Geometry and Topology Seminar

Thursday May 5, 15:00, room MS.03

Francis Bonahon (USC)

Eigenvalue functions

Abstract: Given two $n$ by $n$ matrices $A$ and $B$, what can you say about the eigenvalues of $A^k B$ for a large power $k$, or more generally about the eigenvalues of a large word in $A$ and $B$? In general: not very much. However, under additional hypotheses on $A$ and $B$, one can use Labourie's notion of Anosov representations to obtain first and second order terms for the growth of these eigenvalues. I will discuss results and examples. This is joint work with Guillaume Dreyer.

Thursday May 12, 15:00, room MS.03

Irida Altman (Warwick)

Sutured Floer homology distinguishes between Seifert surfaces

Abstract: Sutured Floer homology (SFH) was introduced by András Juhász as a tool to study an important class of $3$-manifolds with boundary called sutured manifolds. Up until now, evidence had suggested that two minimal genus Seifert surfaces of a knot cannot necessarily be distinguished by using only the (graded) SFH of their complementary manifolds. Nevertheless, using recent work of Friedl, Juhász and Rasmussen, I will exhibit an example where the Euler characteristic of SFH distinguishes between two Seifert surfaces. Time permitting, I will also compute the associated sutured Floer polytopes and show that they too can be used to distinguish between Seifert surfaces.

Thursday May 12, 16:00, room MS.03

Sara Maloni (Warwick)

Bers-Maskit slices of the quasi-Fuchsian space

Abstract: Given a surface $S$, Kra's plumbing construction endows $S$ with a projective structure for which the associated holonomy representation $f$ depends on the `plumbing parameters' $t_i$. In this talk we will describe a more general plumbing construction which gives us a group $G$ in a particular slice of the quasi-Fuchsian space $\QF(S)$ (instead of the Maskit one given by Kra's plumbing construction). Using the complex Fenchel-Nielsen coordinates for $\QF(S)$, we can describe this slice, called the Bers-Maskit slice $\BM(S)$, as a subset of the slice where the length parameters take a fixed real value. Then one can see that, as these values tend to zero, the slices $\BM(S)$ tend to the Maskit slice $\M(S)$. The Bers-Maskit slice are also a connected component of the more general linear slices $\L(S)$. Some results about those slices will be described.

Thursday June 9, 15:00, room MS.03

Graeme Segal (Oxford)

Passing from Riemannian to Lorentzian manifolds: the use of imaginary time in quantum field theory.

Abstract: Physical space-time is a manifold with a Lorentzian (or pseudo-Riemannian) metric, but it is common in quantum field theory to work in an artificial Riemannian manifold, and at the end to return to the actual space-time by a not-too-explicit process of analytic continuation. The possibility of this analytic continuation expresses the positivity of the energy of physical systems. In this talk I shall discuss the geometry of a particular space of complex-valued metrics defined on a general smooth manifold; it is a complexification of the space of Riemannian metrics, and has the Lorentzian metrics on its boundary.