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Homogeneous Dynamics and Geometry in Higher-Rank Lie Groups

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Monday, June 19, 20238:45 AM Welcome coffeeWelcome coffee8:45 AM - 9:30 AM9:30 AM An Introduction to Geometry and Dynamics on Semisimple Lie Groups and Their Quotients by Discrete Subgroups (1/3) - Jean-François Quint (CNRS & Université de Bordeaux)An Introduction to Geometry and Dynamics on Semisimple Lie Groups and Their Quotients by Discrete Subgroups (1/3)
- Jean-François Quint (CNRS & Université de Bordeaux)

9:30 AM - 10:30 AMThe purpose of these lectures is to introduce certain objects and structure results on semisimple Lie groups, their structure theory, their discrete subgroups and the asymptotic behaviour of the latter.10:30 AM Coffee breakCoffee break10:30 AM - 11:00 AM11:00 AM Anosov Representations (1/3) - Tengren Zhang (National University of Singapore)Anosov Representations (1/3)- Tengren Zhang (National University of Singapore)

11:00 AM - 12:00 PMThis minicourse is an introduction to the theory of Anosov representations for non-experts. There will be three lectures in this minicourse. In the first lecture, we will discuss Anosov representations and their various characterizations. In the second lecture, we will discuss the construction of domains of discontinuity for Anosov representations, as well as their relationship with convex projective geometry. In the final lecture, we will discuss recent developments and possible research directions for Anosov representations.12:00 PM Lunch breakLunch break12:00 PM - 1:30 PM1:30 PM An Introduction to Geometry and Dynamics on Semisimple Lie Groups and Their Quotients by Discrete Subgroups (2/3) - Jean-François Quint (CNRS & Université de Bordeaux)An Introduction to Geometry and Dynamics on Semisimple Lie Groups and Their Quotients by Discrete Subgroups (2/3)- Jean-François Quint (CNRS & Université de Bordeaux)

1:30 PM - 2:30 PMThe purpose of these lectures is to introduce certain objects and structure results on semisimple Lie groups, their structure theory, their discrete subgroups and the asymptotic behaviour of the latter.2:30 PM Coffee breakCoffee break2:30 PM - 3:00 PM3:00 PM Divisible Convex Sets with Properly Embedded Cones - Pierre-Louis Blayac (University of Michigan)Divisible Convex Sets with Properly Embedded Cones- Pierre-Louis Blayac (University of Michigan)

3:00 PM - 4:00 PMA divisible convex set is a convex, bounded, and open subset of an affine chart of the real projective space, on which acts cocompactly a discrete group of projective transformations. These objects have a very rich theory, which involves ideas from dynamical systems, geometric group theory, (G,X)-structures, and Riemannian geometry with nonpositive curvature. Moreover, they are an important source of examples of discrete subgroups of Lie groups which are not lattices (although their construction often uses arithmetic lattices). For instance, they have links with Anosov representations. In this talk, we will survey known examples of divisible convex sets and the discrete groups that divide them, and then describe new examples obtained in collaboration with Gabriele Viaggi, of irreducible, non-symmetric, and non-strictly convex divisible convex sets in arbitrary dimensions (at least 3).4:00 PM BreakBreak4:00 PM - 4:15 PM4:15 PM Equidistribution of Periodic Tori - Nguyen-Thi Dang (Université Paris-Saclay)Equidistribution of Periodic Tori- Nguyen-Thi Dang (Université Paris-Saclay)

4:15 PM - 5:15 PMBowen and Margulis independently proved in the 70s that closed geodesics on compact hyperbolic surfaces equidistribute towards the measure of maximal entropy. From a homogeneous dynamics point of view, this measure is the quotient of the Haar measure on $\mathrm{PSL}(2,\mathbb{R})$ modulo some discrete cocompact sugroup. In a joint work with Jialun Li, we investigate the higher rank setting of this problem by taking a higher rank Lie group (like $\mathrm{SL}(d,\mathbb{R})$ for $d\geq 3$) and by studying the dynamical properties of geodesic flows in higher rank: the so-called Weyl chamber flows and their induced diagonal action. We obtain an equidistribution formula of periodic tori (instead of closed orbits of the geodesic flow).5:30 PM CocktailCocktail5:30 PM - 6:30 PM -
Tuesday, June 20, 20239:00 AM Welcome coffeeWelcome coffee9:00 AM - 9:30 AM9:30 AM An Introduction to Geometry and Dynamics on Semisimple Lie Groups and Their Quotients by Discrete Subgroups (3/3) - Jean-François Quint (CNRS & Université de Bordeaux)An Introduction to Geometry and Dynamics on Semisimple Lie Groups and Their Quotients by Discrete Subgroups (3/3)
- Jean-François Quint (CNRS & Université de Bordeaux)

