Higher rank geometric structures, IHP-Paris

14 avr. 2025, 08:30 → 11 juil. 2025, 18:30 Europe/Paris

Amphitheater Darboux (IHP)

Higher Rank Geometric Structures

April 14th -- July 11th, 2025 at the Institut Henri Poincaré, Paris

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Demand for financial support is now closed

Register here for the program.

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Introductory School April 14 -- 18 at CIRM Marseille

With Mini-Courses by

Michel Boileau (Aix-Marseille Université)
Federica Fanoni (CNRS, Université Paris-Est Créteil)
Fanny Kassel (CNRS, Institut des Hautes Etudes Scientifiques)
Qiongling Li (Chern Mathematics Institute, Nankai University)
Barbara Schapira (Université de Montpellier)

Register for the school at Geometric Structures and discrete group actions 

 

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Workshops:

- Higher rank geometric structures, Higgs bundles and physics May 19 -- 27 at IHP Paris
- Low-dimensional phenomena: geometry and dynamics June 23 -- 27 at IHP Paris

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Week Schedule:

Week of:

• April 21st Discrete subgroupos of Lie groups via Coxeter groups 6h mini-course by Sami Douba

• April 28th Around proper actions on homogeneus spaces 6h mini-course by Yosuke Morita
  

• May 5th Geodesic currents and counting curves 6h mini-course by Viveka Erlandsson and Didac Martinez Granado

• May 12th Hyperbolic 3-manifolds 6h mini-course by Richard Canary

• June 2nd Higgs bundles, Slodowy slices and Anosov representations 6h mini-course by Brian Collier

• June 9th Domains of discontinuity for Anosov representations 6h mini-course by Sara Maloni

• June 16th Complex hyperbolic lattices 6h mini-course by Matthew Stover

• June 30th Hausdorff dimension of limit sets 6h mini-course by Pablo Lessa

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Organizing committee:

Kenneth Bromberg (University of Utah)    
Maria Beatrice Pozzetti (Heidelberg Universität)    
Andrès Sambarino (CNRS - Sorbonne Université)    
Nicolas Tholozan (CNRS - Ecole Normale Supérieure)

Scientific committee:

Richard Canary (Michigan)    
Alessandra Iozzi (Zurich)    
François Labourie (Nice)    
Anna Wienhard (Heidelberg)

 

 

Highrags2025-Official Poster.pdf

mercredi 23 avril ¶

Discrete subgroups of Lie groups via Coxeter groups: Discrete subgroups of Lie groups via Coxeter groups I¶

Due to the profound interplay between their algebraic, geometric, and combinatorial properties, Coxeter groups, and their realizations as linear reflection groups, have long been fundamental for those interested in groups and their representations. Apart from their utility in the construction of lattices in isometry groups of real hyperbolic spaces, they also provide a rich source of infinite-covolume discrete subgroups of semisimple Lie groups (where their deformations can often simultaneously be managed yet exhibit interesting/illucidative phenomena), and indeed of infinite-index subgroups of lattices themselves. This mini-course will be an attempt to illustrate these points, with an emphasis on examples and applications.

Président de session: Sami Douba (IHES)

jeudi 24 avril ¶

Discrete subgroups of Lie groups via Coxeter groups: Discrete subgroups of Lie groups via Coxeter groups II¶

Due to the profound interplay between their algebraic, geometric, and combinatorial properties, Coxeter groups, and their realizations as linear reflection groups, have long been fundamental for those interested in groups and their representations. Apart from their utility in the construction of lattices in isometry groups of real hyperbolic spaces, they also provide a rich source of infinite-covolume discrete subgroups of semisimple Lie groups (where their deformations can often simultaneously be managed yet exhibit interesting/illucidative phenomena), and indeed of infinite-index subgroups of lattices themselves. This mini-course will be an attempt to illustrate these points, with an emphasis on examples and applications.

