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Events
20 years of Anosov representations

MPI für Mathematik in den Naturwissenschaften Leipzig (Leipzig)

Inselstr. 22
04103 Leipzig
Germany

E1 05 (Leibniz-Saal)

conference

07.10.24 11.10.24

20 years of Anosov representations

Since their introduction in François Labourie's seminal 2004 paper, Anosov representations have been a driving force in the study of discrete subgroups of higher-rank Lie groups, and have helped to stimulate two decades of research in the area. This conference/workshop, celebrating 20 years of progress in the field, brings together researchers studying the geometric, dynamical, and algebraic aspects of Anosov representations, as well as their connections to other topics.

The event will consist of a two-day introductory workshop for younger mathematicians, intended to provide an overview of Anosov representations for those new to the subject, followed by a three-day research conference focusing on recent developments. Participants should feel free to apply to attend either the workshop or the conference, or both. The workshop will consist mainly of expository talks given by participants; younger mathematicians (especially early PhD students) are invited to apply to give one of these talks.

The registration is closed.

The speakers for the Young Researchers Workshop are (in order of their talks):

Debattam Das (IISER Mohali)
Jennifer Vaccaro (University of Illinois at Chicago)
Enya Hsiao (MPI MiS )
Jordan Grant (University of Michigan)
Victor Jaeck (ETH Zurich)
Blandine Galiay (IHES)
Luis Torres (University of Texas at Austin)
Thimothé Lemistre (Université de Nice)
Joaquin Lejtreger (Sorbonne Université)
Vicky Wen (University of Wisconsin-Madison)

The speakers of the Conference are (in alphabetic order):

Dick Canary (University of Michigan), participation cancelled
Colin Davalo (Université Grenoble-Alpes)
Subhadip Dey (MPI MiS)
Simion Filip (University of Chicago)
Xenia Flamm (MPI MiS)
Vladimir Fock (Université de Strasbourg )
François Labourie (Université Côte d‘Azur)
Sara Maloni (University of Virginia)
Alex Nolte (Rice University)
Hee Oh (Yale University)
Maria Beatrice Pozzetti (Heidelberg University), participation cancelled
Andrés Sambarino (CNRS - Sorbonne Université)
Nicolas Tholozan (CNRS - École Normale Supérieure)
Jérémy Toulisse (Université Côte d‘Azur), participation cancelled
Kostas Tsouvalas (MPI MiS)
Tengren Zhang (National University of Singapore)

Questions for the Future provided by (in alphabetic order):

Thierry Barbot (Université d'Avignon)
Marc Burger (ETH Zurich)
Jeff Danciger (University of Texas)
François Guéritaud (Université de Strasbourg )
Alessandra Iozzi (ETH Zurich)
Rafael Potrie (Universidad de la República)

Organised in partnership with the Clay Mathematics Institute.

WARNING: Please be wary if a company directly contacts you regarding payment for accommodation in Leipzig: this is fraud / phishing.

Program

	Young Researchers Workshop
08:45 - 08:55	Welcome Coffee & Registration
08:55 - 09:00	Welcome Address
09:00 - 09:50	Debattam Das (IISER Mohali, India) Introduction: Rank one case
10:00 - 10:30	Coffee & Tea
10:30 - 11:20	Jennifer Vaccaro (University of Illinois at Chicago, USA) Higher rank symmetric spaces
11:30 - 12:20	Enya Hsiao (MPI Leipzig, Germany) Anosov representations via boundary maps
12:30 - 13:30	Lunch Break
13:30 - 14:30	Open Problems Discussion
14:30 - 15:20	Jordan Grant (University of Michigan, USA) Anosov representations via singular value gaps
15:30 - 16:00	Coffee & Tea
16:00 - 16:50	Victor Jaeck (ETH Zurich, Switzerland) Constructing Anosov representations

	Young Researchers Workshop
08:45 - 09:00	Welcome Coffee
09:00 - 09:50	Blandine Galiay (IHES, France) Hitchin representations
10:00 - 10:30	Coffee & Tea
10:30 - 11:20	Luis Torres (The University of Texas at Austin, USA) Examples: Convex projective manifolds
11:30 - 12:20	Timothé Lemistre (Université de Nice, France) Examples: $\mathbb{H}^{p,q}$-convex cocompact representations
12:30 - 12:35	Group photo
12:35 - 13:30	Lunch Break
13:30 - 14:30	Open Problems Discussion
14:30 - 15:20	Joaquín Lejtreger (Sorbonne Université, France) Domains of discontinuity
15:30 - 16:00	Coffee & Tea
16:00 - 16:50	Rou Wen (University of Wisconsin-Madison, USA) Discrete subgroups beyond Anosov representations
18:00 - 19:00	Dinner and Q&A Session with Nicolas Tholozan (CNRS - École Normale Supérieure)

