# Geometry and Topology Seminar

## Warwick Mathematics Institute, Term III, 2015-2016

 Thursday April 28, 15:00, room MS.04 Thomas Morzadec (Paris-Sud) Geometric compactification of the space of half-translation surfaces Abstract: A half-translation structure on a surface $S$ is a locally Euclidean metric (allowing conical singularities) such that the holonomy of every loop disjoint from the singularities lies in $\{\pm \Id\}$. The space $\Flat(S)$ of half-translation structures on $S$ is endowed with a natural topology which is not compact. In their paper "Length spectra and degeneration of flat metrics", Duchin-Rafi-Leininger built an extrinsic compactification of $\Flat(S)$, using geodesic currents on $S$, following the work of Bonahon on hyperbolic metrics. During my talk, I will describe an intrinsic compactification of $\Flat(S)$, based on geometric tools. I will explain how it allows us to understand degenerations of half-translation structures.

 Thursday April 28, 16:00, room MS.04 Alexey Sossinsky (Independent University of Moscow) Minima of the Euler functional for plane curves Abstract: (Joint work with Oleg Karpenkov and Sergei Avvakumov.) The Euler functional of a smooth plane curve is the integral along the curve of the square of its curvature. In 1774, Euler posed the problem of finding the curves that give the minima of his functional. In the talk, I will give the answer to this problem, sketch its proof, show that it implies the Whitney-Graustein theorem on the classification of regular plane curves up to regular homotopy, and run computer animations showing the gradient descent of curves to their minimal form.

 Thursday May 12, 15:00, room MS.04 Saul Schleimer (Warwick) Veering Dehn surgery Abstract: (Joint work with Henry Segerman.) It is a theorem of Moise that every three-manifold admits a triangulation, and thus infinitely many. Thus, it can be difficult to learn anything really interesting about the three-manifold from any given triangulation. Thurston introduced ideal triangulations'' for studying manifolds with torus boundary; Lackenby introduced taut ideal triangulations'' for studying the Thurston norm ball; Agol introduced veering triangulations'' for studying punctured surface bundles over the circle. Veering triangulations are very rigid; one current conjecture is that any fixed three-manifold admits only finitely many veering triangulations. After giving an overview of these ideas, we will introduce veering Dehn surgery''. We use this to give the first infinite families of veering triangulations with various interesting properties.

 Thursday May 19, 15:00, room MS.04 Antoine Gournay (Dresden) Harmonic functions, group cohomology and Hilbertian compression Abstract: (Joint work with P.-N. Jolissaint.) In this talk, I will try to explain how to use harmonic functions to gain insight on the cohomology of the unitary representations of a group and on the "optimal" equivariant embeddings of a group in a Hilbert space. This will be then applied to compute precisely which is the best equivariant embedding of the group of automorphisms of the regular tree in a Hilbert space.

 Thursday May 26, 15:00, room MS.04 Dawid Kielak (Bielefeld) Nielsen realisation for right-angled Artin groups and free products Abstract: We will look at the realisation problem of Nielsen in the setting of RAAGs and free products of groups, and discuss some recent developments in answering the problem.

 Thursday June 2, 15:00, room MS.04 Yong Hou (IAS) Rigidity of Kleinian groups Abstract: In this talk I will give a general presentation of my recent work, showing that purely loxodromic Kleinian groups, of Hausdorff dimension less than one, are classical Schottky groups. This uses my earlier result on the classification of Kleinian groups of sufficently small Hausdorff dimension. This result, in conjunction with another result (joint with Anderson), resolves the Bers uniformization conjecture. No prior knowledge of the subject will be assumed.

 Thursday June 9, 15:00, room MS.04 Richard Webb (Cambridge) Polynomial-time Nielsen--Thurston type recognition Abstract: (Joint work with Mark Bell.) We shall describe a polynomial-time algorithm that computes a (tight) geodesic in the curve graph between two given vertices. There are several consequences e.g. one can determine the Nielsen--Thurston type and compute the canonical reducing curve system of mapping classes in polynomial time. Time permitting, we shall describe a polynomial-time algorithm to compute the quotient orbifold of a periodic mapping class, and we shall discuss the conjugacy problem for the mapping class group.

 Thursday June 9, 16:00, room MS.04 Vaibhav Gadre (Glasgow) The stratum of random mapping classes Abstract: (Joint work with Joseph Maher.) We consider random walks on the mapping class group. By results of Maher and also Rivin it is known that a random element is pseudo-Anosov. We consider random walks whose support generates a non-elementary subgroup and contains a pseudo-Anosov in the principal stratum of quadratic differentials. We show that the mapping classes along almost every infinite sample path are pseudo-Anosov in the principal stratum.

 Thursday June 16, 15:00, room MS.04 Mark Bell (UIUC) Polynomial-time curve reduction Abstract: A given curve on a surface can be extremely complicated. Thus it can be very hard to determine various of its properties: for example, its topological type. We will discuss a new argument that, when the curve is given by its intersections with the edges of an ideal triangulation, there is always a "reduction" to a simpler configuration. By performing a carefully chosen edge flip or a (power of a) Dehn twist we find a "change of coordinates" which decreases the complexity by a definite fraction. The reduction procedure allows us to simplify curves to have bounded complexity in polynomial time in the bit-size of the initial curve. Using this, in joint work with Richard Webb, we can compute the intersection number of, and the boundary of a regular neighbourhood of, a pair of curves in polynomial time. The latter of these is a key ingredient in being able to compute geodesics in the curve complex in polynomial time.

 Thursday June 16, 16:00, room MS.04 Gareth Jones (Southampton) Dessins d'enfants on Riemann surfaces Abstract: Compact Riemann surfaces are equivalent to complex algebraic curves. Belyi's Theorem, as reinterpreted by Grothendieck and others, shows that such a curve is defined over an algebraic number field if and only if the corresponding Riemann surface is obtained, in a canonical way, from an embedded graph, called a dessin d'enfant. It follows that the absolute Galois group, the large, important and mysterious automorphism group of the field of all algebraic numbers, has a faithful action on these combinatorial objects. It remains faithful even when restricted to such simple examples as plane trees, or such symmetric examples as regular maps, so that one can "see" the Galois theory of algebraic number fields by drawing pictures. I shall give a survey of some recent developments in this area. Only very basic knowledge of the above topics is required.

 Thursday June 23, 15:00, room MS.04 TBA (TBA) TBA Abstract: TBA

 Thursday June 30, 15:00, room MS.04 TBA (TBA) TBA Abstract: TBA

Information on past talks. This line was last edited Tuesday, January 19, 2016 08:35:36 PM GMT