Président de session: Sami Douba (IHES)

vendredi 25 avril ¶

Discrete subgroups of Lie groups via Coxeter groups: Discrete subgroups of Lie groups via Coxeter groups III¶

Due to the profound interplay between their algebraic, geometric, and combinatorial properties, Coxeter groups, and their realizations as linear reflection groups, have long been fundamental for those interested in groups and their representations. Apart from their utility in the construction of lattices in isometry groups of real hyperbolic spaces, they also provide a rich source of infinite-covolume discrete subgroups of semisimple Lie groups (where their deformations can often simultaneously be managed yet exhibit interesting/illucidative phenomena), and indeed of infinite-index subgroups of lattices themselves. This mini-course will be an attempt to illustrate these points, with an emphasis on examples and applications.

Président de session: Sami Douba (IHES)

lundi 28 avril ¶

Around proper actions on homogeneous spaces: Around proper actions on homogeneous spaces I¶

Let G/H be a reductive homogeneous space and Γ be a (torsion-free) discrete subgroup of G. The quotient of G/H by the left Γ-action is always a manifold in the Riemannian setting (i.e. when H is compact), but not always otherwise: the Γ-action on G/H needs to be proper. In the last 40 years or so, the study of proper actions on non-Riemannian homogeneous spaces has been one of the fertile areas in mathematics to which various methods (Lie-theoretic / dynamical / homotopical / ...) were applied and produced partially overlapping but different results. In this mini-course, we plan to discuss the following topics in this area:

• Basics on reductive Lie groups
• Properness criterion in terms of the Cartan projection (Kobayashi, Benoist) and its consequences
• Proper SL(2,R)-actions vs. proper surface group actions (joint work with Maciej Bocheński)
• Necessary conditions for the existence of proper cocompact actions (joint work with Fanny Kassel and Nicolas Tholozan)

If time permits, we will also discuss:

• A conjectural picture of proper cocompact actions (Tholozan's geometric fibration conjecture)
• Relation to Anosov representation theory (Guéritaud-Guichard-Kassel-Wienhard, Kassel-Tholozan, ...)

Présidents de session: Maciej Bocheński (University of Warmia and Mazury in Olsztyn), Yosuke Morita (Kyushu University)

mardi 29 avril ¶

Around proper actions on homogeneous spaces: Around proper actions on homogeneous spaces II¶

Let G/H be a reductive homogeneous space and Γ be a (torsion-free) discrete subgroup of G. The quotient of G/H by the left Γ-action is always a manifold in the Riemannian setting (i.e. when H is compact), but not always otherwise: the Γ-action on G/H needs to be proper. In the last 40 years or so, the study of proper actions on non-Riemannian homogeneous spaces has been one of the fertile areas in mathematics to which various methods (Lie-theoretic / dynamical / homotopical / ...) were applied and produced partially overlapping but different results. In this mini-course, we plan to discuss the following topics in this area:

• Basics on reductive Lie groups
• Properness criterion in terms of the Cartan projection (Kobayashi, Benoist) and its consequences
• Proper SL(2,R)-actions vs. proper surface group actions (joint work with Maciej Bocheński)
• Necessary conditions for the existence of proper cocompact actions (joint work with Fanny Kassel and Nicolas Tholozan)

If time permits, we will also discuss:

• A conjectural picture of proper cocompact actions (Tholozan's geometric fibration conjecture)
• Relation to Anosov representation theory (Guéritaud-Guichard-Kassel-Wienhard, Kassel-Tholozan, ...)

mercredi 30 avril ¶

Around proper actions on homogeneous spaces: Around proper actions on homogeneous spaces III¶

Let G/H be a reductive homogeneous space and Γ be a (torsion-free) discrete subgroup of G. The quotient of G/H by the left Γ-action is always a manifold in the Riemannian setting (i.e. when H is compact), but not always otherwise: the Γ-action on G/H needs to be proper. In the last 40 years or so, the study of proper actions on non-Riemannian homogeneous spaces has been one of the fertile areas in mathematics to which various methods (Lie-theoretic / dynamical / homotopical / ...) were applied and produced partially overlapping but different results. In this mini-course, we plan to discuss the following topics in this area:

• Basics on reductive Lie groups
• Properness criterion in terms of the Cartan projection (Kobayashi, Benoist) and its consequences
• Proper SL(2,R)-actions vs. proper surface group actions (joint work with Maciej Bocheński)
• Necessary conditions for the existence of proper cocompact actions (joint work with Fanny Kassel and Nicolas Tholozan)