	Conference
08:45 - 08:55	Welcome Coffee
08:55 - 09:00	Welcome Address
09:00 - 09:50	Vladimir Fock (Université de Strasbourg, France) Arithmetic invariants of cluster transformations. (Higher) Teichmüller space is known to possess a (pre-)symplectic structure which has a simple expression in any system of cluster coordinates. We will show that this fact implies the existence of mapping class group invariant functions taking values in $(Z/pZ)^*$ for any prime p on the set of rational points of the corresponding character variety.
09:50 - 10:30	Coffee & Tea
10:30 - 11:20	Alexander Nolte (Rice University, Georgia Tech, USA) Foliations in PSL(4,R)-Teichmüller theory We will present a classification of group-invariant foliations of a domain of discontinuity for PSL(4,R)-Hitchin representations whose leaves are properly convex domains in projective spaces. Two foliations of this form are the starting point of Guichard-Wienhard’s beautiful framework to describe the projective geometry of PSL(4,R)-Hitchin representations; the main result is that there is exactly one other similar foliation. We will emphasize how the special structure of Hitchin representations can be used to understand the shape of these domains of discontinuity, and the role this plays in the proof.
11:30 - 12:30	Tengren Zhang (National University of Singapore, Republic of Singapore) Global topology of the space of d-pleated surfaces In this talk, we define the notion of d-pleated surfaces, which is a higher rank generalization of the classical notion of (abstract) pleated surfaces in three dimensional hyperbolic space. We give a description of the global topology of the space of conjugacy classes of d-pleated surfaces. We also prove that every connected component of the character variety contains exactly one connected component of the space of d-pleated surfaces. This is joint work with Sara Maloni, Giuseppe Martone, and Filippo Mazzoli.
12:30 - 14:30	Lunch Break
14:30 - 15:20	Hee Oh (Yale, USA) $L^2$-triviality, Hausdorff dimension and Anosov subgroups For a discrete subgroup $\Gamma$ of a semisimple Lie group $G$, the space $L^2(\Gamma\backslash G)$ plays a crucial role in bridging representation theory with dynamics. We define $\Gamma$ to be $L^2$-trivial if $L^2(\Gamma\backslash G)$ is weakly equivalent to $L^2(G)$. Determining $L^2$-triviality for a given discrete subgroup is challenging. We present a criterion for Anosov subgroups to be $L^2$ trivial based on the Hausdorff dimension of their limit sets. For example, we obtain that the image of any surface subgroup in a real split higher rank simple Lie group under a positive representation is $L^2$-trivial. We also provide an example of projective Anosov subgroups that are not $L^2$-trivial. It remains an open question whether all Borel Anosov subgroups in higher rank are $L^2$-trivial. This talk is based on several independent joint works with S. Edwards, with M. Fraczyk, and with S. Dey and D. Kim.
15:20 - 16:00	Coffee & Tea
16:00 - 16:50	Konstantinos Tsouvalas (Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany) Robust QIE and limits of Anosov representations One of the key properties of Anosov representations into any real semisimple Lie group $G$, is that they are structurally stable (i.e. small perturbations of an Anosov representation of a hyperbolic group remain Anosov). In this talk, we will provide examples of stable, quasi-isometrically embedded hyperbolic subgroups of $\textup{SL}_n(\mathbb{C})$ (n sufficiently large) which fail to be algebraic limits of Anosov representations, showing that structural stability fails to characterize Anosov representations.
18:00 - 22:30	Conference Dinner