9:30 AM - 10:30 AMThe purpose of these lectures is to introduce certain objects and structure results on semisimple Lie groups, their structure theory, their discrete subgroups and the asymptotic behaviour of the latter.10:30 AM Coffee breakCoffee break10:30 AM - 11:00 AM11:00 AM Anosov Representations (2/3) - Tengren Zhang (National University of Singapore)Anosov Representations (2/3)- Tengren Zhang (National University of Singapore)

11:00 AM - 12:00 PMThis minicourse is an introduction to the theory of Anosov representations for non-experts. There will be three lectures in this minicourse. In the first lecture, we will discuss Anosov representations and their various characterizations. In the second lecture, we will discuss the construction of domains of discontinuity for Anosov representations, as well as their relationship with convex projective geometry. In the final lecture, we will discuss recent developments and possible research directions for Anosov representations.12:00 PM Lunch breakLunch break12:00 PM - 1:30 PM1:30 PM Local Mixing of Diagonal Flows on Anosov Homogeneous Spaces - Pratyush Sarkar (UC San Diego)Local Mixing of Diagonal Flows on Anosov Homogeneous Spaces- Pratyush Sarkar (UC San Diego)

1:30 PM - 2:30 PMFor convex cocompact (and more generally, geometrically finite) rank one locally symmetric spaces, Winter proved mixing of the frame flow with respect to the Bowen-Margulis-Sullivan measure. Mixing results in homogeneous dynamics have many applications in counting, equidistribution, and decay of matrix coefficients. For Anosov subgroups of higher rank Lie groups, the analogous Bowen-Margulis-Sullivan measures are infinite and one looks for local mixing. In a joint work with Michael Chow, we prove local mixing of one-parameter diagonal flows on Anosov homogeneous spaces, generalizing the result of Winter. We also discuss some applications including a recent result of Chow-Fromm regarding joint equidistribution of maximal flat cylinders and holonomies.2:30 PM Coffee breakCoffee break2:30 PM - 3:00 PM3:00 PM Geometric Structures Associated to Anosov Representations - Sara Maloni (University of Virginia)Geometric Structures Associated to Anosov Representations- Sara Maloni (University of Virginia)

3:00 PM - 4:00 PMAnosov representations can be considered a generalization of convex-cocompact representations to groups of higher-rank. In this talk, we are considering connected components of Anosov representations from the fundamental group of a closed hyperbolic manifold N, and which contains Fuchsian representations, and their associated domains of discontinuity. We will prove that the quotient of these domains of discontinuity is always smooth fiber bundles over N. Determining the topology of the fiber is hard in general, but we are able to describe it for representations in Sp(4,C), and for the domain of discontinuity in the space of complex Lagrangians in C^4 by using the classification of smooth 4-manifolds. This is joint work with Daniele Alessandrini, Nicolas Tholozan, and Anna Wienhard.4:00 PM BreakBreak4:00 PM - 4:15 PM4:15 PM The Bottom of the $L^2$ Spectrum of Higher-rank Locally Symmetric Spaces - Sam Edwards (Durham University)The Bottom of the $L^2$ Spectrum of Higher-rank Locally Symmetric Spaces- Sam Edwards (Durham University)

4:15 PM - 5:15 PMFor a rank one geometrically finite locally symmetric space Γ\X, the bottom of the L^2 spectrum of the Laplace operator is a simple eigenvalue corresponding to a positive eigenfunction if and only if the critical exponent of Γ is strictly greater than half the volume entropy of X. In particular, there exists infinite volume rank one locally symmetric spaces with square integrable positive Laplace eigenfunctions. In contrast, a higher-rank symmetric space Γ\X without rank one factors has a square-integrable positive Laplace eigenfunction if and only Γ is a lattice. We will explain some aspects of the connection between square integrability of positive Laplace eigenfunctions and Patterson-Sullivan and Bowen-Margulis-Sullivan measures in the higher-rank setting. Based on joint work with Oh and Fraczyk-Lee-Oh. -
Wednesday, June 21, 20238:45 AM Welcome coffeeWelcome coffee8:45 AM - 9:15 AM9:15 AM Anosov Representations (3/3) - Tengren Zhang (National University of Singapore)Anosov Representations (3/3)
- Tengren Zhang (National University of Singapore)