If time permits, we will also discuss:

• A conjectural picture of proper cocompact actions (Tholozan's geometric fibration conjecture)
• Relation to Anosov representation theory (Guéritaud-Guichard-Kassel-Wienhard, Kassel-Tholozan, ...)

lundi 5 mai ¶

Geodesic currents and counting curves: Geodesic currents and counting curves I¶

In this course we will give an introduction to geodesic currents and length functions on them with a view towards Mirzakhani’s curve counting theorems. In her thesis Mirzakhani obtained the asymptotic growth of the number of primitive simple closed geodesics of length bounded by L on a hyperbolic surface and showed that it is a constant times L^{6g-6}, where g is the genus of the surface. In fact, she obtained this growth for each topological type of simple geodesics (i.e. those in a fixed mapping class group orbit of a simple geodesic) and later extended it, through very different methods, to also hold for orbits of non-simple geodesics. We will discuss how rephrasing the counting question to one about measures on the space of currents gives a unified proof of the two theorems which is more combinatorial in nature and has the key feature that it gives flexibility in the choice of length used to measure the geodesics/curves.

Présidents de session: Didac Martinez Granado (University of Luxembourg / NUS Singapore), Viveka Erlandsson (University of Bristol)

mardi 6 mai ¶

Geodesic currents and counting curves: Geodesic currents and counting curves II¶

In this course we will give an introduction to geodesic currents and length functions on them with a view towards Mirzakhani’s curve counting theorems. In her thesis Mirzakhani obtained the asymptotic growth of the number of primitive simple closed geodesics of length bounded by L on a hyperbolic surface and showed that it is a constant times L^{6g-6}, where g is the genus of the surface. In fact, she obtained this growth for each topological type of simple geodesics (i.e. those in a fixed mapping class group orbit of a simple geodesic) and later extended it, through very different methods, to also hold for orbits of non-simple geodesics. We will discuss how rephrasing the counting question to one about measures on the space of currents gives a unified proof of the two theorems which is more combinatorial in nature and has the key feature that it gives flexibility in the choice of length used to measure the geodesics/curves.

mercredi 7 mai ¶

Geodesic currents and counting curves: Geodesic currents and counting curves III¶

In this course we will give an introduction to geodesic currents and length functions on them with a view towards Mirzakhani’s curve counting theorems. In her thesis Mirzakhani obtained the asymptotic growth of the number of primitive simple closed geodesics of length bounded by L on a hyperbolic surface and showed that it is a constant times L^{6g-6}, where g is the genus of the surface. In fact, she obtained this growth for each topological type of simple geodesics (i.e. those in a fixed mapping class group orbit of a simple geodesic) and later extended it, through very different methods, to also hold for orbits of non-simple geodesics. We will discuss how rephrasing the counting question to one about measures on the space of currents gives a unified proof of the two theorems which is more combinatorial in nature and has the key feature that it gives flexibility in the choice of length used to measure the geodesics/curves.

lundi 12 mai ¶

1 Research seminar at IHES¶

mardi 13 mai ¶

Hyperbolic 3-manifolds: Hyperbolic 3-manifolds I¶

We will begin by introducing crucial concepts and definitions in the study of hyperbolic
3-manifolds with finitely generated fundamental groups. One fundamental object of study is the convex core of a hyperbolic 3-manifold (which is the smallest closed, non-empty convex submanifold). When the convex core is compact the manifold is said to be convex cocompact and these are the most easily studied manifolds. When the convex core has finite volume, we say the manifold is geometrically finite.

We will discuss the quasiconformal deformation theory of convex cocompact hyperbolic 3-manifolds which gives rise to an open subset of the character variety of (conjugacy classes) of representations of the fundamental group into PSL(2,C). This open neighborhood is parametrized by conformal data at infinity.

We will then introduce the structure theory of geometrically infinite ends of hyperbolic surface. Here, the theory of simplicial hyperbolic surfaces and/or pleated surfaces will play a crucial role. If time permits we will discuss Thurston's Covering Theorem.

