2025 IHES Summer School - Discrete Subgroups of Lie Groups: Dynamics, Actions, Rigidity

7 juil. 2025, 08:30 → 18 juil. 2025, 18:00 Europe/Paris

Marilyn and James Simons Conference Center (IHES)

2025 IHES SUMMER SCHOOL

Theme: Recent rigidity results for discrete subgroups of Lie groups and their actions on manifolds, at the intersection of dynamics with Lie theory and geometry. 

The Summer School will be held at the Institut des Hautes Études Scientifiques (IHES) from July 7-18, 2025. IHES is located in Bures-sur-Yvette, south of Paris (40 minutes by train from Paris) - Access map

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Recently there has been remarkable progress on several important problems broadly centered around the study of discrete subgroups of Lie groups. The primary goal of this summer school is to allow young reseachers to come together and learn about a number of these exciting developments. 

Activities will be centered around lecture series by established experts known both for their strong contributions to the field and for the high quality of their mathematical exposition. We also plan to foster an environment where these young mathematicians are able to learn from each other and have opportunities to begin new collaborations that will drive the future of the subject.

The programme of the school will consist of nine mini-courses each ranging between 3 to 5 hours of lecture, and will include evening problem sessions.

Organizing Committee/Scientific Committe: David Fisher (Rice University), Fanny Kassel (CNRS & IHES), Ralf Spatzier (University of Michigan) and Matthew Stover (Temple University).

This school is open to everybody but intended primarily for young participants, including Ph.D. students and postdoctoral fellows. 

Application is open until March 16, 2025

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Mini-courses speakers:

• Simion Filip, University of Chicago
• Homin Lee, Northwestern University
• Sam Mellick, Jagiellonian University and Amanda Wilkens, Carnegie Mellon University 
  
• Daniel Monclair, Université Paris-Saclay
• Maria Beatrice Pozzetti, Universitá di Bologna
• Roman Sauer, Karlsruher Institut für Technologie
• Barbara Schapira, IMAG, Université de Montpellier
  
• Antoine Song, California Institute of Technology
• Nattalie Tamam, Imperial College London

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This is an IHES Summer School organized in partnership with the following institutions and sponsors:

Exercise-Filip-1.pdf

Exercises - Lee.pdf

Exercises - Monclair.pdf

Exercises - Wilkens.pdf

Exercises-Pozzetti.pdf

Exercises-Sauer.pdf

Exercises-Schapira.pdf

Exercises-Song.pdf

Exercises-Taman.pdf

SummerSchool-Lie-2025_webPoster.pdf

lun. 7 juillet

08:45 Registration and welcome coffee

1 Higher Property T (1/5)

Kazhdan’s Property T is a fundamental analytic invariant of discrete or, more generally, locally compact groups that is defined in terms of unitary representations. According to Delorme-Guichardet it can be characterized by the vanishing of the group cohomology in degree 1 for arbitrary unitary coefficients. This suggests an obvious generalizations to a higher degree Property T. We will discuss the first higher property T result by Garland from the 1970s. We will then turn to the Lie group case and their lattices, which is based on joint work with Uri Bader. As time permits, we will also discuss applications of (higher) property T.

Orateur: Roman Sauer (Karlsruher Institut für Technologie)

Vidéo

Kazhdan’s Property T is a fundamental analytic invariant of discrete or, more generally, locally compact groups that is defined in terms of unitary representations. 
According to Delorme-Guichardet it can be characterized by the vanishing of the group cohomology in degree 1 for arbitrary unitary coefficients. This suggests an obvious generalizations to a higher degree Property T. We will discuss the first higher property T result by Garland from the 1970s. We will then turn to the Lie group case and their lattices, which is based on joint work with Uri Bader. As time permits, we will also discuss applications of (higher) property T.

10:15 Coffee break

2 Thermodynamical formalism and geometric applications (1/3)

In these lectures, I will first present a construction of good invariant measures for the geodesic flow of a hyperbolic surface, the so-called Gibbs measures.
I will explain some of their important ergodic properties. Afterwards, I will present some geometric questions that can be solved thanks to these Gibbs measures, for example the regularity of entropy under a change of metrics, or the study of horocyclic flow on abelian covers of compact hyperbolic surfaces, or....

Orateur: Barbara Schapira (IMAG, Université de Montpellier)

Vidéo

In these lectures, I will first present a construction of good invariant measures for the geodesic flow of a hyperbolic surface, the so-called Gibbs measures.

I will explain some of their important ergodic properties. Afterwards, I will present some geometric questions that can be solved thanks to these Gibbs measures, for example the regularity of entropy under a change of metrics, or the study of horocyclic flow on abelian covers of compact hyperbolic surfaces, or....

11:30 Exercises

12:30 Buffet-lunch at IHES

3 An introduction to higher rank Teichmüller theory (1/4)

The minicourse will focus on discrete subgroups of semisimple Lie groups G isomorphic to fundamental groups $\Gamma$ of surfaces. These typically admit a rich deformation theory and can be parametrized as subset of the character variety $X=Hom(\Gamma, G)/G$. I will first discuss the Anosov condition, describing open subsets of $X$ and then discuss higher rank Teichmüller theories: connected components of $X$ only consisting of discrete and faithful representations. We proved with Beyrer-Guichard-Labourie-Wienhard that for classical groups G these are explained by $\Theta$-positivity, a Lie algebraic framework introduced by Guichard-Wienhard. After introducing this concept I will explain how closedness in the character variety is ultimately due to a collar lemma, generalizing a key geometric feature of hyperbolic surfaces.

Orateur: Maria Beatrice Pozzetti (Universitá di Bologna)

Vidéo

The minicourse will focus on discrete subgroups of semisimple Lie groups G isomorphic to fundamental groups $\Gamma$ of surfaces. These typically admit a rich deformation theory and can be parametrized as subset of the character variety $X=Hom(\Gamma, G)/G$. I will first discuss the Anosov condition, describing open subsets of $X$ and then discuss higher rank Teichmüller theories: connected components of $X$ only consisting of discrete and faithful representations. We proved with Beyrer-Guichard-Labourie-Wienhard that for classical groups G these are explained by $\Theta$-positivity, a Lie algebraic framework introduced by Guichard-Wienhard. After introducing this concept I will explain how closedness in the character variety is ultimately due to a collar lemma, generalizing a key geometric feature of hyperbolic surfaces.

15:30 Coffee break

4 Higher Property T (2/5)

Kazhdan’s Property T is a fundamental analytic invariant of discrete or, more generally, locally compact groups that is defined in terms of unitary representations. According to Delorme-Guichardet it can be characterized by the vanishing of the group cohomology in degree 1 for arbitrary unitary coefficients. This suggests an obvious generalizations to a higher degree Property T. We will discuss the first higher property T result by Garland from the 1970s. We will then turn to the Lie group case and their lattices, which is based on joint work with Uri Bader. As time permits, we will also discuss applications of (higher) property T.