9:15 AM - 10:15 AMThis minicourse is an introduction to the theory of Anosov representations for non-experts. There will be three lectures in this minicourse. In the first lecture, we will discuss Anosov representations and their various characterizations. In the second lecture, we will discuss the construction of domains of discontinuity for Anosov representations, as well as their relationship with convex projective geometry. In the final lecture, we will discuss recent developments and possible research directions for Anosov representations.10:15 AM Coffee breakCoffee break10:15 AM - 10:45 AM10:45 AM Recent Developments - Discrete Subgroups with Finite Bowen-Margulis-Sullivan Measure in Higher Rank (1/2) - Minju Lee (University of Chicago)Recent Developments - Discrete Subgroups with Finite Bowen-Margulis-Sullivan Measure in Higher Rank (1/2)- Minju Lee (University of Chicago)

10:45 AM - 11:45 AMLet G be a connected semisimple real algebraic group and D be its Zariski dense discrete subgroup. We prove that if D\G admits any finite Bowen-Margulis-Sullivan measure, then D is virtually a product of higher rank lattices and discrete subgroups of rank one factor of G. This may be viewed as a measure-theoretic analog of classification of convex cocompact actions by Kleiner-Leeb and Quint, which was conjectured by Corlette in 1994. This is joint work with Mikolaj Fraczyk. We will then discuss its application on the bottom of the L^2 spectrum, in joint work with Samuel Edwards, Mikolaj Fraczyk, and Hee Oh.11:45 AM BreakBreak11:45 AM - 12:00 PM12:00 PM Dynamics Associated to Anosov Representations: Some Geometric Consequences (1/2) - Andrés Sambarino (CNRS & IMJ-PRG)Dynamics Associated to Anosov Representations: Some Geometric Consequences (1/2)- Andrés Sambarino (CNRS & IMJ-PRG)

12:00 PM - 1:00 PMWe will review some dynamical systems associated to an Anosov representation and draw some geometric conclusions. More precisely, we will review topics such as the Patterson-Sullivan Theory for these representations, the critical hypersurface, dynamical intersection, dynamics of the \theta-Weyl-chamber flow, and finally directional-conicality and generalizations. We will then consider the approach of dominated sequences and draw conclusions on the regularity of limit sets. Some relevant references are: lecture 1: Babillot-Ledrappier, Bridgeman-Canary-Labourie-S., Burger-Landesberg-Lee-Oh, Carvajales, Chow-Sarkar, Dey-Kapovich, Ledrappier, Lee-Oh, S. lecture 2: Bochi-Potrie-S., Pozzetti-S., Pozzetti-S.-Wienhard, Zhang-Zimmer. -
Thursday, June 22, 20239:00 AM Welcome coffeeWelcome coffee9:00 AM - 9:30 AM9:30 AM Homogeneous Dynamics and u-Gibbs States - Rafael Potrie (Universidad de la Republica, Montevideo)Homogeneous Dynamics and u-Gibbs States
- Rafael Potrie (Universidad de la Republica, Montevideo)

9:30 AM - 10:30 AMOrbit closure and measure classification results are quite central results and tools in homogeneous dynamics. Recently, new techniques provided more 'robust' approaches to this kind of results and it makes sense to try to extend some results to the non-homogeneous setting. I will try to explain what would be the natural questions in the non-linear setting and report some progress in this direction.10:30 AM Coffee breakCoffee break10:30 AM - 11:00 AM11:00 AM Recent Developments - Horospherically Invariant Measures and a Rank Dichotomy for Anosov Groups (2/2) - Minju Lee (University of Chicago)Recent Developments - Horospherically Invariant Measures and a Rank Dichotomy for Anosov Groups (2/2)- Minju Lee (University of Chicago)

11:00 AM - 12:00 PMLet $G$ be a product of simple real algebraic groups of rank one and $\Gamma$ be a Zariski dense and Anosov subgroup with respect to a minimal parabolic subgroup $P$. Let $N$ be the unipotent radical of $P$. For each direction $u$ in the interior of Weyl chamber, we show that there exists at most one $N$-invariant measure in $\Gamma\backslash G$ which is supported on the forward recurrent subset for the $\exp(tu)$-action. This can be viewed as a generalization of the unique ergodicity result for the horospherical action due to Furstenberg, Burger, Roblin and Winter for $\Gamma$ convex cocompact. This is joint work with Or Landesberg, Elon Lindenstrauss and Hee Oh.12:00 PM Lunch breakLunch break12:00 PM - 1:30 PM1:30 PM Orbit Counting Theorem for Cusped Hitchin Representations - Giuseppe Martone (Yale University)Orbit Counting Theorem for Cusped Hitchin Representations- Giuseppe Martone (Yale University)

