Past meetings
April 29, 2021: main talk at 2:30 CET
Ivo Slegers (University of Bonn/Max Planck Institute of Mathematics)
The energy functional of a Hitchin representation
In this talk we will discuss the energy functionals on Teichmüller space that are associated to Hitchin representations. We will show that these functionals are strictly plurisubharmonic functions on Teichmüller space. In the second half of this talk we consider the question whether a Hitchin representation is uniquely determined by its energy functional. We answer a similar question in a simpler setting and discuss work in progress on how these results can be expanded to include Hitchin representations.
May 6, 2021: main talk at 2:30 CET
Pierre-Louis Blayac (Université Paris-Sud)
The measure of maximal entropy for the geodesic flow on compact convex projective manifolds.
Among discrete subgroups of Lie groups, the Anosov ones, introduced by Labourie and Guichard--Wienhard, have drawn a lot of attention. To each Anosov group is associated a natural Anosov flow, and results from the theory of dynamical systems, in particular thermodynamical formalism, have been applied to this flow, with success, to obtain informations on the Anosov group.
Let V be a real vector space. Another interesting and famous class of discrete subgroups of Lie groups are subgroups of PGL(V) that preserve an open subset of P(V) which is properly convex, i.e. convex and bounded in some affine chart. The quotient by a discrete subgroup of PGL(V) of an invariant properly convex open subset of P(V) is a convex projective manifold, and its unit tangent bundle carries a natural geodesic flow. By work of Benoist, if the discrete group is word-hyperbolic and the quotient is compact, then the group and the geodesic flow are Anosov. We will discuss about the dynamics of the geodesic flow on compact convex projective manifolds whose fundamental group is not necessarily word-hyperbolic; more precisely we will be interested in the measure of maximal entropy. For this, we will use the similarities these manifolds have with non-positively curved Riemannian manifolds, and the theory of Patterson--Sullivan densities.
May 13, 2021 Holiday: no seminar
May 20, 2021: main talk at 5:30 CET
Konstantinos Tsouvalas (University of Michigan)
Quasi-isometric embeddings inapproximable by Anosov representations into SL(d,R).
Anosov representations were introduced by Labourie for fundamental groups of closed negatively curved Riemannian manifolds and further generalized by Guichard-Wienhard for more general Gromov hyperbolic groups. They form a rich and stable class of discrete subgroups of Lie groups and today are recognized as a higher rank analogue of classical convex cocompact subgroups of simple rank 1 Lie groups. In this talk, we are going to exhibit examples of quasi-isometric embeddings of hyperbolic groups into SL(d,R), d \geq 5, which fail to be in the closure of Anosov representations into SL(d,R).
May 27, 2021: main talk at 5:30 CET
Minju Lee (Yale University)
Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends.
This is joint work with Hee Oh. We establish an analogue of Ratner's orbit closure theorem for any connected closed subgroup generated by unipotent elements in SO(d,1) acting on the space \Gamma\SO(d,1), assuming that the associated hyperbolic manifold M=\Gamma\H^d is a convex cocompact manifold with Fuchsian ends. For d = 3, this was proved earlier by McMullen, Mohammadi and Oh. In a higher dimensional case, the possibility of accumulation on closed orbits of intermediate subgroups causes serious issues, but in the end, all orbit closures of unipotent flows are relatively homogeneous. Our results imply the following: for any k\geq 1,
(1) the closure of any k-horosphere in M is a properly immersed submanifold;
(2) the closure of any geodesic (k+1)-plane in M is a properly immersed submanifold;
(3) an infinite sequence of maximal properly immersed geodesic (k+1)-planes intersecting core(M) becomes dense in M.
June 3, 2021 Holiday: no seminar
June 10, 2021: main talk at 5:30 CET
Filippo Mazzoli (University of Virginia)
A para-hyperKähler structure on the space of GHMC AdS-manifolds.
A celebrated result by Donaldson asserts that the space of almost-Fuchsian manifolds admits a natural hyperKähler structure invariant under the action of the mapping class group, extending the Weil-Petersson Kähler structure of Teichmüller space. In this talk we will discuss the occurrence of a similar phenomenon for the deformation space of globally hyperbolic Anti-de Sitter 3-manifolds. In particular we will see how such space carries a parahyperKähler structure, where a pseudo-Riemannian metric and 3 symplectic structures coexist with a integrable complex structure and two para-complex structures, satisfying the relations of para-quaternionic numbers. This project is a joint work with Andrea Seppi (Université Grenoble Alpes) and Andrea Tamburelli (Rice University).
June 17, 2021: main talk at 2:30 CET
Andrew Zimmer (University of Wisconsin-Madison)
Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups
In this work, we develop a theory of Anosov representation of geometrically finite Fuchsian groups in SL(d,R) and show that "cusped Hitchin representations" are Borel Anosov in this sense. We establish analogues of many properties of traditional Anosov representations. In particular, we show that our Anosov representations are stable under type-preserving deformations and that their limit maps vary analytically. We also observe that our Anosov representations fit into the previous frameworks of relatively Anosov and relatively dominated representations developed by Kapovich-Leeb and Zhu. This is joint work with Richard Canary and Tengren Zhang.
June 24, 2021: main talk at 2:30 CET
Vincent Delecroix (Bordeaux)
Random multicurves in high genera
A multicurve on a closed surface S of genus g >= 2 is a homotopy class of a disjoint union of simple non-contractible curves. There are countably many multicurves on S. Thanks to the work of M. Mirzakhani, F. Arana-Herrera and M. Liu one can make sense of the shape of a random multicurve on S. In this talk, I will recall the definition and the construction of this random multicurve and will (partly) answer to the question: what is the asymptotic shape of a random multicurve as the genus tends to infinity?
Friday July 2, 2021 main talk at 10:00 am CET
Alexander Thomas (MPI Bonn)
Punctual Hilbert schemes, q-deformations and geometric structures
In the first part of the talk, I present various punctual Hilbert schemes and their q-deformations. We then use this tool to construct geometric structures on surfaces, called higher complex structures, whose moduli space share multiple properties with Hitchin components. Partially joint work with Vladimir Fock.
July 8, 2021: main talk at 2:30 CET
Jane Wang (Indiana)
Slope gap distributions of Veech translation surfaces
Translation surfaces are surfaces that are locally Euclidean except at finitely many points called cone points. A saddle connection is then a straight trajectory that begins and ends at a cone point. It is known that on almost every translation surface, the set of angles of saddle connections on the surface is equidistributed in the circle. A finer notion of how random the saddle connection directions are is given by something called the gap distribution of the surface.
In this talk, we will explain what the slope gap distribution of a translation surface is and survey some known results about slope gap distributions, including how one can use properties of the horocycle flow to compute the slope gap distributions of special translation surfaces called Veech surfaces. We'll then discuss results showing that the slope gap distributions of Veech surfaces have to satisfy some nice properties. This project is joint work with Luis Kumanduri and Anthony Sanchez.
July 15, 2021: main talk at 2:30 CET
Julien Marché (IMJ-PRG)
The valuative viewpoint on the space of measured laminations
Measured laminations on surfaces already have a lot of different interpretations like measured foliations, actions on real trees and valuations on the character variety. The two latter have been much studied in the context of Morgan-Shalen compactification of the Teichmüller space but the description of the valuation associated to a measured lamination was rather inexplicit. In this talk, I will describe our joint work with Christopher-Lloyd Simon where we provide a simple characterization of valuations coming from measured laminations. We used it to compute the automorphism group of the character variety but beyond this result, I will explain that this gives a convenient framework for studying the space of measured laminations, including its symplectic structure.