Orateur: Roman Sauer (Karlsruher Institut für Technologie)

Vidéo

Kazhdan’s Property T is a fundamental analytic invariant of discrete or, more generally, locally compact groups that is defined in terms of unitary representations. 
According to Delorme-Guichardet it can be characterized by the vanishing of the group cohomology in degree 1 for arbitrary unitary coefficients. This suggests an obvious generalizations to a higher degree Property T. We will discuss the first higher property T result by Garland from the 1970s. We will then turn to the Lie group case and their lattices, which is based on joint work with Uri Bader. As time permits, we will also discuss applications of (higher) property T.

17:00 Exercises

mar. 8 juillet

09:00 Welcome coffee

5 Thermodynamical formalism and geometric applications (2/3)

In these lectures, I will first present a construction of good invariant measures for the geodesic flow of a hyperbolic surface, the so-called Gibbs measures.
I will explain some of their important ergodic properties. Afterwards, I will present some geometric questions that can be solved thanks to these Gibbs measures, for example the regularity of entropy under a change of metrics, or the study of horocyclic flow on abelian covers of compact hyperbolic surfaces, or ...

Orateur: Barbara Schapira (IMAG, Université de Montpellier)

Vidéo

In these lectures, I will first present a construction of good invariant measures for the geodesic flow of a hyperbolic surface, the so-called Gibbs measures.

I will explain some of their important ergodic properties. Afterwards, I will present some geometric questions that can be solved thanks to these Gibbs measures, for example the regularity of entropy under a change of metrics, or the study of horocyclic flow on abelian covers of compact hyperbolic surfaces, or ...

10:15 Coffee break

6 Higher Property T (3/5)

Kazhdan’s Property T is a fundamental analytic invariant of discrete or, more generally, locally compact groups that is defined in terms of unitary representations. According to Delorme-Guichardet it can be characterized by the vanishing of the group cohomology in degree 1 for arbitrary unitary coefficients. This suggests an obvious generalizations to a higher degree Property T. We will discuss the first higher property T result by Garland from the 1970s. We will then turn to the Lie group case and their lattices, which is based on joint work with Uri Bader. As time permits, we will also discuss applications of (higher) property T.

Orateur: Roman Sauer (Karlsruher Institut für Technologie)

Vidéo

Kazhdan’s Property T is a fundamental analytic invariant of discrete or, more generally, locally compact groups that is defined in terms of unitary representations. 
According to Delorme-Guichardet it can be characterized by the vanishing of the group cohomology in degree 1 for arbitrary unitary coefficients. This suggests an obvious generalizations to a higher degree Property T. We will discuss the first higher property T result by Garland from the 1970s. We will then turn to the Lie group case and their lattices, which is based on joint work with Uri Bader. As time permits, we will also discuss applications of (higher) property T.

11:30 Exercises

12:30 Lunch-break

7 An introduction to higher rank Teichmüller theory (2/4)

The minicourse will focus on discrete subgroups of semisimple Lie groups G isomorphic to fundamental groups $\Gamma$ of surfaces. These typically admit a rich deformation theory and can be parametrized as subset of the character variety $X=Hom(\Gamma, G)/G$. I will first discuss the Anosov condition, describing open subsets of $X$ and then discuss higher rank Teichmüller theories: connected components of $X$ only consisting of discrete and faithful representations. We proved with Beyrer-Guichard-Labourie-Wienhard that for classical groups G these are explained by $\Theta$-positivity, a Lie algebraic framework introduced by Guichard-Wienhard. After introducing this concept I will explain how closedness in the character variety is ultimately due to a collar lemma, generalizing a key geometric feature of hyperbolic surfaces.

Orateur: Maria Beatrice Pozzetti (Universitá di Bologna)

Vidéo

The minicourse will focus on discrete subgroups of semisimple Lie groups G isomorphic to fundamental groups $\Gamma$ of surfaces. These typically admit a rich deformation theory and can be parametrized as subset of the character variety $X=Hom(\Gamma, G)/G$. I will first discuss the Anosov condition, describing open subsets of $X$ and then discuss higher rank Teichmüller theories: connected components of $X$ only consisting of discrete and faithful representations. We proved with Beyrer-Guichard-Labourie-Wienhard that for classical groups G these are explained by $\Theta$-positivity, a Lie algebraic framework introduced by Guichard-Wienhard. After introducing this concept I will explain how closedness in the character variety is ultimately due to a collar lemma, generalizing a key geometric feature of hyperbolic surfaces.

15:30 Coffee break

8 Measure rigidity in higher rank lattice actions (1/3)

In this mini-course, we will discuss about actions of higher rank lattices, focusing on how measures and measure rigidity play important roles in various settings.
First, we introduce some definitions and properties related to actions of higher rank lattices and measure rigidity results.
Then, using measure rigidity, we will discuss classical theorems, Margulis’ normal subgroup theorem and Margulis’ superrigidity theorem, as well as, recent works on smooth actions of higher rank lattices on manifolds, so-called Zimmer program.

Orateur: Homin Lee (Northwestern University)

Vidéo

In this mini-course, we will discuss about actions of higher rank lattices, focusing on how measures and measure rigidity play important roles in various settings. 

First, we introduce some definitions and properties related to actions of higher rank lattices and measure rigidity results.

Then, using measure rigidity, we will discuss classical theorems, Margulis’ normal subgroup theorem and Margulis’ superrigidity theorem, as well as, recent works on smooth actions of higher rank lattices on manifolds, so-called Zimmer program.

17:00 Exercises

mer. 9 juillet

09:00 Welcome coffee

9 Higher Property T (4/5)

Kazhdan’s Property T is a fundamental analytic invariant of discrete or, more generally, locally compact groups that is defined in terms of unitary representations. According to Delorme-Guichardet it can be characterized by the vanishing of the group cohomology in degree 1 for arbitrary unitary coefficients. This suggests an obvious generalizations to a higher degree Property T. We will discuss the first higher property T result by Garland from the 1970s. We will then turn to the Lie group case and their lattices, which is based on joint work with Uri Bader. As time permits, we will also discuss applications of (higher) property T.

Orateur: Roman Sauer (Karlsruher Institut für Technologie)

Vidéo

Kazhdan’s Property T is a fundamental analytic invariant of discrete or, more generally, locally compact groups that is defined in terms of unitary representations. 
According to Delorme-Guichardet it can be characterized by the vanishing of the group cohomology in degree 1 for arbitrary unitary coefficients. This suggests an obvious generalizations to a higher degree Property T. We will discuss the first higher property T result by Garland from the 1970s. We will then turn to the Lie group case and their lattices, which is based on joint work with Uri Bader. As time permits, we will also discuss applications of (higher) property T.