	Conference
08:45 - 09:00	Welcome Coffee
09:00 - 09:50	Subhadip Dey (Max Planck Institute for Mathematics in the Sciences, Germany) Discrete subgroups with full limit sets In this talk, we'll address the question of whether a discrete subgroup of a higher-rank Lie group G with a full limit set in its Furstenberg boundary must necessarily be a lattice. We'll demonstrate the existence of non-lattice discrete subgroups of G with full limit sets. These groups are infinitely generated, and we'll discuss how they can be constructed using an appropriate sequence of Anosov subgroups. This leaves open the intriguing question of whether finitely generated examples can exist. This is based on joint work with Sebastian Hurtado.
09:50 - 10:30	Coffee & Tea
10:30 - 11:20	Colin Davalo (Université Grenoble-Alpes, France) Finite-sided Dirichlet domains and Anosov representations In his study of uniform lattices in SL(n,R), Selberg described a construction of a polyhedral fundamental domain for any discrete subgroup of SL(n,R) in the projective model for the associated symmetric space. This construction is closely related to the construction of Dirichlet domains. However, for non-uniform lattices, these polyhedral domains can have infinitely many sides. As Dirichlet domains are always finite-sided for convex-cocompact subgroups in rank one Lie groups, it is natural to ask if the Dirichlet-Selberg domains are finite-sided for Anosov subgroups. Very simple Anosov subgroups in SL(3,R) can have infinite-sided Dirichlet-Selberg domains. We introduce a condition that in general is stronger than the Anosov property that implies that all Dirichlet-Selberg domains are finite sided in a strong sense. This condition turns out to be equivalent to some Anosov property if the subgroup lies in Sp(2n,R) or O(n,n). The proof of this result can be generalized to show that certain Dirichlet domains for Finsler metrics are finite-sided for certain Anosov subgroups. This is a joint work with Max Riestenberg.
11:30 - 12:30	Simion Filip (University of Chicago, USA) Anosov representations from Variations of Hodge structure Variations of Hodge Structure (VHS) arise naturally in algebraic geometry when studying families of algebraic manifolds, especially when looking at moduli spaces. The associated monodromy representations encode the topology of the family and control many algebro-geometric features. I will explain some local analytic conditions on the VHS which imply that the monodromy representation is Anosov. These local conditions are satisfied for interesting families of algebraic manifolds and also give examples of Anosov representations in Sp(2g) and SO(g,g+1) (for all g>1) outside the space of Theta-positive representations.
12:30 - 12:35	Group photo
12:35 - 14:30	Lunch Break
14:30 - 15:20	Sara Maloni (University of Virginia, USA) Geometric structures associated to Anosov representations. In this talk we are considering connected components of Anosov representations from the fundamental group of a closed hyperbolic manifold N, and which contains Fuchsian representations, and their associated domains of discontinuity. We will prove that the quotient of these domains of discontinuity are always smooth fiber bundles over N. Determining the topology of the fiber is hard in general, but we are able to describe it for representations in Sp(4,C), and for the domain of discontinuity in the space of complex Lagrangians in C^4 by using the classification of smooth 4-manifolds. This is joint work with Daniele Alessandrini, Nicolas Tholozan and Anna Wienhard. Time permitting, we will also discuss current work in progress with Mazzoli and Tholozan on the description of the fiber for SL(3, C) when we embed SL(2,C) reducibly and irreducibly.
15:20 - 16:00	Coffee & Tea
16:00 - 16:50	Clay Lecture: François Labourie (University Côte d'Azur, France) Double uniformisations’ cousin: complex projective spaces and anti de Sitter space. After recalling the double uniformisation for the complex projective spaces and related notions, we explain the similar picture for anti de Sitter space, where the unit disk/hyperbolic space is replaced by the annulus/de Sitter space. We also prove the main properties of the cousins of Epstein surface, W-volume. We define finally a Liouville action for metrics, smooth enough crossratios and positive curves. This is a joint work with Jérémy Toulisse and Yilin Wang.
17:00 - 22:00	Future Questions

