Abstracts

Yves Benoist

Convolution and square on cyclic groups

We will construct functions on cyclic groups of odd order whose convolution square is proportional to their square.

Francis Bonahon

The symplectic structure of the SL_n(R)-Hitchin component

In earlier work, Guillaume Dreyer and the speaker introduced a parametrization of the SL_n(R)-Hitchin component of a closed surface, well-behaved with respect to a maximal geodesic lamination on the surface. We compute the Atiyah-Bott-Goldman symplectic form in these coordinates.

 This is joint work with Yaşar Sözen and Hatice Zeybeck. 

Martin Bridgeman

The Weil-Petersson gradient flow of renormalized volume uniformizes relatively acylindrical manifolds

We consider the Weil-Petersson gradient vector field of renormalized volume on the deformation space of convex cocompact hyperbolic structures of a (relatively) acylindrical manifold. Using a toy model for the flow, we show that the flow has a global attracting fixed point at the structure M_geod the unique structure with totally geodesic convex core boundary. 

This is joint work with Kenneth Bromberg, and Franco Vargas Pallete

Brian Collier 

A Higgs bundle parameterization of positive representations.

Guichard-Labourie-Wienhard's work on positivity has identified the source of conjecturally all connected components of the character variety of a closed surface into a real Lie group which consist entirely of discrete-faithful representations. In this talk I will present a parameterization of these spaces using Higgs bundles. This parameterization is a Higgs bundle version of a Slodowy slice through a special class of sl2 triples.  I will also describe some special properties of the representations in these components and explain what remains to be done to verify that this Slodowy slice construction captures all higher Teichmuller spaces.

Bertrand Deroin 

Holonomy of conical hyperbolic metrics on surfaces

A conjecture of Goldman states that a generic representation of a closed surface group into PSL(2,R) of negative Euler class is the holonomy of a branched hyperbolic metric. I will explain a proof of this in the case of representation of co-Euler class equal to one. 

This is a joint work with Nicolas Tholozan.

David Dumas

Families of complex manifolds arising from Anosov representations

We discuss families of compact, complex manifolds associated to components of Anosov representations from pi_1(S) to G, where S is a closed orientable surface and G is a complex semisimple Lie group.  These families arise from quotients of domains of discontinuity in flag varieties constructed by Guichard-Wienhard and Kapovich-Leeb-Porti.  We present results on the deformation theory of these manifolds, showing in particular that the families arising from Anosov representations are universal when G has large enough rank and when the limit curve of the representation is assumed to have small Hausdorff dimension.  In particular such universality holds in a neighborhood of the Hitchin component, giving a local higher-rank analogue of the Bers simultaneous uniformization theorem.

This is joint work with Andrew Sanders (arxiv:2010.05147).

Viveka Erlandson

Dihedral subgroups of lattices in PSL(2, R)

I will discuss the growth of the number of infinite dihedral subgroups of lattices in PSL(2, R). Such subgroups exist whenever the lattice has 2-torsion and they are closely related to so-called reciprocal geodesics on the corresponding quotient orbifold.  These are closed geodesics passing through the order two orbifold point, or equivalently, homotopy classes of closed curves having a representative in the fundamental group that’s conjugate to its own inverse. We obtain the asymptotic growth of infinite dihedral subgroups (or reciprocal geodesics) in any lattice, generalizing earlier work of Sarnak and Bourgain-Kontorivich on the growth of the number of reciprocal geodesics on the modular surface. Time allowing, I will explain how our methods also show that reciprocal geodesics are equidistributed in the unit tangent bundle. 

This is joint work with Juan Souto. 

James Farre

Long curves and random hyperbolic surfaces

We will fix some topological data, a pants decomposition, of a closed surface of genus g and build hyperbolic structures by gluing hyperbolic pairs of pants along their boundary. The set of all hyperbolic metrics with a pants decomposition having a given set of lengths defines a (3g-3)-dimensional immersed torus in the (6g-6)-dimensional moduli space of hyperbolic metrics, a twist torus. Mirzakhani conjectured that as the lengths of the pants curves tend to infinity, that the corresponding twist torus equidistributes in the moduli space. In joint work-in-progress with Aaron Calderon, we confirm Mirzakhani's conjecture. In the talk, we explain how to import tools in Teichmüller dynamics on the moduli space of flat surfaces with cone points to dynamics on the moduli space of hyperbolic surfaces with geodesic laminations.

Vladimir Fock

Satake correspondence and cluster duality

Satake correspondence is an isomorphism between the representation ring of a finite dimensional Lie group and the Hecke algebra corresponding to the dual affine Lie algebra. On the other hand the space of integer measured laminations generate an algebra isomorphic to the space SL(2) local systems. We going to suggest a construction interpreting higher laminations as local systems with values in an affine Weyl group and relating both isomorphisms as a particular cases of cluster duality.