10:15 Coffee break

10 The Geometry of Diverging Orbits (1/3)

The study of diagonal group actions on homogeneous spaces occupies a central role in modern homogeneous dynamics, with deep connections to number theory, ergodic theory, and the geometry of locally symmetric spaces. In this talk, we focus on the ones which escape to infinity. We explore recent advances in the analysis of such orbits, emphasizing the use of representation theory and their geometric properties as powerful tools to probe their structure, and more specifically to distinguish between certain types of them.

Orateur: Nattalie Tamam (Imperial College London)

Vidéo

The study of diagonal group actions on homogeneous spaces occupies a central role in modern homogeneous dynamics, with deep connections to number theory, ergodic theory, and the geometry of locally symmetric spaces. In this talk, we focus on the ones which escape to infinity. We explore recent advances in the analysis of such orbits, emphasizing the use of representation theory and their geometric properties as powerful tools to probe their structure, and more specifically to distinguish between certain types of them.

 

11:30 Exercises

12:30 Lunch-break

11 An introduction to higher rank Teichmüller theory (3/4)

The minicourse will focus on discrete subgroups of semisimple Lie groups G isomorphic to fundamental groups $\Gamma$ of surfaces. These typically admit a rich deformation theory and can be parametrized as subset of the character variety $X=Hom(\Gamma, G)/G$. I will first discuss the Anosov condition, describing open subsets of $X$ and then discuss higher rank Teichmüller theories: connected components of $X$ only consisting of discrete and faithful representations. We proved with Beyrer-Guichard-Labourie-Wienhard that for classical groups G these are explained by $\Theta$-positivity, a Lie algebraic framework introduced by Guichard-Wienhard. After introducing this concept I will explain how closedness in the character variety is ultimately due to a collar lemma, generalizing a key geometric feature of hyperbolic surfaces.

Orateur: Maria Beatrice Pozzetti (Universitá di Bologna)

Vidéo

The minicourse will focus on discrete subgroups of semisimple Lie groups G isomorphic to fundamental groups $\Gamma$ of surfaces. These typically admit a rich deformation theory and can be parametrized as subset of the character variety $X=Hom(\Gamma, G)/G$. I will first discuss the Anosov condition, describing open subsets of $X$ and then discuss higher rank Teichmüller theories: connected components of $X$ only consisting of discrete and faithful representations. We proved with Beyrer-Guichard-Labourie-Wienhard that for classical groups G these are explained by $\Theta$-positivity, a Lie algebraic framework introduced by Guichard-Wienhard. After introducing this concept I will explain how closedness in the character variety is ultimately due to a collar lemma, generalizing a key geometric feature of hyperbolic surfaces.

15:30 Coffee break

12 Thermodynamical formalism and geometric applications (3/3)

In these lectures, I will first present a construction of good invariant measures for the geodesic flow of a hyperbolic surface, the so-called Gibbs measures.
I will explain some of their important ergodic properties. Afterwards, I will present some geometric questions that can be solved thanks to these Gibbs measures, for example the regularity of entropy under a change of metrics, or the study of horocyclic flow on abelian covers of compact hyperbolic surfaces, or ...

Orateur: Barbara Schapira (IMAG, Université de Montpellier)

Vidéo

In these lectures, I will first present a construction of good invariant measures for the geodesic flow of a hyperbolic surface, the so-called Gibbs measures.

I will explain some of their important ergodic properties. Afterwards, I will present some geometric questions that can be solved thanks to these Gibbs measures, for example the regularity of entropy under a change of metrics, or the study of horocyclic flow on abelian covers of compact hyperbolic surfaces, or ...

17:00 Exercises

jeu. 10 juillet

09:00 Welcome coffee

13 The Geometry of Diverging Orbits (2/3)

The study of diagonal group actions on homogeneous spaces occupies a central role in modern homogeneous dynamics, with deep connections to number theory, ergodic theory, and the geometry of locally symmetric spaces. In this talk, we focus on the ones which escape to infinity. We explore recent advances in the analysis of such orbits, emphasizing the use of representation theory and their geometric properties as powerful tools to probe their structure, and more specifically to distinguish between certain types of them.

Orateur: Nattalie Tamam (Imperial College London)

Vidéo

The study of diagonal group actions on homogeneous spaces occupies a central role in modern homogeneous dynamics, with deep connections to number theory, ergodic theory, and the geometry of locally symmetric spaces. In this talk, we focus on the ones which escape to infinity. We explore recent advances in the analysis of such orbits, emphasizing the use of representation theory and their geometric properties as powerful tools to probe their structure, and more specifically to distinguish between certain types of them.

 

10:15 Coffee break

14 An introduction to higher rank Teichmüller theory (4/4)

The minicourse will focus on discrete subgroups of semisimple Lie groups G isomorphic to fundamental groups $\Gamma$ of surfaces. These typically admit a rich deformation theory and can be parametrized as subset of the character variety $X=Hom(\Gamma, G)/G$. I will first discuss the Anosov condition, describing open subsets of $X$ and then discuss higher rank Teichmüller theories: connected components of $X$ only consisting of discrete and faithful representations. We proved with Beyrer-Guichard-Labourie-Wienhard that for classical groups G these are explained by $\Theta$-positivity, a Lie algebraic framework introduced by Guichard-Wienhard. After introducing this concept I will explain how closedness in the character variety is ultimately due to a collar lemma, generalizing a key geometric feature of hyperbolic surfaces.

Orateur: Maria Beatrice Pozzetti (Universitá di Bologna)

Vidéo

The minicourse will focus on discrete subgroups of semisimple Lie groups G isomorphic to fundamental groups $\Gamma$ of surfaces. These typically admit a rich deformation theory and can be parametrized as subset of the character variety $X=Hom(\Gamma, G)/G$. I will first discuss the Anosov condition, describing open subsets of $X$ and then discuss higher rank Teichmüller theories: connected components of $X$ only consisting of discrete and faithful representations. We proved with Beyrer-Guichard-Labourie-Wienhard that for classical groups G these are explained by $\Theta$-positivity, a Lie algebraic framework introduced by Guichard-Wienhard. After introducing this concept I will explain how closedness in the character variety is ultimately due to a collar lemma, generalizing a key geometric feature of hyperbolic surfaces.

11:30 Exercises

12:30 Lunch-break

15 Measure rigidity in higher rank lattice actions (2/3)

In this mini-course, we will discuss about actions of higher rank lattices, focusing on how measures and measure rigidity play important roles in various settings.
First, we introduce some definitions and properties related to actions of higher rank lattices and measure rigidity results.
Then, using measure rigidity, we will discuss classical theorems, Margulis’ normal subgroup theorem and Margulis’ superrigidity theorem, as well as, recent works on smooth actions of higher rank lattices on manifolds, so-called Zimmer program.