In the last portion of the lecture series, we will introduce and motivate the major accomplishments of the last two decades in Thurston's program to understand infinite volume hyperbolic 3-manifolds. Here, the major results include the proofs of
Marden's Tameness Conjecture, Ahlfors' Measure conjecture, Thurston's Ending Lamination Conjecture and the Bers-Sullivan-Thurston Density conjecture.

Président de session: Richard Canary (University of Michigan)

jeudi 15 mai ¶

Hyperbolic 3-manifolds: Hyperbolic 3-manifolds II¶

We will begin by introducing crucial concepts and definitions in the study of hyperbolic
3-manifolds with finitely generated fundamental groups. One fundamental object of study is the convex core of a hyperbolic 3-manifold (which is the smallest closed, non-empty convex submanifold). When the convex core is compact the manifold is said to be convex cocompact and these are the most easily studied manifolds. When the convex core has finite volume, we say the manifold is geometrically finite.

We will discuss the quasiconformal deformation theory of convex cocompact hyperbolic 3-manifolds which gives rise to an open subset of the character variety of (conjugacy classes) of representations of the fundamental group into PSL(2,C). This open neighborhood is parametrized by conformal data at infinity.

We will then introduce the structure theory of geometrically infinite ends of hyperbolic surface. Here, the theory of simplicial hyperbolic surfaces and/or pleated surfaces will play a crucial role. If time permits we will discuss Thurston's Covering Theorem.

In the last portion of the lecture series, we will introduce and motivate the major accomplishments of the last two decades in Thurston's program to understand infinite volume hyperbolic 3-manifolds. Here, the major results include the proofs of
Marden's Tameness Conjecture, Ahlfors' Measure conjecture, Thurston's Ending Lamination Conjecture and the Bers-Sullivan-Thurston Density conjecture.

Président de session: Richard Canary (University of Michigan)

vendredi 16 mai ¶

Hyperbolic 3-manifolds: Hyperbolic 3-manifolds III¶

We will begin by introducing crucial concepts and definitions in the study of hyperbolic
3-manifolds with finitely generated fundamental groups. One fundamental object of study is the convex core of a hyperbolic 3-manifold (which is the smallest closed, non-empty convex submanifold). When the convex core is compact the manifold is said to be convex cocompact and these are the most easily studied manifolds. When the convex core has finite volume, we say the manifold is geometrically finite.

We will discuss the quasiconformal deformation theory of convex cocompact hyperbolic 3-manifolds which gives rise to an open subset of the character variety of (conjugacy classes) of representations of the fundamental group into PSL(2,C). This open neighborhood is parametrized by conformal data at infinity.

We will then introduce the structure theory of geometrically infinite ends of hyperbolic surface. Here, the theory of simplicial hyperbolic surfaces and/or pleated surfaces will play a crucial role. If time permits we will discuss Thurston's Covering Theorem.

In the last portion of the lecture series, we will introduce and motivate the major accomplishments of the last two decades in Thurston's program to understand infinite volume hyperbolic 3-manifolds. Here, the major results include the proofs of
Marden's Tameness Conjecture, Ahlfors' Measure conjecture, Thurston's Ending Lamination Conjecture and the Bers-Sullivan-Thurston Density conjecture.

Président de session: Richard Canary (University of Michigan)

mardi 3 juin ¶

Higgs bundles, Slodowy slices and Anosov representations: Higgs bundles, Slodowy slices and Anosov representations I¶

This minicourse will focus on Slodowy slices in the moduli space of G-Higgs bundles on a closed Riemann surface, where G is a complex semisimple Lie group. Under the nonabelian Hodge correspondence, these spaces describe specific ways of deforming Fuchsian representations of a closed surface group in the G-character variety via a group homomorphism from SL(2,C) into G. For the principal SL(2,C) in G, the Slodowy slice is the well known and extensively studied Hitchin section, and the associated representations (known as Hitchin representations) define ``higher rank Teichmuller components" of the character variety for the split real form of G. That is, components consisting entirely of Anosov representations. For other special groups homomorphisms of SL(2,C) into G, known as magical sl2s, the Slodowy slice parameterizes the higher rank Teichmuller spaces associated to positive representations, in the sense of Guichard-Labourie-Wienhard. Conjecturally, this describes all higher rank Teichmuller spaces for closed surfaces. I will spend the first 2 talks setting up these objects, discussing their relations to the component count of the character variety of representations of a closed surface group into real semisimple Lie groups.
In general, the Slodowy slice is closed in the moduli space but not open. In the final talk, I will discuss a conjectural picture where the Slodowy slice is a generalization of the Fuchsian locus in Quasifuchsian space. One aim of this conjectural picture is to prove that, under the nonabelian Hodge correspondence, the representations associated to Higgs bundles in general Slodowy slices are all Anosov. In the end, I will focus on the aspects of this picture which have been proven when one restricts to real forms of G in low rank, specifically recent work by S. Filip, J. Zhang, S. Bronstien and C. Davalo, and also myself.

