Winter 2022

October 20, 2022: main talk at 14:00 in SR 8 at the MI

Frieder Jäckel (Bonn)

Stability of hyperbolic metrics 

We show that if a 3-manifold admits a metric of finite volume that is almost hyperbolic in a suitable, then there exists a hyperbolic metric that is close to the given metric in the C^{2,\alpha}-topology. We then discuss an application of this result to the drilling and filling of hyperbolic 3-manifolds. The talk is based on joint work with Ursula Hamenstädt. 

November 3, 2022: main talk at 14:00 in SR 8 at the MI

Clarence Kineider (Strasbourg)

Partial abelianization of local systems using spectral networks

Spectral networks are combinatorial tools introduced by Gaiotto, Moore and Neitzke for the study of TQFTs. In this talk, I will explain how these tools can be used to "split" a high rank local system over a surface S into a local system of smaller rank over a ramified cover of S. Using this, I will give a non-commutative generalization of Fock-Goncharov coordinates for GL_2(A)-local systems, where A is a non-commutative unital ring.

November 10, 2022: main talk at 14:00 in SR 8 at the MI

Gonzalo Ruiz Stolowicz (EPFL)

Hyperbolic representations of groups and functions of hyperbolic type: the case SL_2(R)

Some generalities about group representations by isometries on  the complex hyperbolic space of infinite dimensions will be introduced. For this purpose the functions of  hyperbolic type, which play a role analogous to that of positive type functions with respect to unitary representations, will be discussed. 

In particular the case of SL_2 will be treated in detail. A new family of representations will be described as well as the questions it raises about the structure of the set of hyperbolic type functions defined on a given group.   

November 11, 2022: main talk at 15:00 in SR 10 at the MI

Chris Connell (Indiana University, Bloomington)

Local (semi-)rigidity for boundary actions

We build a machine for establishing local rigidity up to semi-conjugacy of group actions on spaces that can be formulated into the framework of a bundle structure with some additional properties. A principal component of this machine involves extending a construction of Bowden-Mann using barycenter maps. As a key application, we establish $C^0$ local rigidity, up to semi-conjugacy, for the generalized projective actions of higher rank lattices on their boundary $G/Q$ (for an arbitrary parabolic subgroups $Q$). This is joint work with Mitul Islam, Thang Nguyen and Ralf Spatzier.

November 24, 2022: main talk at 14:00 in SR 8 at the MI

Yohann Bouilly

Dynamics on compact character variety: an ergodic action of a subgroup

The character variety of a surface encodes geometric structures as Teichmüller space or holomorphic vector bundles. The mapping class group of the surface acts on the character variety and preserves a measure. This action is known to be properly discontinuous on the Teichmüller space but its behavor is different for character varieties in a compact group : the action is ergodic. 

In this talk we will explain why the Torelli subgroup, which is the kernel of the action of the mapping class group on the first homology space, acts ergodically on character varieties in a compact Lie group.

December 1, 2022: main talk at 14:00 in SR 8 at the MI

Johannes Walcher (Heidelberg University)

Exponential networks and representations of quivers

Some time ago, with R. Eager and S. Selmani, we introduced the notion of “exponential networks” as a tool to study special Lagrangian submanifolds in local mirror symmetry. Recently, with R. Senghaas, M. Romo, and S. Banerjee, we have described a collection of anomaly-free networks on the punctured plane in correspondence with linear partitions, and a variety of other relations.

January 12, 2023: main talk at 14:00 in SR 8 at the MI

Davide Spriano (Oxford)

Hyperbolic models for CAT(0) spaces 

A very successful approach in geometric group theory is to construct ``hyperbolic models'' for interesting groups, namely a hyperbolic space on which a (non-hyperbolic) group acts in a nice enough way. The earliest example of this philosophy is the Bass-Serre, and other more recent examples include the curve graph for mapping class groups, contact graph for cubical groups, free factor/free splitting/cyclic splitting complex for Out(F_n) and so on.

In this talk we will describe the curtain model, a hyperbolic space on which CAT(0) groups act, and discuss several results about it.