Orateur: Homin Lee (Northwestern University)

Vidéo

In this mini-course, we will discuss about actions of higher rank lattices, focusing on how measures and measure rigidity play important roles in various settings. 

First, we introduce some definitions and properties related to actions of higher rank lattices and measure rigidity results.

Then, using measure rigidity, we will discuss classical theorems, Margulis’ normal subgroup theorem and Margulis’ superrigidity theorem, as well as, recent works on smooth actions of higher rank lattices on manifolds, so-called Zimmer program.

15:30 Coffee break

16 Higher Property T (5/5)

Kazhdan’s Property T is a fundamental analytic invariant of discrete or, more generally, locally compact groups that is defined in terms of unitary representations. According to Delorme-Guichardet it can be characterized by the vanishing of the group cohomology in degree 1 for arbitrary unitary coefficients. This suggests an obvious generalizations to a higher degree Property T. We will discuss the first higher property T result by Garland from the 1970s. We will then turn to the Lie group case and their lattices, which is based on joint work with Uri Bader. As time permits, we will also discuss applications of (higher) property T.

Orateur: Roman Sauer (Karlsruher Institut für Technologie)

Vidéo

Kazhdan’s Property T is a fundamental analytic invariant of discrete or, more generally, locally compact groups that is defined in terms of unitary representations. 
According to Delorme-Guichardet it can be characterized by the vanishing of the group cohomology in degree 1 for arbitrary unitary coefficients. This suggests an obvious generalizations to a higher degree Property T. We will discuss the first higher property T result by Garland from the 1970s. We will then turn to the Lie group case and their lattices, which is based on joint work with Uri Bader. As time permits, we will also discuss applications of (higher) property T.

17:00 Exercises

ven. 11 juillet

09:00 Welcome coffee

17 Locally homogeneous flows and Anosov representations (1/5)

Anosov representations form an open set of homomorphisms of a discrete hyperbolic group into a semi-simple Lie group G. Labourie introduced them in a dynamical language, requiring that a section of an associate flat bundle should provide a hyperbolic set for some flow. Later, several equivalent characterisations of Anosov representations that do not involve the dynamics of a flow were identified, often with the goal of producing geometric structures associated to these representations in the form of compact quotients of open subsets of flag manifolds.
This course will present recent work with B. Delarue and A. Sanders that uses (non-compact) quotients of open subsets of appropriate homogeneous spaces (not flag manifolds) for the Lie group G, equipped with a flow that commutes with the action of G. This quotient produces a locally homogeneous flow with uniformly hyperbolic dynamics (Smale's axiom A). This approach allows for the use of modern analytic techniques of smooth dynamics that were not applicable to Anosov representations so far.
The first lectures will focus on the case of projective Anosov representations into SL(d,R). After introducing Anosov representations in this setting, we will describe the construction of the locally homogeneous flow and its dynamical properties. In a second part of the course, we will study the case of a general semi-simple Lie group G (and arbitrary flag manifolds used to define Anosov representations). We will see how the linear algebra of the SL(d,R) case can be replaced with differential geometric notions in the general situation. Several examples will be discussed, with an emphasis on those that can be described as some non-Riemannian geodesic flow.
The course will use notions from Lie theory, dynamical systems, differential geometry and a touch of geometric group theory. The necessary background on these subjects will be kept to a minimum.

Orateur: Daniel Monclair (Université Paris-Saclay)

Vidéo

Anosov representations form an open set of homomorphisms of a discrete hyperbolic group into a semi-simple Lie group G. Labourie introduced them in a dynamical language, requiring that a section of an associate flat bundle should provide a hyperbolic set for some flow. Later, several equivalent characterisations of Anosov representations that do not involve the dynamics of a flow were identified, often with the goal of producing geometric structures associated to these representations in the form of compact quotients of open subsets of flag manifolds.
This course will present recent work with B. Delarue and A. Sanders that uses (non-compact) quotients of open subsets of appropriate homogeneous spaces (not flag manifolds) for the Lie group G, equipped with a flow that commutes with the action of G. This quotient produces a locally homogeneous flow with uniformly hyperbolic dynamics (Smale's axiom A). This approach allows for the use of modern analytic techniques of smooth dynamics that were not applicable to Anosov representations so far.
The first lectures will focus on the case of projective Anosov representations into SL(d,R). After introducing Anosov representations in this setting, we will describe the construction of the locally homogeneous flow and its dynamical properties. In a second part of the course, we will study the case of a general semi-simple Lie group G (and arbitrary flag manifolds used to define Anosov representations). We will see how the linear algebra of the SL(d,R) case can be replaced with differential geometric notions in the general situation. Several examples will be discussed, with an emphasis on those that can be described as some non-Riemannian geodesic flow.
The course will use notions from Lie theory, dynamical systems, differential geometry and a touch of geometric group theory. The necessary background on these subjects will be kept to a minimum.

10:15 Coffee break

18 Poisson–Voronoi tessellations and fixed price in higher rank (1/5)

We will start by defining and motivating the Poisson point process, which is, informally, a "maximally random" scattering of points in space, and discussing the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group (for higher rank semisimple Lie groups, the minimum number of generators in a lattice is sublinear in the covolume). In this minicourse we will unpack the proof. No prior knowledge on Poisson--Voronoi tessellations, fixed price or higher rank will be assumed.

Orateurs: Amanda Wilkens (Carnegie Mellon University), Sam Mellick (Jagiellonian University)

Vidéo

We will start by defining and motivating the Poisson point process, which is, informally, a "maximally random" scattering of points in space, and discussing the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group (for higher rank semisimple Lie groups, the minimum number of generators in a lattice is sublinear in the covolume). In this minicourse we will unpack the proof. No prior knowledge on Poisson--Voronoi tessellations, fixed price or higher rank will be assumed.

 

11:30 Exercises

12:30 Lunch-break

19 Measure rigidity in higher rank lattice actions (3/3)

In this mini-course, we will discuss about actions of higher rank lattices, focusing on how measures and measure rigidity play important roles in various settings.
First, we introduce some definitions and properties related to actions of higher rank lattices and measure rigidity results.
Then, using measure rigidity, we will discuss classical theorems, Margulis’ normal subgroup theorem and Margulis’ superrigidity theorem, as well as, recent works on smooth actions of higher rank lattices on manifolds, so-called Zimmer program.

Orateur: Homin Lee (Northwestern University)

Vidéo

In this mini-course, we will discuss about actions of higher rank lattices, focusing on how measures and measure rigidity play important roles in various settings. 

First, we introduce some definitions and properties related to actions of higher rank lattices and measure rigidity results.

Then, using measure rigidity, we will discuss classical theorems, Margulis’ normal subgroup theorem and Margulis’ superrigidity theorem, as well as, recent works on smooth actions of higher rank lattices on manifolds, so-called Zimmer program.

