Geometry and Topology Seminar

Thursday April 24, 16:00, room B3.03

Edmund Harriss (Imperial)

Some aspects of the Penrose tiling

Abstract: The Penrose tiling was discovered in the 1970's and is one of the simplest known sets of aperiodic tiles. These are sets of tiles which tile the plane but only in a non-periodic manner. They are also very beautiful as they have a five-fold rotational symmetry.

In this talk I will discuss the Penrose tiling and the links between two methods of constructing it: the substitution method used by Penrose and the projection method discovered by De Bruijn.

Thursday May 1, 16:00, room B1.01

Saul Schleimer (Warwick)

Compressed words in hyperbolic groups

Abstract: Suppose that G is a Gromov hyperbolic group. Then the "compressed word problem" in G has a polynomial-time solution. (When G is a free group this result is due to Markus Lohrey.) As a consequence, the word problems in Aut(G) and Out(G) are also polynomial-time. Since surface groups are hyperbolic, this gives a new solution to the word problem in the mapping class group.

Tuesday May 20, 16:00, room MS.05

Adam Epstein (Warwick)

Thurston Endomorphisms of Teichmuller Spaces: some worked examples arising from holomorphic dynamics

Abstract: Let f be an rational endomorphism of the Riemann Sphere, P¹. The postcritical set of f, P_f, is by definition the smallest forward invariant set containing every critical value. Maps for which P_f is finite are the subject of much study. Over 25 years ago, Thurston proved a fundamental existence and uniqueness theorem concerning such maps. A key consideration is the Thurston endomorphism &sigma_f of the Teichmuller space Teich(P¹,P_f), defined via pullback of complex structures. In recent work with Xavier Buff, Sarah Koch and Kevin Pilgrim, we have obtained explicit descriptions of &sigma_f in various cases.

Thursday May 22, 15:15, room B1.01

François Guéritaud (Orsay)

Which arborescent links are hyperbolic?

Abstract: Arborescent links in the 3-sphere form an interesting class with several topological characterizations: for example, these are the links whose associated double branched covers are graph manifolds. It has been known since unpublished work of Bonahon and Siebenmann which arborescent links are hyperbolic. I will outline a new, self-contained proof of this result, using a generalization of the notion of angle structures for a triangulation. (Joint work with David Futer.)

Thursday May 22, 16:30, room B1.01

Jeremy Kahn (Stony Brook)

The Ehrenpreis conjecture for punctured surfaces

Abstract: The Ehrenpreis conjecture states that any two Riemann surface of the same general type have finite covers that are arbitrarily close in the Teichmuller metric. Let S be a non-compact finite-area hyperbolic Riemann surface, and let &epsilon > 0. We normalize the Weil-Peterssen metric on hyperbolic surfaces by dividing the standard inner product by the area of the surface; this makes it invariant under passing to a finite cover and strictly smaller than the Teichmuller metric. We prove that there is a finite degree cover S' of S such that the normalized Weil-Peterssen distance between S' and a cover of the modular surface is less than &epsilon. (Joint work with Vladamir Markovic.)

Thursday May 29, 16:00, room B1.01

Andrew Lobb (Imperial)

Khovanov homologies and the slice genus

Abstract: We will give an overview of the combinatorial knot invariants known as Khovanov homologies. We shall discuss applications of these invariants to low-dimensional topology, and in particular see how some perturbations of the homologies give rise to lower bounds on the "slice genus" of a knot. The slice genus of a knot is the minimal genus among smoothly embedded surfaces in a 4-ball with boundary the knot on the bounding 3-sphere.