Président de session: Brian Collier (University of California Riverside)

jeudi 5 juin ¶

Higgs bundles, Slodowy slices and Anosov representations: Higgs bundles, Slodowy slices and Anosov representations II¶

This minicourse will focus on Slodowy slices in the moduli space of G-Higgs bundles on a closed Riemann surface, where G is a complex semisimple Lie group. Under the nonabelian Hodge correspondence, these spaces describe specific ways of deforming Fuchsian representations of a closed surface group in the G-character variety via a group homomorphism from SL(2,C) into G. For the principal SL(2,C) in G, the Slodowy slice is the well known and extensively studied Hitchin section, and the associated representations (known as Hitchin representations) define ``higher rank Teichmuller components" of the character variety for the split real form of G. That is, components consisting entirely of Anosov representations. For other special groups homomorphisms of SL(2,C) into G, known as magical sl2s, the Slodowy slice parameterizes the higher rank Teichmuller spaces associated to positive representations, in the sense of Guichard-Labourie-Wienhard. Conjecturally, this describes all higher rank Teichmuller spaces for closed surfaces. I will spend the first 2 talks setting up these objects, discussing their relations to the component count of the character variety of representations of a closed surface group into real semisimple Lie groups.
In general, the Slodowy slice is closed in the moduli space but not open. In the final talk, I will discuss a conjectural picture where the Slodowy slice is a generalization of the Fuchsian locus in Quasifuchsian space. One aim of this conjectural picture is to prove that, under the nonabelian Hodge correspondence, the representations associated to Higgs bundles in general Slodowy slices are all Anosov. In the end, I will focus on the aspects of this picture which have been proven when one restricts to real forms of G in low rank, specifically recent work by S. Filip, J. Zhang, S. Bronstien and C. Davalo, and also myself.

Président de session: Brian Collier (University of California Riverside)

vendredi 6 juin ¶

Higgs bundles, Slodowy slices and Anosov representations: Higgs bundles, Slodowy slices and Anosov representations III¶

This minicourse will focus on Slodowy slices in the moduli space of G-Higgs bundles on a closed Riemann surface, where G is a complex semisimple Lie group. Under the nonabelian Hodge correspondence, these spaces describe specific ways of deforming Fuchsian representations of a closed surface group in the G-character variety via a group homomorphism from SL(2,C) into G. For the principal SL(2,C) in G, the Slodowy slice is the well known and extensively studied Hitchin section, and the associated representations (known as Hitchin representations) define ``higher rank Teichmuller components" of the character variety for the split real form of G. That is, components consisting entirely of Anosov representations. For other special groups homomorphisms of SL(2,C) into G, known as magical sl2s, the Slodowy slice parameterizes the higher rank Teichmuller spaces associated to positive representations, in the sense of Guichard-Labourie-Wienhard. Conjecturally, this describes all higher rank Teichmuller spaces for closed surfaces. I will spend the first 2 talks setting up these objects, discussing their relations to the component count of the character variety of representations of a closed surface group into real semisimple Lie groups.
In general, the Slodowy slice is closed in the moduli space but not open. In the final talk, I will discuss a conjectural picture where the Slodowy slice is a generalization of the Fuchsian locus in Quasifuchsian space. One aim of this conjectural picture is to prove that, under the nonabelian Hodge correspondence, the representations associated to Higgs bundles in general Slodowy slices are all Anosov. In the end, I will focus on the aspects of this picture which have been proven when one restricts to real forms of G in low rank, specifically recent work by S. Filip, J. Zhang, S. Bronstien and C. Davalo, and also myself.