15:30 Coffee break

20 The Geometry of Diverging Orbits (3/3)

The study of diagonal group actions on homogeneous spaces occupies a central role in modern homogeneous dynamics, with deep connections to number theory, ergodic theory, and the geometry of locally symmetric spaces. In this talk, we focus on the ones which escape to infinity. We explore recent advances in the analysis of such orbits, emphasizing the use of representation theory and their geometric properties as powerful tools to probe their structure, and more specifically to distinguish between certain types of them.

Orateur: Nattalie Taman (Imperial College London)

Vidéo

The study of diagonal group actions on homogeneous spaces occupies a central role in modern homogeneous dynamics, with deep connections to number theory, ergodic theory, and the geometry of locally symmetric spaces. In this talk, we focus on the ones which escape to infinity. We explore recent advances in the analysis of such orbits, emphasizing the use of representation theory and their geometric properties as powerful tools to probe their structure, and more specifically to distinguish between certain types of them.

 

17:00 Exercises

mar. 15 juillet

09:00 Welcome coffee

21 Poisson–Voronoi tessellations and fixed price in higher rank (2/5)

We will start by defining and motivating the Poisson point process, which is, informally, a "maximally random" scattering of points in space, and discussing the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group (for higher rank semisimple Lie groups, the minimum number of generators in a lattice is sublinear in the covolume). In this minicourse we will unpack the proof. No prior knowledge on Poisson--Voronoi tessellations, fixed price or higher rank will be assumed.

Orateurs: Amanda Wilkens (Carnegie Mellon University), Sam Mellick (Jagiellonian University)

Vidéo

We will start by defining and motivating the Poisson point process, which is, informally, a "maximally random" scattering of points in space, and discussing the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group (for higher rank semisimple Lie groups, the minimum number of generators in a lattice is sublinear in the covolume). In this minicourse we will unpack the proof. No prior knowledge on Poisson--Voronoi tessellations, fixed price or higher rank will be assumed.

 

10:15 Coffee break

22 Geometry of Anosov flows and Rigidity (1/4)

The geodesic flow on a manifold of negative sectional curvature is an archetypal example of an Anosov flow, a dynamical system under which every vector gets uniformly expanded or uniformly contracted. We will begin with an introduction to the geometry of these dynamical systems, including invariant manifolds, ergodicity, and various regularity questions.
We will then introduce flows that are compact group extensions of Anosov flows, discuss the associated "Brin group", a sort of Galois group of the extension. We will then apply these techniques to explain a result jointly obtained with David Fisher and Ben Lowe, saying that if a compact negatively curved real-analytic Riemannian manifold has infinitely many totally geodesic hypersurfaces, then it must be of constant sectional curvature.

Orateur: Simion Filip (University of Chicago)

Vidéo

The geodesic flow on a manifold of negative sectional curvature is an archetypal example of an Anosov flow, a dynamical system under which every vector gets uniformly expanded or uniformly contracted. We will begin with an introduction to the geometry of these dynamical systems, including invariant manifolds, ergodicity, and various regularity questions.

We will then introduce flows that are compact group extensions of Anosov flows, discuss the associated "Brin group", a sort of Galois group of the extension. We will then apply these techniques to explain a result jointly obtained with David Fisher and Ben Lowe, saying that if a compact negatively curved real-analytic Riemannian manifold has infinitely many totally geodesic hypersurfaces, then it must be of constant sectional curvature.

11:30 Exercises

12:30 Lunch-break

23 Harmonic maps in high-dimensional spheres, representations and random matrices (1/4)

This course will be about harmonic maps from 2d surfaces to spheres of high dimensions, coming from unitary representations of surface groups. This topic falls under the common theme in geometric analysis of studying geometric objects from topological data and vice versa. We will discuss rigidity phenomena for the shape of such harmonic maps into spheres, focusing on both the high-dimensional asymptotic regime, where random matrix theory plays a role, and the infinite dimensional case, where representation theory of PSL2(R) is central.

Orateur: Antoine Song (California Institute of Technology)

Vidéo

This course will be about harmonic maps from 2d surfaces to spheres of high dimensions, coming from unitary representations of surface groups. This topic falls under the common theme in geometric analysis of studying geometric objects from topological data and vice versa. We will discuss rigidity phenomena for the shape of such harmonic maps into spheres, focusing on both the high-dimensional asymptotic regime, where random matrix theory plays a role, and the infinite dimensional case, where representation theory of PSL2(R) is central.

15:30 Coffee break

24 Locally homogeneous flows and Anosov representations (2/5)

Anosov representations form an open set of homomorphisms of a discrete hyperbolic group into a semi-simple Lie group G. Labourie introduced them in a dynamical language, requiring that a section of an associate flat bundle should provide a hyperbolic set for some flow. Later, several equivalent characterisations of Anosov representations that do not involve the dynamics of a flow were identified, often with the goal of producing geometric structures associated to these representations in the form of compact quotients of open subsets of flag manifolds.
This course will present recent work with B. Delarue and A. Sanders that uses (non-compact) quotients of open subsets of appropriate homogeneous spaces (not flag manifolds) for the Lie group G, equipped with a flow that commutes with the action of G. This quotient produces a locally homogeneous flow with uniformly hyperbolic dynamics (Smale's axiom A). This approach allows for the use of modern analytic techniques of smooth dynamics that were not applicable to Anosov representations so far.
The first lectures will focus on the case of projective Anosov representations into SL(d,R). After introducing Anosov representations in this setting, we will describe the construction of the locally homogeneous flow and its dynamical properties. In a second part of the course, we will study the case of a general semi-simple Lie group G (and arbitrary flag manifolds used to define Anosov representations). We will see how the linear algebra of the SL(d,R) case can be replaced with differential geometric notions in the general situation. Several examples will be discussed, with an emphasis on those that can be described as some non-Riemannian geodesic flow.
The course will use notions from Lie theory, dynamical systems, differential geometry and a touch of geometric group theory. The necessary background on these subjects will be kept to a minimum.

Orateur: Daniel Monclair (Université Paris-Saclay)

Vidéo

Anosov representations form an open set of homomorphisms of a discrete hyperbolic group into a semi-simple Lie group G. Labourie introduced them in a dynamical language, requiring that a section of an associate flat bundle should provide a hyperbolic set for some flow. Later, several equivalent characterisations of Anosov representations that do not involve the dynamics of a flow were identified, often with the goal of producing geometric structures associated to these representations in the form of compact quotients of open subsets of flag manifolds.
This course will present recent work with B. Delarue and A. Sanders that uses (non-compact) quotients of open subsets of appropriate homogeneous spaces (not flag manifolds) for the Lie group G, equipped with a flow that commutes with the action of G. This quotient produces a locally homogeneous flow with uniformly hyperbolic dynamics (Smale's axiom A). This approach allows for the use of modern analytic techniques of smooth dynamics that were not applicable to Anosov representations so far.
The first lectures will focus on the case of projective Anosov representations into SL(d,R). After introducing Anosov representations in this setting, we will describe the construction of the locally homogeneous flow and its dynamical properties. In a second part of the course, we will study the case of a general semi-simple Lie group G (and arbitrary flag manifolds used to define Anosov representations). We will see how the linear algebra of the SL(d,R) case can be replaced with differential geometric notions in the general situation. Several examples will be discussed, with an emphasis on those that can be described as some non-Riemannian geodesic flow.
The course will use notions from Lie theory, dynamical systems, differential geometry and a touch of geometric group theory. The necessary background on these subjects will be kept to a minimum.

