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# Geometry and Topology Seminar

## Warwick Mathematics Institute, Term II, 2018-2019

 Thursday January 10, 15:00, room MS.03 None (None) None Abstract: None

 Thursday January 17, 15:00, room MS.03 Artem Dudko (IMPAN, Warsaw) On computational complexity of Cremer Julia sets Abstract: Informally speaking, a compact subset of a plane is called computable if there is an algorithm which can draw arbitrarily good approximations of this set. Computational complexity measures how long does it take to draw these approximations. It is known that for some classes of rational maps (e.g. hyperbolic, parabolic and Collet-Eckmann) the Julia sets have polynomial complexity. For others (e.g. Siegel) the Julia sets can have arbitrarily high computational complexity and even may be uncomputable. However, not much is known in the case of presence of Cremer periodic points. We show that there exist abundant Cremer quadratic polynomials with Julia sets of arbitrarily high complexity. The talk is based on a joint work with Michael Yampolsky.

 Thursday January 24, 15:00, room MS.03 Dmitry Jakobson (McGill) On small gaps in the length spectrum Abstract: We discuss upper and lower bounds for the size of gaps in the length spectrum of negatively curved manifolds. For manifolds with algebraic generators for the fundamental group, we establish the existence of exponential lower bounds for the gaps. On the other hand, we show that the existence of arbitrary small gaps is topologically generic: this is established both for surfaces of constant negative curvature, and for the space of negatively curved metrics. While arbitrary small gaps are topologically generic, it is plausible that the gaps are not too small for almost every metric; we discuss one result. This is joint work with Dima Dolgopyat.

 Thursday January 31, 15:00, room MS.03 None (None) None Abstract: None

 Thursday February 7, 15:00, room MS.03 Stavros Garoufalidis (Georgia Tech) Counting incompressible surfaces in three-manifolds Abstract: I will give a method for deciding if a normal surface is incompressible, and if two such surfaces are isotopic, and apply this to count and list incompressible surfaces in 3-manifolds. We will illustrate our results with examples, and connect them with measured laminations in 3-manifolds. Joint work with Nathan Dunfield and Hyam Rubinstein.

 Thursday February 14, 15:00, room MS.03 Raphael Zentner (Regensburg) Irreducible $\SL(2,\CC)$-representations of homology three-spheres Abstract: We prove that the splicing of any two non-trivial knots in the three-sphere admits an irreducible $\SU(2)$-representation of its fundamental group. This uses instanton gauge theory, and in particular a non-vanishing result of Kronheimer-Mrowka and some new results that we establish for holonomy perturbations of the ASD equation. Using a result of Boileau, Rubinstein, and Wang (which builds on the geometrization theorem of three-manifolds), it follows that the fundamental group of any integer homology three-sphere (different from the three-sphere itself) admits irreducible representations of its fundamental group in $\SL(2,\CC)$. As a corollary, three-sphere recognition is in co-NP if the generalized Riemann hypothesis holds. It was known to be in NP by a result of Schleimer.

 Thursday February 21, 15:00, room MS.03 Montse Casals (Basque Country) Real cubings Abstract: Real trees are a class of metric spaces that generalises simplicial trees. The study of real trees and in particular, the structure of groups that act on them, was crucial for the description of the JSJ decomposition as well as a key tool in the solution of Tarski problems for free groups. In this talk, I will review real trees and some key results of this theory. I will then introduce a new class of metric spaces that generalises real trees, called real cubings, explain some structural results and formulate some open questions. Joint work with Mark Hagen and Ilya Kazachkov.

 Thursday February 28, 15:00, room MS.03 None (None) None Abstract: None

 Thursday March 7, 15:00, room MS.03 None (None) None Abstract: None

 Thursday March 14, 15:00, room MS.03 Charles Fougeron (Max Planck) Lyapunov exponents and surface groups representations Abstract: On a hyperbolic surface, we consider a representation of its fundamental group in a matrix group; in other words, a flat bundle above the surface. Under some integrability hypothesis we can associate to theses objects a set of Lyapunov exponents and a flag decomposition of the bundle that will characterize the dynamics of the vectors in the bundle transported along hyperbolic geodesics on the surface. This decomposition is sometimes called dynamical variation of Hodge structure. In fact, some recent results relate this dynamical decomposition with the holomorphic subbundles of the flat bundle. During the seminar, I will explain this link, and will consider a specific case induced by hypergeometric differential equations. Finally, I will present some advances in a work in progress with S. Filip on a family of examples that was remarked to be special through the computation of their Lyapunov exponents, and whose monodromy groups give new conjectural examples of discrete or thin subgroups of matrix groups.

Information on past talks. This line was last edited 2019-01-11.