Summer 2023

April 20, 2023: main talk at 15:00 in SR 8 (note the special time)

Gabriele Viaggi (Heidelberg/Univ. of Rome)

Divisible convex sets with properly embedded cones

An open subset of real projective space is said to be properly convex if it is contained in an affine chart where it is convex and bounded. Together with their natural Hilbert metric and group of projective symmetries, properly convex sets are a rich source of geometry, dynamics, and group theory. Of particular interest are those that are divisible, that is, they admit a compact quotient by a discrete group of symmetries. In general, it is a challenging problem to construct such examples. After reviewing the global classification scheme, I will describe a new class of examples of divisible convex sets with special geometric properties (non-symmetric and non-strictly convex) that completes a missing part of the picture. This is joint work with Pierre-Louis Blayac.

April 27, 2023: main talk at 14:00 in SR 8 at the MI

Tommaso Scognamiglio (Université Paris Cité)

Character varieties for non-orientable surfaces

Character varieties for a Riemann surface X are affine varieties parametrising representations of the fundamental group of X and so the local systems on the variety. The study of the geometry of character varieties and the computation of their cohomology is extremely interesting, especially because of their connection with the moduli spaces of Higgs bundles via non abelian Hodge correspondence. In this talk, I will explain how this setting translates for non-orientable real surfaces and how this is related to some interesting involutions on the moduli space of Higgs bundles. The cohomology of character varieties for non-orientable surfaces is way less known and I will explain also some of the recent advances in this subject

May 4, 2023: main talk at 14:00 in SR 8 at the MI

(no seminar)

May 25, 2023: main talk at 14:00 in SR 8 at the MI

Katrin Tent (Münster)

Burnside groups and iterated small cancellation theory

In 1902 Burnside asked whether any finitely generated torsion group is necessarily finite. By now there is a long line of negative answers, albeit not necessarily accessible. I will explain the basics of small cancellation theory and our approach to the Burnside problem. (joint with A. Atkarskaya and E. Rips)

June 15, 2023: main talk at 14:00 in SR 8 at the MI

Frank Ferrari (Université Libre de Bruxelles)

Random Geometries of Constant Curvature: from Condensed Matter Physics to the Combinatorics of Self-Overlapping Curves through Black Holes.

Abstract: Two-dimensional quantum gravity models fall in three classes: Liouville gravity, for which the geometry is wildly random in the bulk; topological gravity, for which the geometries, having constant curvature and geodesic boundaries, have a finite number of moduli; and an intermediate class of models, which has attracted a lot of attention recently, for which the metrics have constant curvature but the boundaries can fluctuate wildly. These so-called Jackiw-Teitelboim theories, mainly in negative curvature, have been intensively studied in the physics literature in recent years. They build a bridge between several different problems, from strongly coupled electron systems in condensed matter theory to quantum black hole models. They are analysed using a variety of techniques, including matrix models, Mirzakhani/Eynard recursion relations, integration over Diff(S1)/PSL(2,R), etc.

June 16, 2023: main talk at 15.00 in SR 7 at the MI (note special time and location)

Swarnava Mukhopadhyay (Tata Institute of Fundamental Research)

Hitchin Connection for parabolic bundles

In this talk, we will discuss a construction of a Hitchin-type connection on the bundle of theta functions for the moduli of parabolic bundles. We will start with a more general framework of constructing connections from heat operators with a given symbol map due to van Geeman-de Jong.

This is joint work with Indranil Biswas and Richard Wentworth

June 29, 2023: main talk at 14:00 in SR 8 at the MI

Tianqi Wang (National University of Singapore)

Restricted Anosov representations via flows

There are many examples of non-Anosov representations with good geometric properties, such as Minsky's primitive-stable representations. We introduce the notion of restricted Anosov representations, characterized by their dominated splitting behavior over associated flows, to encompass these examples. Here are two applications. Firstly, for a closed hyperbolic surface group, we show that the collection of representations which are Anosov in restriction to the simple geodesics flow gives a domain of discontinuity for the mapping class group action (joint with Nicolas Tholozan). Secondly, for a relatively hyperbolic group, we show that a representation being both divergent and Weisman's extended geometrically finite is equivalent to being Anosov in restriction to a flow associated with the boundary extension.

July 5, 2023: virtual talk at 19:00 (note the special time and location)

Erick Daniel Gordillo Herrerias (UNAM, Mexico)
Random Wind-Tree models

This presentation will explore the dynamics of a family of infinite translation surfaces known as random wind-tree models. These models serve as a natural generalization of the previously studied periodic wind-tree models. The talk will provide an overview of the key concepts and techniques used in studying these intriguing structures.

July 6, 2023: main talk at 14:00 in SR 8 at the MI

No talks (Mercator Workshop)

July 13, 2023: main talk at 14:00 in SR 8 at the MI

(no talks)

July 20, 2023: main talk at 14:00 in SR 8 at the MI

Andrea Tamburelli (University of Pisa)

Asymptotics of minimal surfaces in SL(3,R)/SO(3) along rays

Hitchin’s theory of Higgs bundles associated holomorphic differentials on a Riemann surface to representations of the fundamental group of the surface into a Lie group. We study the geometry common to representations whose associated holomorphic differentials lie on a ray. In the setting of SL(3,R), we provide a formula for the asymptotic holonomy of the representations in terms of the local geometry of the differential. Alternatively, we show how the associated equivariant harmonic maps to a symmetric space converge to a harmonic map to a building, with geometry determined by the differential. This is joint work with John Loftin and Mike Wolf.

July 27, 2023: virtual talk at 16:00 (note special time and location)

Yuping Ruan (Univ. of Michigan/Northwestern Univ.)

Boundary rigidity and filling minimality via the barycenter method

A compact manifold with a smooth boundary is boundary rigid if its boundary and boundary distance function uniquely determine its interior up to boundary preserving isometries. Under certain natural conditions, the notion of boundary rigidity is closely related to Gromov's filling minimality. In this talk, we will give a brief overview of Burago-Ivanov's approach to prove filling minimality and boundary rigidity for almost Euclidean and almost real hyperbolic metrics. Then we will explain how we generalize their results to regions in a rank-1 symmetric space equipped with an almost symmetric metric. We will also explain the relations to Besson-Courtois-Gallot's barycenter constructions used in their celebrated volume entropy rigidity theorem. 

July 28, 2023: main talk at 14.00 in SR 8 at the MI (note special time and location)

Shinpei Baba (Osaka University, Japan)

Bending maps of Teichmüller spaces and character varieties

Poincare holonomy varieties (or SL(2, C)-opers) are half-dimensional symplectic slices in the PSL(2, C)-character varieties of surfaces. The vector space of holomorphic quadratic differentials on a Riemann surface corresponds to such a slice by the holonomy map of complex projective structures. We construct analuogues of such slices from the viewpoint of Thurston's parametrization of complex projective structures. 

September 7, 2023: main talk at 14:00 in SR 8 at the MI

Alex Nolte (Rice University)

Leaves of properly convex foliated projective structures

In one of the few cases where qualitative descriptions of geometric structures giving rise to a higher Teichmüller space is known, Guichard and Wienhard showed in 2008 that PSL(4,R) Hitchin representations can be understood in terms of certain projective structures on unit tangent bundles of surfaces. These projective structures come with natural equivariant foliations of their developed images by properly convex domains in projective lines and planes. The family of leaves of the codimension-1 foliation gives an interesting and mysterious invariant of these Hitchin representations. I will discuss some recent results on these leaves. For instance, except in the (Fuchsian) case, there must be many projectively non-equivalent leaves and there is a strong sense in which no leaf can have many symmetries.