Abstract: The lengths of geodesics on hyperbolic surfaces satisfy intriguing equations, known as identities, relating these lengths to geometric quantities of the surface. The talk will be about a family of identities that relate lengths of closed geodesics and orthogeodesics to boundary lengths or the number of cusps. These include, as particular cases, identities due to Basmajian, to McShane and to Mirzakhani and Tan-Wong-Zhang. In contrast to previously studied cases, the new identities include lengths taken among all closed geodesics.

Title: Quasiconformally Homogenous Riemann Surfaces

Abstract: A Riemann Surface X is said to be K-quasiconformally homogeneous if for any two points x and y on it, there exists a K quasiconformal self-mapping taking x to y. If such a K exists we say that X is a QCH surface. It is not difficult to show that the regular cover of a closed Riemann surface is QCH, and hence the quasiconformal deformation of such a regular cover is also QCH. This leads us to the question to what extent being qc equivalent to a regular cover characterizes QCH surfaces. Bonfert-Taylor, Canary, Souto, and Taylor showed that there are QCH Riemann surfaces that are not qc-equivalent to the regular cover of a closed surface. On the other hand, in joint work with Nick Vlamis we show that all QCH ladder surfaces are qc-equivalent to the regular cover of a closed surface.

After an introduction to the basics we will discuss the proof of this theorem which involves the hyperbolic geometry of the surface.

Title: Curves on the torus with controlled intersection

Abstract: A well known counting problem in surface topology asks for the maximum number of simple closed curves that can fit on a closed surface of genus g so that every pair intersect at most k times -- such a collection of curves is called a "k-system". Answers to this problem, even just for order of growth in k or g, are still unknown. The case k=1 is a notoriously difficult problem, with notable recent progress from Przytycki and Greene.

Somewhat surprisingly, the complementary case g=1 remains mysterious. Ian Agol made the elegant observation that, on the torus, the size of a k-system is bounded by one more than the smallest prime larger than k, and via the Prime Number Theorem one can deduce that this quantity is asymptotic to k. We will discuss joint work with Tarik Aougab, in which we tighten the available upper bounds for g=1, showing that a k-system on the torus has size at most k+O(\sqrt{k}\log k). (Curiously, this matches the bound one would obtain via Agol's bound with the assumption of the Riemann hypothesis.) Our methods involve analysis of some combinatorial aspects of the associahedron and the hyperbolic geometry of the Farey complex.

Title: Constraining mapping class group homomorphisms using finite subgroups

Abstract: We will start by reviewing the connection between isometries of the hyperbolic plane and finite subgroups of mapping class groups. Then we will discuss a new short proof of a theorem of Aramayona--Souto that constrains homomorphisms between mapping class groups of closed surfaces, as well as new results on constraining homomorphisms from mapping class groups to homeomorphism groups of spheres. This is joint work with Lei Chen.

Title: Weyl problem for convex surfaces in the hyperbolic 3-space and the Koebe circle domain conjecture

Abstract: The classical Weyl problem (in the hyperbolic geometric setting) asks if every complete hyperbolic surface of genus zero can be isometrically embedded in to the hyperbolic 3-space as a convex surface. We show that this problem is closely related to Koebe’s circle domain conjecture. Schramm’s transboundary extremal length plays a key role in the proof. This is a joint work with Tianqi Wu.

Title: Hidden symmetries of the dodecahedron

Abstract: The dodecahedron has some hidden affine symmetries that can be seen after passing to a 10-fold translation surface cover. This symmetry group also acts isometrically on the hyperbolic plane. The quotient of the hyperbolic plane modulo this group is a finite but large surface, with genus 131, with 19 cone singularities and featuring 362 cusps. I'll explain how these features of this hyperbolic surface can be used to answer questions about the flat geometry of the dodecahedron. This is joint work with Jayadev Athreya and David Aulicino.

Title: On Maskit's commutator rigidity

Abstract: In 1986, Bernie Maskit proved that the isomorphism class of a closed Riemann surface of genus at least two is uniquely determined by its homology cover. In terms of Fuchsian groups, this result can be states as follows: If \Gamma_{1}, \Gamma_{2}<PSL(2,R) are both torsion free cocompact Fuchsian groups, then \Gamma_{1}=\Gamma_{2} if and only if \Gamma'_{1}=\Gamma'_{2} where \Gamma_{j}' denotes the commutator subgroup of \Gamma_{j}. This result can be thought of as a kind of Torelli’s theorem from the Fuchsian group point of view. In this talk we will discuss commutator rigidity and some of its generalizations to other classes of Kleinian groups.

Title: Rigidity of hyperbolic cone surfaces.

Abstract: : It was proved by Otal that negatively curved Riemannian metrics are determined by their marked length spectrum by showing that this spectrum determines the metric's Liouville current, which in turns determines the metric. The same is true for any negatively curved cone metric (in fact, for any non-positively curved metric). In this talk we will discuss to what extent we can use only a topological property of the Liouville current--namely its support--to determine hyperbolic cone metrics; equivalently, to what extent a hyperbolic cone metric is determined by the endpoints (in the universal cover) of it's non-singular geodesics.This is joint work with Chris Leininger and Chandrika Sadanand.

Title: Isospectral hyperbolic surfaces of infinite type

Abstract: Two hyperbolic surfaces are said to be (length) isospectral if they have the same collection of lengths of primitive closed geodesics, counted with multiplicity (i.e. if they have the same length spectrum). For closed surfaces, there is an upper bound on the size of isospectral hyperbolic structures depending only on the topology. We will show that the situation is different for infinite-type surfaces, by constructing large families of isospectral hyperbolic structures on surfaces of infinite genus.