	Conference
08:45 - 09:00	Welcome Coffee
09:00 - 09:50	Andrés Sambarino (CNRS - Sorbonne Université, France) The set of normalized variations and pressure forms on the Hitchin component Given a non-zero tangent vector $v$ to the character variety $\mathfrak X(\Gamma, G)$ of a discrete semi-group $\Gamma$ with values on a semi-simple Lie group $G$ we introduce, by analogy with Benoist's limit cone, the cone of Jordan variations and the set of normalized variations. The latter depends on the choice of a functional of the Cartan space of $G$ that is positive on Benoist's limit cone (of the base-point of $v$). We will explain results regarding convexity and non-empty interior for these sets and explain their implications on the non-degeneracy problem for pressure forms on the Hitchin component.
09:50 - 10:30	Coffee & Tea
10:30 - 11:20	Xenia Flamm (MPI-MiS Leipzig, Germany) Hilbert geometry over non-Archimedean ordered fields Convex projective surfaces arise as a geometric interpretation of Hitchin representations in SL(3, R). Their Hilbert metric encodes important information about the representation. Understanding degenerations of convex projective structures on a surface naturally leads to the study of the Hilbert geometry of subsets of the projective plane over a non-Archimedean ordered field. The goal of this talk is to introduce Hilbert geometry over such fields, and to show that degenerations of convex projective structures can in fact be viewed as length functions for these Hilbert metric spaces. This is joint work with Anne Parreau.
11:30 - 12:30	Nicolas Tholozan (ENS-PSL, France) Compact quotients of reductive homogeneous spaces. Let $G$ be be a semisimple Lie group, $H$ a reductive subgroup of $G$ and $\Gamma$ a discrete subgroup acting properly and cocompactly on $G/H$. I will present a joint work with Fanny Kassel, where we prove that $\Gamma$ satisfies a sharp form of the Benoist—Kobayashi properness criterion. When $H$ has co-rank $1$ in $G$, it follows that $\Gamma$ is an Anosov subgroup of $G$ and that small deformations of $\Gamma$ keep acting properly.
12:30 - 12:45	Closing Remarks

Participants (as of today)

