Geometry and Topology Seminar

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

Jessica Banks (Oxford)

The Kakimizu complex of a link

Abstract: We give an introduction to the Kakimizu complex of a link, covering a number of recent results. In particular we will see that the Kakimizu complex of a knot may be locally infinite, that the Alexander polynomial of an alternating link carries information about its Seifert surfaces, and that the Kakimizu complex of a special alternating link is understood.

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

None (None)

None

Abstract: None.

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

Michael Farber (Warwick)

Stochastic algebraic topology

Abstract: Talk cancelled.

Thursday February 2, 15:00, room MS.04

Stefan Friedl (Köln)

Complexity of surfaces in circle bundles over closed 3-manifolds

Abstract: Let $M$ be a circle bundle over a closed $3$-manifold $N$. Let $S$ be a Thurston norm minimizing surface in $N$ which lifts a surface $T$ in $M$. Using recent work of Ian Agol and Dani Wise we will show that in many cases $T$ has minimal complexity in $M$ as well.

This is joint work with Stefano Vidussi.

Thursday February 9, 15:00, room MS.04

Ian Short (Open)

Ford circles and continued fractions

Abstract: Continued fractions can be represented by chains of horospheres in hyperbolic space. We discuss the application of this representation of continued fractions in proving a variety of theorems on convergence, graph geodesics, and Diophantine approximation.

It will be an accessible talk with plenty of diagrams.

Thursday February 16, 15:00, room MS.04

Chris Leininger (UIUC)

Mapping class groups, Kleinian groups and convex cocompactness

Abstract: For mapping class groups there is a notion of convex cocompactness, due to Farb and Mosher, defined by way of analogy with the concept of the same name in Kleinian groups. On the other hand, there are certain Kleinian groups which can themselves naturally be thought of as subgroups of mapping class groups. After describing some of the background, I will discuss a direct relationship between convex cocompactness in the two settings for this special class of groups. This is joint work with Spencer Dowdall and Richard Kent.

Thursday February 23, 15:00, room MS.04

Karen Vogtmann (Cornell)

Automorphisms of free groups, hairy graphs and modular forms

Abstract: The group of outer automorphisms of a free group acts on a space of finite graphs known as outer space, and a classical theorem of Hurwicz implies that the homology of the quotient by this action is an invariant of the group. A more recent theorem of M.Kontsevich relates the homology of this quotient to the Lie algebra cohomology of a certain infinite-dimensional symplectic Lie algebra. Using this connection, S.Morita discovered a series of new homology classes for $\Out(F_n)$. In joint work with J.Conant and M.Kassabov, we reinterpret Morita's classes in terms of hairy graphs, and show how this graphical picture then leads to the construction of large numbers of new classes, including some based on classical modular forms for $\SL(2,\ZZ)$.

Thursday March 1, 15:00, room MS.04

Daryl Cooper (UCSB)

Convex projective manifolds and cusps

Abstract: A properly convex real projective manifold or orbifold is $W/G$ where $W$ is the interior of a compact convex set in real projective space disjoint from some hyperplane and $G$ is a discrete group of projective transformations which preserve $W$. The strictly convex case requires there is no line segment in the boundary of $W$. Strictly convex structures have many similarities to hyperbolic (constant curvature minus one) structures, particularly in the finite volume case. By contrast, properly convex structures are far more general. We will discuss aspects of these objects from the Kleinian groups perspective. This material is in arXiv:1109.0585: On convex projective manifolds and cusps.

Thursday March 8, 15:00, room MS.04

John Hunton (Leicester)

The shape of an attractor

Abstract: Suppose we have a compact differentiable manifold $M$ with a diffeomorphism $d: M \to M$, defining a hyperbolic dynamical system. Suppose $A$ is an attractor for this system: what can the space $A$ look like, even through the eyes of basic topological invariants, cohomology, K-theory etc? Of course $A$ can be very complicated indeed, but I will look at the case of those attractors of dimension one less than that of $M$, and show that there is a nice connection between each such attractor and certain moduli spaces of aperiodic tilings, objects for which there is already powerful machinery available for their analysis. Conversely, we derive constraints and obstructions for when a tiling space has a codimension-one embedding in a manifold.

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

Alina Vdovina (Newcastle)

Trivalent expanders and Riemann surfaces

Abstract: We introduce a family of trivalent expanders which tessellate compact hyperbolic surfaces with large isometry groups. We compare this family with Platonic graphs and modifications of them and prove topological and spectral properties of these families. This is joint work with Ioannis Ivrissimtzis and Norbert Peyerimhoff.