Président de session: Brian Collier (University of California Riverside)

mardi 10 juin ¶

Domains of Discontinuity for Anosov Representations: Domains of Discontinuity for Anosov Representations I¶

This minicourse will delve into the theory of cocompact domains of discontinuity for Anosov representations, with a particular focus on their construction and properties. We will introduce the necessary background on flag varieties, Lie theory, and Anosov representations. We will then explore the foundational work of Guichard and Wienhard, and, time permitting, also the work of Kapovich-Leeb-Porti. We will demonstrate that the spaces constructed are indeed cocompact domains of discontinuity. The course will then highlight joint work with Alessandrini, Tholozan, and Wienhard, showing that, in many cases, the quotient of these domains by the Anosov representation is homeomorphic to a fiber bundle. Finally, we will examine the topology of the fiber in some specific cases. The course will feature guest lectures by León Carvajales, who will discuss his joint work with Stecker on non-cocompact domains of discontinuity on some homogeneous spaces, including (pseudo-Riemannian) symmetric spaces, and Mason Hart, who will present his research on the topology of the fiber using Fintushel's classification of circle actions on 4-manifolds.

Président de session: Sara Maloni (University of Virginia)

jeudi 12 juin ¶

Domains of Discontinuity for Anosov Representations: Domains of Discontinuity for Anosov Representations II¶

This minicourse will delve into the theory of cocompact domains of discontinuity for Anosov representations, with a particular focus on their construction and properties. We will introduce the necessary background on flag varieties, Lie theory, and Anosov representations. We will then explore the foundational work of Guichard and Wienhard, and, time permitting, also the work of Kapovich-Leeb-Porti. We will demonstrate that the spaces constructed are indeed cocompact domains of discontinuity. The course will then highlight joint work with Alessandrini, Tholozan, and Wienhard, showing that, in many cases, the quotient of these domains by the Anosov representation is homeomorphic to a fiber bundle. Finally, we will examine the topology of the fiber in some specific cases. The course will feature guest lectures by León Carvajales, who will discuss his joint work with Stecker on non-cocompact domains of discontinuity on some homogeneous spaces, including (pseudo-Riemannian) symmetric spaces, and Mason Hart, who will present his research on the topology of the fiber using Fintushel's classification of circle actions on 4-manifolds.

Président de session: Sara Maloni (University of Virginia)

2 Conférence Carte Blanche "Trace vives" à la maison Poincaré¶

Comment les formes se relient-elles les unes aux autres ? Là où l'artiste saisit intuitivement l'équilibre entre les formes, les mathématiciennes et mathématiciens cartographient ces relations en incorporant des notions de distance et de transformation dans ce qu'ils nomment les espaces de modules — de vastes paysages structurés de toutes les formes géométriques possibles.

Avec Bruno Teheux, nous avons imaginé le projet « Life Lines », et recueilli des milliers de dessins à travers le monde afin d’échantillionner la créativité collective. Le jeu Quadratis, conçu avec Paul Turner, permet d’explorer et visualiser ces espaces de modules. Ces projets seront le coeur de l’exposé où vous pourrez aussi les expérimenter en direct.

Par Hugo Parlier, mathématicien, Université de Fribourg (Suisse).

https://billetterie-maison-poincare.ihp.fr/evenements/conferences-cartes-blanche/traces-vives

Orateur: Hugo Parlier (University of Luxembourg)

vendredi 13 juin ¶

Domains of Discontinuity for Anosov Representations: Domains of Discontinuity for Anosov Representations III¶

This minicourse will delve into the theory of cocompact domains of discontinuity for Anosov representations, with a particular focus on their construction and properties. We will introduce the necessary background on flag varieties, Lie theory, and Anosov representations. We will then explore the foundational work of Guichard and Wienhard, and, time permitting, also the work of Kapovich-Leeb-Porti. We will demonstrate that the spaces constructed are indeed cocompact domains of discontinuity. The course will then highlight joint work with Alessandrini, Tholozan, and Wienhard, showing that, in many cases, the quotient of these domains by the Anosov representation is homeomorphic to a fiber bundle. Finally, we will examine the topology of the fiber in some specific cases. The course will feature guest lectures by León Carvajales, who will discuss his joint work with Stecker on non-cocompact domains of discontinuity on some homogeneous spaces, including (pseudo-Riemannian) symmetric spaces, and Mason Hart, who will present his research on the topology of the fiber using Fintushel's classification of circle actions on 4-manifolds.