17:00 Exercises

mer. 16 juillet

09:00 Welcome coffee

25 Geometry of Anosov flows and Rigidity (2/4)

The geodesic flow on a manifold of negative sectional curvature is an archetypal example of an Anosov flow, a dynamical system under which every vector gets uniformly expanded or uniformly contracted. We will begin with an introduction to the geometry of these dynamical systems, including invariant manifolds, ergodicity, and various regularity questions.
We will then introduce flows that are compact group extensions of Anosov flows, discuss the associated "Brin group", a sort of Galois group of the extension. We will then apply these techniques to explain a result jointly obtained with David Fisher and Ben Lowe, saying that if a compact negatively curved real-analytic Riemannian manifold has infinitely many totally geodesic hypersurfaces, then it must be of constant sectional curvature.

Orateur: Simion Filip (University of Chicago)

Vidéo

The geodesic flow on a manifold of negative sectional curvature is an archetypal example of an Anosov flow, a dynamical system under which every vector gets uniformly expanded or uniformly contracted. We will begin with an introduction to the geometry of these dynamical systems, including invariant manifolds, ergodicity, and various regularity questions.

We will then introduce flows that are compact group extensions of Anosov flows, discuss the associated "Brin group", a sort of Galois group of the extension. We will then apply these techniques to explain a result jointly obtained with David Fisher and Ben Lowe, saying that if a compact negatively curved real-analytic Riemannian manifold has infinitely many totally geodesic hypersurfaces, then it must be of constant sectional curvature.

10:15 Coffee break

26 Harmonic maps in high-dimensional spheres, representations and random matrices (2/4)

This course will be about harmonic maps from 2d surfaces to spheres of high dimensions, coming from unitary representations of surface groups. This topic falls under the common theme in geometric analysis of studying geometric objects from topological data and vice versa. We will discuss rigidity phenomena for the shape of such harmonic maps into spheres, focusing on both the high-dimensional asymptotic regime, where random matrix theory plays a role, and the infinite dimensional case, where representation theory of PSL2(R) is central.

Orateur: Antoine Song (California Institute of Technology)

Vidéo

This course will be about harmonic maps from 2d surfaces to spheres of high dimensions, coming from unitary representations of surface groups. This topic falls under the common theme in geometric analysis of studying geometric objects from topological data and vice versa. We will discuss rigidity phenomena for the shape of such harmonic maps into spheres, focusing on both the high-dimensional asymptotic regime, where random matrix theory plays a role, and the infinite dimensional case, where representation theory of PSL2(R) is central.

11:30 Exercises

12:30 Lunch-break

27 Locally homogeneous flows and Anosov representations (3/5)

Anosov representations form an open set of homomorphisms of a discrete hyperbolic group into a semi-simple Lie group G. Labourie introduced them in a dynamical language, requiring that a section of an associate flat bundle should provide a hyperbolic set for some flow. Later, several equivalent characterisations of Anosov representations that do not involve the dynamics of a flow were identified, often with the goal of producing geometric structures associated to these representations in the form of compact quotients of open subsets of flag manifolds.
This course will present recent work with B. Delarue and A. Sanders that uses (non-compact) quotients of open subsets of appropriate homogeneous spaces (not flag manifolds) for the Lie group G, equipped with a flow that commutes with the action of G. This quotient produces a locally homogeneous flow with uniformly hyperbolic dynamics (Smale's axiom A). This approach allows for the use of modern analytic techniques of smooth dynamics that were not applicable to Anosov representations so far.
The first lectures will focus on the case of projective Anosov representations into SL(d,R). After introducing Anosov representations in this setting, we will describe the construction of the locally homogeneous flow and its dynamical properties. In a second part of the course, we will study the case of a general semi-simple Lie group G (and arbitrary flag manifolds used to define Anosov representations). We will see how the linear algebra of the SL(d,R) case can be replaced with differential geometric notions in the general situation. Several examples will be discussed, with an emphasis on those that can be described as some non-Riemannian geodesic flow.
The course will use notions from Lie theory, dynamical systems, differential geometry and a touch of geometric group theory. The necessary background on these subjects will be kept to a minimum.

Orateur: Daniel Monclair (Université Paris-Saclay)

Vidéo

Anosov representations form an open set of homomorphisms of a discrete hyperbolic group into a semi-simple Lie group G. Labourie introduced them in a dynamical language, requiring that a section of an associate flat bundle should provide a hyperbolic set for some flow. Later, several equivalent characterisations of Anosov representations that do not involve the dynamics of a flow were identified, often with the goal of producing geometric structures associated to these representations in the form of compact quotients of open subsets of flag manifolds.
This course will present recent work with B. Delarue and A. Sanders that uses (non-compact) quotients of open subsets of appropriate homogeneous spaces (not flag manifolds) for the Lie group G, equipped with a flow that commutes with the action of G. This quotient produces a locally homogeneous flow with uniformly hyperbolic dynamics (Smale's axiom A). This approach allows for the use of modern analytic techniques of smooth dynamics that were not applicable to Anosov representations so far.
The first lectures will focus on the case of projective Anosov representations into SL(d,R). After introducing Anosov representations in this setting, we will describe the construction of the locally homogeneous flow and its dynamical properties. In a second part of the course, we will study the case of a general semi-simple Lie group G (and arbitrary flag manifolds used to define Anosov representations). We will see how the linear algebra of the SL(d,R) case can be replaced with differential geometric notions in the general situation. Several examples will be discussed, with an emphasis on those that can be described as some non-Riemannian geodesic flow.
The course will use notions from Lie theory, dynamical systems, differential geometry and a touch of geometric group theory. The necessary background on these subjects will be kept to a minimum.

15:30 Coffee break

28 Poisson–Voronoi tessellations and fixed price in higher rank (3/5)

We will start by defining and motivating the Poisson point process, which is, informally, a "maximally random" scattering of points in space, and discussing the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group (for higher rank semisimple Lie groups, the minimum number of generators in a lattice is sublinear in the covolume). In this minicourse we will unpack the proof. No prior knowledge on Poisson--Voronoi tessellations, fixed price or higher rank will be assumed.