This is joint work with H. Petyt and A. Zalloum. 

January 19, 2023: main talk at 14:00 in SR 8 at the MI

Martin Mion-Mouton (Technion)

Geometry of invariant distributions and rigidity of partially hyperbolic diffeomorphisms

The stable, unstable (and central) distributions of (partially) hyperbolic dynamics are a priori only Hölder continuous, and several works seem to suggest that their lack of regularity is in fact the main obstacle to their rigidity. Concerning contact-Anosov flows, successive works of Ghys (in dimension three) and Benoist-Foulon-Labourie (in higher dimensions) have for instance proved that the smoothness of the stable and unstable distributions forces the system to be algebraic.

In this talk, I will present an analog rigidity result for three-dimensional volume-preserving partially hyperbolic diffeomorphisms whose stable, unstable and central distributions are smooth, and whose stable-unstable plane field is a contact distribution. In this situation, the invariant distributions of the partially hyperbolic diffeomorphism define a Cartan geometry, whose interaction with the dynamics of the diffeomorphism yields the rigidity phenomenon in question.

February 2, 2023: main talk at 14:00 in SR 8 at the MI

Menelaos Zikidis (Univ. of Sheffield)
Geometric Structures on Moduli of Stability Conditions

In this lecture I will discuss aspects of an ongoing program, aiming to encode Donaldson-Thomas enumerative invariants associated to triangulated Calabi-Yau-3 categories into Geometric Structures on their moduli of Bridgeland stability conditions. I will concentrate on a class of a examples associated to theories of class-S[A_1], where the construction of the Geometric Structure admits an explicit moduli-theoretic description in terms of isomonodromic deformations of Riemann-Hilbert data over projective algebraic curves. We see that this Complex Hyperkähler Structure arises algebraically from a family of non-linear, flat and symplectic connections on a stack parameterizing Higgs Bundles and Flat Connections on curves, fibering over the moduli space of quadratic differentials. If time permits, I will comment on the nature of the associated Twistor Space.

(SPECIAL TIME/PLACE) February 3, 2023: main talk at 14:00 in SR 8 at the MI

Michael Landry (Washington University St. Louis)
Endperiodic maps via pseudo-Anosov flows

This is joint work with Y. Minsky and S. Taylor.  An endperiodic map of an infinite-type surface L with finitely many ends is a homeomorphism such that each end of L is either attracting or repelling.

We show that when the endperiodic map is atoroidal (i.e. fixes no multicurves up to isotopy), it is isotopic to a “spun pseudo-Anosov“ (spA) map f: L  L, i.e. the first return map of a pseudo-Anosov suspension flow phi on a closed 3-manifold M. Here, L is a leaf of a depth one foliation of M that is transverse to phi.

As the return map to L under the flow phi, f inherits many of phi's dynamical properties; for example, it preserves a pair of singular foliations of L and acts with pseudo-Anosov-like dynamics at its periodic points. We show how to use this picture to recover the lamination theory of Handel-Miller, although our arguments are independent of that theory. Using our structure, we define and characterize stretch factors of endperiodic maps, relate Cantwell-Conlon's foliation cones to Thurston's fibered cones, and define a convex entropy function on foliation cones that extends log(stretch factor). In total, this shows that the theory of atoroidal endperiodic maps and their mapping tori is strongly analogous to, and inseparable from, the Thurston-Fried-McMullen fibered face theory of pseudo-Anosov homeomorphisms and their closed mapping tori.

February 9, 2023: main talk at 14:00 in SR 8 at the MI

Antonin Guilloux (Sorbonne Université)

Slimness in the 3-sphere 

Viewed as the boundary at infinity of the complex hyperbolic plane, the 3-sphere is equipped with a contact structure. The interplay between this contact structure and limit sets of subgroups of PU(2,1) has deep consequences on the properties of these subgroups. Some limit sets enjoy the property of slimness, that we introduce. Using this property, one can shed new lights on known results, and better describe deformations of subgroups. I will present this notion through simple examples and pictures, and then describe some results we obtain. This is a joint work with E. Falbel and P. Will.