Président de session: Sara Maloni (University of Virginia)

lundi 16 juin ¶

3 Research seminar at IHES¶

mardi 17 juin ¶

Complex hyperbolic lattices: Complex hyperbolic lattices I¶

Complex hyperbolic lattices occupy an interesting space in between fundamental groups of hyperbolic manifolds and higher-rank lattices, and they also are a key case for a number of open problems on discrete subgroups of Lie groups. This mini-course will discuss (0) basic geometry (1) rigidity and a geometric viewpoint on arithmeticity (2) explicit constructions via uniformization and applications and (3) a potpourri of open problems.

Président de session: Matthew Stover (Department of Mathematics, Temple University)

jeudi 19 juin ¶

Complex hyperbolic lattices: Complex hyperbolic lattices II¶

Complex hyperbolic lattices occupy an interesting space in between fundamental groups of hyperbolic manifolds and higher-rank lattices, and they also are a key case for a number of open problems on discrete subgroups of Lie groups. This mini-course will discuss (0) basic geometry (1) rigidity and a geometric viewpoint on arithmeticity (2) explicit constructions via uniformization and applications and (3) a potpourri of open problems.

Président de session: Matthew Stover (Department of Mathematics, Temple University)

vendredi 20 juin ¶

Complex hyperbolic lattices: Complex hyperbolic lattices III¶

Complex hyperbolic lattices occupy an interesting space in between fundamental groups of hyperbolic manifolds and higher-rank lattices, and they also are a key case for a number of open problems on discrete subgroups of Lie groups. This mini-course will discuss (0) basic geometry (1) rigidity and a geometric viewpoint on arithmeticity (2) explicit constructions via uniformization and applications and (3) a potpourri of open problems.

Président de session: Matthew Stover (Department of Mathematics, Temple University)

mardi 1 juillet ¶

Hausdorff dimension of limit sets: Hausdorff dimension of limit sets I¶

We will consider the problem of calculating the Hausdorff dimension of limit sets of discrete subgroups of SL_2(R) and SL_3(R) with respect to the natural geometric distances on the corresponding spaces of flags. The plan is to begin with basic results on dimension of sets and measures in Euclidean space, including the Moran-Hutchinson theorem. Later discuss the classical techniques and results for SL_2(R) of Bowen, Patterson, and Sullivan. And finally give some idea of what is known for SL_3(R) where active research is still ongoing.

Président de session: Pablo Lessa (Universidad de la República)

jeudi 3 juillet ¶

Hausdorff dimension of limit sets: Hausdorff dimension of limit sets II¶

We will consider the problem of calculating the Hausdorff dimension of limit sets of discrete subgroups of SL_2(R) and SL_3(R) with respect to the natural geometric distances on the corresponding spaces of flags. The plan is to begin with basic results on dimension of sets and measures in Euclidean space, including the Moran-Hutchinson theorem. Later discuss the classical techniques and results for SL_2(R) of Bowen, Patterson, and Sullivan. And finally give some idea of what is known for SL_3(R) where active research is still ongoing.

Président de session: Pablo Lessa (Universidad de la República)

vendredi 4 juillet ¶

Hausdorff dimension of limit sets: Hausdorff dimension of limit sets III¶

We will consider the problem of calculating the Hausdorff dimension of limit sets of discrete subgroups of SL_2(R) and SL_3(R) with respect to the natural geometric distances on the corresponding spaces of flags. The plan is to begin with basic results on dimension of sets and measures in Euclidean space, including the Moran-Hutchinson theorem. Later discuss the classical techniques and results for SL_2(R) of Bowen, Patterson, and Sullivan. And finally give some idea of what is known for SL_3(R) where active research is still ongoing.

Président de session: Pablo Lessa (Universidad de la República)