Orateurs: Amanda Wilkens (Carnegie Mellon University), Sam Mellick (Jagiellonian University)

Vidéo

We will start by defining and motivating the Poisson point process, which is, informally, a "maximally random" scattering of points in space, and discussing the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group (for higher rank semisimple Lie groups, the minimum number of generators in a lattice is sublinear in the covolume). In this minicourse we will unpack the proof. No prior knowledge on Poisson--Voronoi tessellations, fixed price or higher rank will be assumed.

17:00 Exercises

jeu. 17 juillet

09:00 Welcome coffee

29 Harmonic maps in high-dimensional spheres, representations and random matrices (3/4)

This course will be about harmonic maps from 2d surfaces to spheres of high dimensions, coming from unitary representations of surface groups. This topic falls under the common theme in geometric analysis of studying geometric objects from topological data and vice versa. We will discuss rigidity phenomena for the shape of such harmonic maps into spheres, focusing on both the high-dimensional asymptotic regime, where random matrix theory plays a role, and the infinite dimensional case, where representation theory of PSL2(R) is central.

Orateur: Antoine Song (California Institute of Technology)

Vidéo

This course will be about harmonic maps from 2d surfaces to spheres of high dimensions, coming from unitary representations of surface groups. This topic falls under the common theme in geometric analysis of studying geometric objects from topological data and vice versa. We will discuss rigidity phenomena for the shape of such harmonic maps into spheres, focusing on both the high-dimensional asymptotic regime, where random matrix theory plays a role, and the infinite dimensional case, where representation theory of PSL2(R) is central.

10:15 Coffee break

30 Locally homogeneous flows and Anosov representations (4/5)

Anosov representations form an open set of homomorphisms of a discrete hyperbolic group into a semi-simple Lie group G. Labourie introduced them in a dynamical language, requiring that a section of an associate flat bundle should provide a hyperbolic set for some flow. Later, several equivalent characterisations of Anosov representations that do not involve the dynamics of a flow were identified, often with the goal of producing geometric structures associated to these representations in the form of compact quotients of open subsets of flag manifolds.
This course will present recent work with B. Delarue and A. Sanders that uses (non-compact) quotients of open subsets of appropriate homogeneous spaces (not flag manifolds) for the Lie group G, equipped with a flow that commutes with the action of G. This quotient produces a locally homogeneous flow with uniformly hyperbolic dynamics (Smale's axiom A). This approach allows for the use of modern analytic techniques of smooth dynamics that were not applicable to Anosov representations so far.
The first lectures will focus on the case of projective Anosov representations into SL(d,R). After introducing Anosov representations in this setting, we will describe the construction of the locally homogeneous flow and its dynamical properties. In a second part of the course, we will study the case of a general semi-simple Lie group G (and arbitrary flag manifolds used to define Anosov representations). We will see how the linear algebra of the SL(d,R) case can be replaced with differential geometric notions in the general situation. Several examples will be discussed, with an emphasis on those that can be described as some non-Riemannian geodesic flow.
The course will use notions from Lie theory, dynamical systems, differential geometry and a touch of geometric group theory. The necessary background on these subjects will be kept to a minimum.

Orateur: Daniel Monclair (Université Paris-Saclay)

Vidéo

Anosov representations form an open set of homomorphisms of a discrete hyperbolic group into a semi-simple Lie group G. Labourie introduced them in a dynamical language, requiring that a section of an associate flat bundle should provide a hyperbolic set for some flow. Later, several equivalent characterisations of Anosov representations that do not involve the dynamics of a flow were identified, often with the goal of producing geometric structures associated to these representations in the form of compact quotients of open subsets of flag manifolds.
This course will present recent work with B. Delarue and A. Sanders that uses (non-compact) quotients of open subsets of appropriate homogeneous spaces (not flag manifolds) for the Lie group G, equipped with a flow that commutes with the action of G. This quotient produces a locally homogeneous flow with uniformly hyperbolic dynamics (Smale's axiom A). This approach allows for the use of modern analytic techniques of smooth dynamics that were not applicable to Anosov representations so far.
The first lectures will focus on the case of projective Anosov representations into SL(d,R). After introducing Anosov representations in this setting, we will describe the construction of the locally homogeneous flow and its dynamical properties. In a second part of the course, we will study the case of a general semi-simple Lie group G (and arbitrary flag manifolds used to define Anosov representations). We will see how the linear algebra of the SL(d,R) case can be replaced with differential geometric notions in the general situation. Several examples will be discussed, with an emphasis on those that can be described as some non-Riemannian geodesic flow.
The course will use notions from Lie theory, dynamical systems, differential geometry and a touch of geometric group theory. The necessary background on these subjects will be kept to a minimum.

11:30 Exercises

12:30 Buffet-lunch at IHES

31 Poisson–Voronoi tessellations and fixed price in higher rank (4/5)

We will start by defining and motivating the Poisson point process, which is, informally, a "maximally random" scattering of points in space, and discussing the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group (for higher rank semisimple Lie groups, the minimum number of generators in a lattice is sublinear in the covolume). In this minicourse we will unpack the proof. No prior knowledge on Poisson--Voronoi tessellations, fixed price or higher rank will be assumed.

Orateurs: Amanda Wilkens (Carnegie Mellon University), Sam Mellick (Jagiellonian University)

Vidéo

We will start by defining and motivating the Poisson point process, which is, informally, a "maximally random" scattering of points in space, and discussing the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group (for higher rank semisimple Lie groups, the minimum number of generators in a lattice is sublinear in the covolume). In this minicourse we will unpack the proof. No prior knowledge on Poisson--Voronoi tessellations, fixed price or higher rank will be assumed.

15:30 Coffee break

32 Geometry of Anosov flows and Rigidity (3/4)

The geodesic flow on a manifold of negative sectional curvature is an archetypal example of an Anosov flow, a dynamical system under which every vector gets uniformly expanded or uniformly contracted. We will begin with an introduction to the geometry of these dynamical systems, including invariant manifolds, ergodicity, and various regularity questions.
We will then introduce flows that are compact group extensions of Anosov flows, discuss the associated "Brin group", a sort of Galois group of the extension. We will then apply these techniques to explain a result jointly obtained with David Fisher and Ben Lowe, saying that if a compact negatively curved real-analytic Riemannian manifold has infinitely many totally geodesic hypersurfaces, then it must be of constant sectional curvature.

Orateur: Simion Filip (University of Chicago)

Vidéo

The geodesic flow on a manifold of negative sectional curvature is an archetypal example of an Anosov flow, a dynamical system under which every vector gets uniformly expanded or uniformly contracted. We will begin with an introduction to the geometry of these dynamical systems, including invariant manifolds, ergodicity, and various regularity questions.