Sofia Amontova

Université de Genève, Switzerland

Jacques Audibert

Max Planck Institute for Mathematics in the Sciences, Germany

Thierry Barbot

avignon university, France

Martin Bobb

MPI Leipzig, Germany

Jonathan Boretsky

MPI Leipzig, Germany

Samuel Bronstein

Ecole Normale Superieure, France

Marc Burger

ETH Zurich, Switzerland

Fernando Camacho Cadena

Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

Anunoy Chakraborty

George Mason University, USA

Adam Chalumeau

Strasbourg, France

Jeffrey Danciger

University of Texas-Austin, USA

Debattam Das

IISER Mohali, India

Colin Davalo

Université Grenoble-Alpes, France

Subhadip Dey

Max Planck Institute for Mathematics in the Sciences, Germany

Farid Diaf

Université Greoble Alpes, France

Barthelemy Doba

University of Sorbonne Paris Nord, France

Sami Douba

Institut des Hautes Études Scientifiques, France

Bruno Dular

University of Luxembourg, Luxembourg

Christian El Emam

University of Luxembourg, Luxembourg

Ekaterina Eremenko

Max Planck Institute for Mathematics in the Sciences, Germany

Parker Evans

Rice University, USA

Simion Filip

University of Chicago, USA

Xenia Flamm

MPI-MiS Leipzig, Germany

Segev Gonen Cohen

ETH Zurich, Switzerland

Jordan Grant

University of Michigan, USA

François Guéritaud

CNRS / Université de Strasbourg, France

Olivier Guichard

Université de Strasbourg, France

Paula Heim

MPI MiS Leipzig, Germany

Enya Hsiao

MPI Leipzig, Germany

Alessandra Iozzi

ETH, Switzerland

Mitul Islam

Max Planck Institute for Mathematics in the Sciences

Victor Jaeck

ETH Zurich, Switzerland

Fanny Kassel

Institut des Hautes Études Scientifiques, France

Kanishka Katipearachchi

Humboldt University, Berlin, Germany

Nat Kendal-Freedman

Max Planck Institute for Mathematics in the Sciences, Germany

Clarence Kineider

Max Planck Institute for Mathematics in the Sciences, Germany

François Labourie

University Côte d'Azur, France

Fabian Lander

Max Planck Institute for Mathematics in the Sciences, Germany

Laura Lankers

MPI MiS Leipzig, Germany

Joaquín Lejtreger

Sorbonne Université, France

Timothé Lemistre

Université de Nice, France

Sara Maloni

University of Virginia, USA

Arnaud Maret

Sorbonne Université, France

Casandra Monroe

The University of Texas at Austin, USA

Andrea Egidio Monti

MPI for Mathematics, Bonn, Germany

Merik Juljan Niemeyer

Max Planck Institute for Mathematics in the Sciences, Germany

Alexander Nolte

Rice University, Georgia Tech, USA

Hee Oh

Yale, USA

Rafael Potrie

Universidad de la República, Uruguay

Roman Prosanov

University of Vienna, Austria

Grazia Rago

Università di Bologna, Italy

Charlie Ried

Max Planck Institute for Mathematics in the Sciences, Germany

Max Riestenberg

Max Planck Institute for Mathematics in the Sciences

Vasilii Rogov

Max Planck Institute for Mathematics in the Sciences, Germany

Andrés Sambarino

CNRS - Sorbonne Université, France

Suzanne Schlich

Institut Fourier, Université Grenoble Alpes, France

Jiajun Shi

MPI MiS, Germany

Rym SMAÏ

University of Strasbourg, France

Ilia Smilga

University of Oxford, United Kingdom

Peter Smillie

Max Planck Institute for Mathematics in the Sciences, Germany

Mireille Soergel

Max Planck Institute for Mathematics in the Sciences, Germany

Manvendra Somvanshi

Max Planck Institute for Mathematics in the Sciences, Germany

Giorgos Stamatiou

Université Paris Dauphine, France

Nicolas Tholozan

ENS-PSL, France

Luis Torres

The University of Texas at Austin, USA

Marie Trin

MPI Leipzig, Germany

Roméo Troubat

Institut de Recherche Mathématique Avancée, Université de Strasbourg, France

Konstantinos Tsouvalas

Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany

Jennifer Vaccaro

University of Illinois at Chicago, USA

Theodore Weisman

University of Michigan, USA

Rou Wen

University of Wisconsin-Madison, USA

Anna Wienhard

Max Planck Institute for Mathematics in the Sciences, Germany

Jingyi Xue

National University of Singapore, Republic of Singapore

Zhufeng Yao

National University of Singapore, Republic of Singapore

Neza Zager Korenjak

University of Michigan, USA

Tengren Zhang

National University of Singapore, Republic of Singapore

Feng Zhu

University of Wisconsin Madison, USA

Michael Zshornack

Northwestern University, USA

Organizers of the Conference

Olivier Guichard

Université de Strasbourg

Fanny Kassel

Institut des Hautes Études Scientifiques

Anna Wienhard

Max Planck Institute for Mathematics in the Sciences

Organizers of the Young Researcher's Workshop

Jacques Audibert

Max Planck Institute for Mathematics in the Sciences

Xenia Flamm

Max Planck Institute for Mathematics in the Sciences

Kostas Tsouvalas

Max Planck Institute for Mathematics in the Sciences

Teddy Weisman

University of Michigan

Contact

Antje Vandenberg (administrative contact)

Max Planck Institute for Mathematics in the Sciences Contact via Mail

J. Audibert, X. Flamm, K. Tsouvalas, T. Weisman (organizational contact)

Contact via Mail

20 years of Anosov representations

WARNING: Please be wary if a company directly contacts you regarding payment for accommodation in Leipzig: this is fraud / phishing.

Links

Program

Participants (as of today)

Sofia Amontova

Jacques Audibert

Thierry Barbot

Martin Bobb

Jonathan Boretsky

Samuel Bronstein

Marc Burger

Fernando Camacho Cadena

Anunoy Chakraborty

Adam Chalumeau

Jeffrey Danciger

Debattam Das

Colin Davalo

Subhadip Dey

Farid Diaf

Barthelemy Doba

Sami Douba

Bruno Dular

Christian El Emam

Ekaterina Eremenko

Parker Evans

Simion Filip

Xenia Flamm

Segev Gonen Cohen

Jordan Grant

François Guéritaud

Olivier Guichard

Paula Heim

Enya Hsiao

Alessandra Iozzi

Mitul Islam

Victor Jaeck

Fanny Kassel

Kanishka Katipearachchi

Nat Kendal-Freedman

Clarence Kineider

François Labourie

Fabian Lander

Laura Lankers

Joaquín Lejtreger

Timothé Lemistre

Sara Maloni

Arnaud Maret

Casandra Monroe

Andrea Egidio Monti

Merik Juljan Niemeyer

Alexander Nolte

Hee Oh

Rafael Potrie

Roman Prosanov

Grazia Rago

Charlie Ried

Max Riestenberg

Vasilii Rogov

Andrés Sambarino

Suzanne Schlich

Jiajun Shi

Rym SMAÏ

Ilia Smilga

Peter Smillie

Mireille Soergel

Manvendra Somvanshi

Giorgos Stamatiou

Nicolas Tholozan

Luis Torres

Marie Trin

Roméo Troubat

Konstantinos Tsouvalas

Jennifer Vaccaro

Theodore Weisman

Rou Wen

Anna Wienhard

Jingyi Xue

Zhufeng Yao

Neza Zager Korenjak