software kbmag developed by D. Holt, and state our conjecture describing the growth series. Then, we will discuss three topics. (1) The location of the poles of these growth series. (2) Relation to the existence of epimorphisms between two different two-bridge link groups conjectured by Ohtsuki-Riley-Sakuma. (3) Relation to hyperbolic volume.

Topology Colloquium

Thursday November 13, 14:00, room B3.02

Ulrike Tillmann (Oxford)

The topology of manifolds embedded in Euclidean space

Abstract: Some 25 years ago, Mumford conjectured that the rational cohomology of Riemann's moduli space is a polynomial ring in degrees roughly half the genus of the underlying surface. This was proved by Madsen and Weiss. The key of a simplified proof of this theorem - which also gives a nice statement for dimensions other than 2 - which studies the cobordism category of (d-1) and d-dimensional manifolds embedded in infinite dimensional Euclidean space. The talk will mainly be based on a joint paper with Galatius, Madsen and Weiss.

Thursday November 13, 16:00, room B3.01

Andras Juhasz (Cambridge)

Sutured Floer homology

Abstract: Sutured Floer homology (in short, SFH) is an invariant of balanced sutured manifolds that provides a powerful tool for studying problems in Heegaard Floer homology. For example, using SFH one can easily show that knot Floer homology detects the genus of a knot and that it detects fibred knots. It can also be used to distinguish Seifert surfaces up to isotopy. These results all follow from a decomposition formula for SFH.

Thursday November 20, 16:00, room B3.01

Jacob Rasmussen (Cambridge)

Low genus knots in lens spaces

Abstract: Let K be a knot in the lens space L(p,q) representing a nontrivial homology class. We say K has low genus if there is an essential surface in the complement of K whose genus is small relative to p. I'll explain how the existence of such knots is related to classical problems like "which L(p,q) are surgery on nontrivial knots in S^3," and "which L(p,q) bound rational homology balls?"

Thursday November 27, 11:00, room MS.05

Loretta Bartolini (Oklahoma)

One-sided Heegaard splittings of 3-manifolds

Abstract: Heegaard splittings along orientable surfaces are well-known in 3- manifold theory: the manifold is split into a pair of handlebodies, the embedded discs for which can be used combinatorially to obtain information about both the splitting and the manifold. However, when a non-orientable surface is used in an orientable manifold, the associated Heegaard splitting is one-sided and a single handebody is obtained.

There are many natural parallels between one- and two-sided Heegaard splittings, however there are striking and far-reaching differences: the presence of singular meridian discs; and, the connection with Z_2 homology. Both properties serve to hamper existing methods, while offering new approaches.

Given the direct connection between geometrically incompressible splittings and Z_2 homology classes of the manifold, a finer degree of control of one-sided splitting surfaces can be established over their two-sided counterparts. In particular, the geometrically incompressible one-sided Heegaard splittings of even Dehn fillings of Figure 8 knot space can be explicitly constructed. This involves a result about the behavior of incompressible non-orientable surfaces under Dehn filling, which shows a marked difference from that of either two-sided splittings or incompressible surfaces.

Thursday November 27, 16:00, room B3.01

Sarah Koch (Warwick)

Thurston's pullback map

Abstract: Let ƒ: S^2 → S^2 be a ramified cover of the topological 2-sphere. If all ramification points have finite orbits under the map ƒ, then ƒ is called a Thurston map. Thurston's topological characterization of rational maps says that a Thurston map ƒ: S^2 → S^2 is either equivalent to a rational map F: P^1 → P^1, or there is a topological obstruction. One proves Thurston's theorem by defining a holomorphic map

σ_ƒ: Teich(S^2, P_ƒ) → Teich(S^2, P_ƒ)

on the corresponding Teichmuller spaces. This is called Thurston's pullback map. The branched cover ƒ is equivalent to a rational map if and only if σ_ƒ has a fixed point.

By presenting different examples, we will show that the behavior of this pullback map can be rather varied.