We will then introduce flows that are compact group extensions of Anosov flows, discuss the associated "Brin group", a sort of Galois group of the extension. We will then apply these techniques to explain a result jointly obtained with David Fisher and Ben Lowe, saying that if a compact negatively curved real-analytic Riemannian manifold has infinitely many totally geodesic hypersurfaces, then it must be of constant sectional curvature.

17:00 Exercises

ven. 18 juillet

09:00 Welcome coffee

33 Poisson–Voronoi tessellations and fixed price in higher rank (5/5)

We will start by defining and motivating the Poisson point process, which is, informally, a "maximally random" scattering of points in space, and discussing the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group (for higher rank semisimple Lie groups, the minimum number of generators in a lattice is sublinear in the covolume). In this minicourse we will unpack the proof. No prior knowledge on Poisson--Voronoi tessellations, fixed price or higher rank will be assumed.

Orateurs: Amanda Wilkens (Carnegie Mellon University), Sam Mellick (Jagiellonian University)

Vidéo

We will start by defining and motivating the Poisson point process, which is, informally, a "maximally random" scattering of points in space, and discussing the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group (for higher rank semisimple Lie groups, the minimum number of generators in a lattice is sublinear in the covolume). In this minicourse we will unpack the proof. No prior knowledge on Poisson--Voronoi tessellations, fixed price or higher rank will be assumed.

10:15 Coffee break

34 Locally homogeneous flows and Anosov representations (5/5)

Anosov representations form an open set of homomorphisms of a discrete hyperbolic group into a semi-simple Lie group G. Labourie introduced them in a dynamical language, requiring that a section of an associate flat bundle should provide a hyperbolic set for some flow. Later, several equivalent characterisations of Anosov representations that do not involve the dynamics of a flow were identified, often with the goal of producing geometric structures associated to these representations in the form of compact quotients of open subsets of flag manifolds.
This course will present recent work with B. Delarue and A. Sanders that uses (non-compact) quotients of open subsets of appropriate homogeneous spaces (not flag manifolds) for the Lie group G, equipped with a flow that commutes with the action of G. This quotient produces a locally homogeneous flow with uniformly hyperbolic dynamics (Smale's axiom A). This approach allows for the use of modern analytic techniques of smooth dynamics that were not applicable to Anosov representations so far.
The first lectures will focus on the case of projective Anosov representations into SL(d,R). After introducing Anosov representations in this setting, we will describe the construction of the locally homogeneous flow and its dynamical properties. In a second part of the course, we will study the case of a general semi-simple Lie group G (and arbitrary flag manifolds used to define Anosov representations). We will see how the linear algebra of the SL(d,R) case can be replaced with differential geometric notions in the general situation. Several examples will be discussed, with an emphasis on those that can be described as some non-Riemannian geodesic flow.
The course will use notions from Lie theory, dynamical systems, differential geometry and a touch of geometric group theory. The necessary background on these subjects will be kept to a minimum.

Orateur: Daniel Monclair (Université Paris-Saclay)

Vidéo

Anosov representations form an open set of homomorphisms of a discrete hyperbolic group into a semi-simple Lie group G. Labourie introduced them in a dynamical language, requiring that a section of an associate flat bundle should provide a hyperbolic set for some flow. Later, several equivalent characterisations of Anosov representations that do not involve the dynamics of a flow were identified, often with the goal of producing geometric structures associated to these representations in the form of compact quotients of open subsets of flag manifolds.
This course will present recent work with B. Delarue and A. Sanders that uses (non-compact) quotients of open subsets of appropriate homogeneous spaces (not flag manifolds) for the Lie group G, equipped with a flow that commutes with the action of G. This quotient produces a locally homogeneous flow with uniformly hyperbolic dynamics (Smale's axiom A). This approach allows for the use of modern analytic techniques of smooth dynamics that were not applicable to Anosov representations so far.
The first lectures will focus on the case of projective Anosov representations into SL(d,R). After introducing Anosov representations in this setting, we will describe the construction of the locally homogeneous flow and its dynamical properties. In a second part of the course, we will study the case of a general semi-simple Lie group G (and arbitrary flag manifolds used to define Anosov representations). We will see how the linear algebra of the SL(d,R) case can be replaced with differential geometric notions in the general situation. Several examples will be discussed, with an emphasis on those that can be described as some non-Riemannian geodesic flow.
The course will use notions from Lie theory, dynamical systems, differential geometry and a touch of geometric group theory. The necessary background on these subjects will be kept to a minimum.

11:30 Exercises

12:30 Lunch-break

35 Geometry of Anosov flows and Rigidity (4/4)

The geodesic flow on a manifold of negative sectional curvature is an archetypal example of an Anosov flow, a dynamical system under which every vector gets uniformly expanded or uniformly contracted. We will begin with an introduction to the geometry of these dynamical systems, including invariant manifolds, ergodicity, and various regularity questions.
We will then introduce flows that are compact group extensions of Anosov flows, discuss the associated "Brin group", a sort of Galois group of the extension. We will then apply these techniques to explain a result jointly obtained with David Fisher and Ben Lowe, saying that if a compact negatively curved real-analytic Riemannian manifold has infinitely many totally geodesic hypersurfaces, then it must be of constant sectional curvature.

Orateur: Simion Filip (University of Chicago)

Vidéo

The geodesic flow on a manifold of negative sectional curvature is an archetypal example of an Anosov flow, a dynamical system under which every vector gets uniformly expanded or uniformly contracted. We will begin with an introduction to the geometry of these dynamical systems, including invariant manifolds, ergodicity, and various regularity questions.

We will then introduce flows that are compact group extensions of Anosov flows, discuss the associated "Brin group", a sort of Galois group of the extension. We will then apply these techniques to explain a result jointly obtained with David Fisher and Ben Lowe, saying that if a compact negatively curved real-analytic Riemannian manifold has infinitely many totally geodesic hypersurfaces, then it must be of constant sectional curvature.

15:30 Coffee break

36 Harmonic maps in high-dimensional spheres, representations and random matrices (4/4)

This course will be about harmonic maps from 2d surfaces to spheres of high dimensions, coming from unitary representations of surface groups. This topic falls under the common theme in geometric analysis of studying geometric objects from topological data and vice versa. We will discuss rigidity phenomena for the shape of such harmonic maps into spheres, focusing on both the high-dimensional asymptotic regime, where random matrix theory plays a role, and the infinite dimensional case, where representation theory of PSL2(R) is central.

Orateur: Antoine Song (California Institute of Technology)

Vidéo

This course will be about harmonic maps from 2d surfaces to spheres of high dimensions, coming from unitary representations of surface groups. This topic falls under the common theme in geometric analysis of studying geometric objects from topological data and vice versa. We will discuss rigidity phenomena for the shape of such harmonic maps into spheres, focusing on both the high-dimensional asymptotic regime, where random matrix theory plays a role, and the infinite dimensional case, where representation theory of PSL2(R) is central.

17:00 Exercises