Ursula Hamenstädt 

A geometric approach to surface subgroups of PSL(n,R)

We use partial conjugation and positivity to study some explicit classes of surface subgroups of PSL(n,R). This gives a unified approach to the construction of surface subgroups of any genus g\geq 2 of cocompact lattices in PSL(n,R), as well as points in the Hitchin component with controlled length function and controlled entropy.

This is partially joint work with the geometry group in Bonn.

Kathryn Mann

Spectral rigidity for Anosov flows in dimension 3

The "free homotopy spectrum" of a flow on a manifold is the set of free homotopy classes of periodic orbits.   I will describe an ongoing research program with Thomas Barthelmé and Steven Frankel to show that, for transitive Anosov flows on 3-manifolds, the algebraic-topological data of the free homotopy spectrum completely determines the orbit structure of the flow.  

Shahar Mozes

Locally testable codes

A locally testable code is an error correcting code that has a property-tester who when receiving a word reads q bits of it that are randomly chosen, and rejects the word with probability proportional to its distance from the code. The parameter q is called the locality of the tester.

In a joint work with Irit Dinur, Shai Evra, Ron Livne and Alex Lubotzky, we introduce and use a certain family of complexes associated with groups and which have good expansion properties to construct an infinite family of  locally testable codes which have constant rate, constant distance and constant locality.

Andrew Neitzke

Higher symmetries and the hyperkahler geometry of Higgs bundles

I will review some recent progress in the study of the hyperkahler geometry of moduli spaces of Higgs bundles, emphasizing the role of infinitesimal holomorphic symplectomorphisms of the cotangent bundle of a Riemann surface, which give a useful "higher symmetry" of Hitchin's equations. Closely related higher symmetries also appear in the recent work of Fock-Thomas on higher complex structures. Parts of the talk will be based on joint work with David Dumas and work in progress with Laura Fredrickson.

Duc-Manh Nguyen

Volumes of the moduli spaces of flat surfaces with conical singularities

Since the seminal works by Thurston and Veech, it has been known that the moduli spaces of flat surfaces with prescribed cone angles at the singularities carry some natural volume forms. The values of these volumes have meaningful interpretations in many cases. The goal of this talk is to explain how tools from complex analytic and algebraic geometry can be used to compute efficiently such values, especially in the case of genus 0.

Anne Parreau

The real spectrum compactification of representation spaces

Building on the work of Brumfiel for Teichmuller space, I will explain how to construct, using tools from real algebraic geometry, a compactification for the space of conjugacy classes of representations of a finitely generated group in a semisimple real Lie group, whose boundary points can be seen as classes of representations over non archimedean real closed fields. This compactification naturally dominates the Weyl chamber valued length compactification, and this allows for example to prove the points coming from non archimedean fields with discrete valuation are dense in the latter. This is joint work with Marc Burger, Alessandra Iozzi et Beatrice Pozzetti.

Maria Beatrice Pozzetti

The Hilbert entropy for convex projective structures on surfaces

Classical Patterson-Sullivan theory establishes a beautiful link between the entropy of the geodesic flow of a convex cocompact hyperbolic manifold and the Riemannian Hausdorff dimension of the associated limit set in the boundary at infinity. For Anosov representations the situation is more complicated, since many different geodesic flows can be studied, and the action on the boundary is highly non-conformal. In joint work with Andrés Sambarino we give an interpretation of the Hilbert entropy for convex projective structures on surfaces in terms of the geometry of the associated limit set in the flag manifold.

Barbara Schapira

Strong positive recurrence and applications

For the dynamics of the geodesic flow on a noncompact negatively curved manifold, I will present different notions of entropy and pressure at infinity. These definitions coincide, and, when this pressure at infinity of a potential is strictly smaller than its topological pressure, the potential is said strongly positively recurrent. I will give examples and applications of this notion.

This is a joined work with S. Gouëzel and S. Tapie

Graham Smith

Finite type k-surfaces in 3-dimensional hyperbolic space.

We study the asymptotic structure of ends of finite type k-surfaces in H^3. We use these properties to study the geometries of individual k-surfaces, on the one hand, and their moduli spaces on the other. This work is available on arXiv at https://arxiv.org/abs/1908.04834.

Jeremy Toulisse

A universal space for maximal representations in rank 2

The universal Teichmüller space, introduced by Bers in 1965, is an infinite dimensional space that naturally contains all the Teichmüller spaces. In the same spirit, we propose a universal space for maximal representations in rank 2, discuss its properties and its relations with the theory of maximal surfaces in the pseudohyperbolic space. This is a joint work with François Labourie.

Tengren Zhang

Pleated surfaces in PSL(d,C)

In this talk, we define a notion of d-pleated surfaces that generalizes the classical notion of an (abstract) pleated surface in a hyperbolic space. We then prove a that the space of d-pleated surfaces is holomorphically parameterized by cocyclic pairs, thus generalizing a theorem of Bonahon for classical pleated surfaces. This is joint work with Sara Maloni, Giuseppe Martone, and Filippo Mazzoli.