1:30 PM - 2:30 PMThe dynamics of a cusped Hitchin representation can be encoded by a locally Holder continuous, eventually positive, non-arithmetic potential with an entropy gap at infinity on a topologically mixing countable Markov shifts with the BIP property. After introducing and motivating these notions, I will present an orbit counting theorem for this class of shifts and potentials that, in turn, gives an orbit counting theorem for cusped Hitchin representations. This is joint work with Harry Bray, Dick Canary, and Nyima Kao.2:30 PM Coffee breakCoffee break2:30 PM - 3:00 PM3:00 PM Transverse Groups - Andrew Zimmer (University of Wisconsin, Madison)Transverse Groups- Andrew Zimmer (University of Wisconsin, Madison)

3:00 PM - 4:00 PMIn this talk, I will discuss transverse groups (also called antipodal regular groups), a discrete group class containing the Anosov and relatively Anosov ones. I will describe a metric and flow space such a group acts on, which are analogous to the Cayley graph and the geodesic flow space of a word hyperbolic group. Then I will discuss how to use these spaces to prove new results. This represents joint work with Richard Canary and Tengren Zhang.4:00 PM BreakBreak4:00 PM - 4:15 PM4:15 PM Dynamics Associated to Anosov Representations: Some Geometric Consequences (2/2) - Andrés Sambarino (CNRS & IMJ-PRG)Dynamics Associated to Anosov Representations: Some Geometric Consequences (2/2)- Andrés Sambarino (CNRS & IMJ-PRG)

4:15 PM - 5:15 PMWe will review some dynamical systems associated to an Anosov representation and draw some geometric conclusions. More precisely, we will review topics such as the Patterson-Sullivan Theory for these representations, the critical hypersurface, dynamical intersection, dynamics of the \theta-Weyl-chamber flow, and finally directional-conicality and generalizations. We will then consider the approach of dominated sequences and draw conclusions on the regularity of limit sets. Some relevant references are: lecture 1: Babillot-Ledrappier, Bridgeman-Canary-Labourie-S., Burger-Landesberg-Lee-Oh, Carvajales, Chow-Sarkar, Dey-Kapovich, Ledrappier, Lee-Oh, S. lecture 2: Bochi-Potrie-S., Pozzetti-S., Pozzetti-S.-Wienhard, Zhang-Zimmer. -
Friday, June 23, 20238:45 AM Welcome coffeeWelcome coffee8:45 AM - 9:15 AM9:15 AM On the Dimension of Limit Sets on the Real Projective Plane via Stationary Measures - Wenyu Pan (University of Toronto)On the Dimension of Limit Sets on the Real Projective Plane via Stationary Measures
- Wenyu Pan (University of Toronto)

9:15 AM - 10:15 AMI will present a dimension jump result of limit sets on RP^2 for representations of surface groups in SL(3,R). For Anosov representations, we prove the equality between the Hausdorff dimension and the affinity dimension. In particular, it reveals a dimension jump under perturbation. The core of the proof is to study the stationary measures of finitely supported random walks on SL(3,R). We show the Hausdorff dimensions of the measures are equal to their Lyapunov dimensions under certain assumptions. This is based on ongoing joint work with Jialun Li and Disheng Xu.10:15 AM Coffee breakCoffee break10:15 AM - 10:45 AM10:45 AM Thurston's Asymmetric Metrics for Anosov Representations - León Carvajales (Universidad de la Republica, Montevideo)Thurston's Asymmetric Metrics for Anosov Representations- León Carvajales (Universidad de la Republica, Montevideo)

10:45 AM - 11:45 AMThe Thurston metric is an asymmetric distance on the Teichmüller space of a surface, which is computed by comparing the lengths of closed curves in the two hyperbolic structures. Thurston introduced this metric and proved many interesting properties of it, which we will briefly summarize. The theory of Anosov representations aims to generalize several aspects of the classical Teichmüller-Thurston theory to higher-rank representations of hyperbolic groups. For instance, Bridgeman-Canary-Labourie-Sambarino applied the Thermodynamical Formalism to the underlying geodesic flow to construct pressure metrics on some spaces of Anosov representations, which generalize the Weil-Petersson metric on Teichmüller space. In this talk, we will apply similar techniques to show that Thurston's asymmetric distance also generalizes to this setting. This is joint work with Xian Dai, Beatrice Pozzetti, and Anna Wienhard.11:45 AM BreakBreak11:45 AM - 12:00 PM12:00 PM Ghost Polygons, Poisson Bracket and Convexity - François Labourie (Université Côte d'Azur)Ghost Polygons, Poisson Bracket and Convexity- François Labourie (Université Côte d'Azur)

12:00 PM - 1:00 PMThe moduli space of Anosov representations of a surface group in a group $\mathsf G$, which is an open set in the character variety, admits many more natural functions than the regular functions: length functions, correlation functions. We compute the Poisson bracket of those functions using some combinatorial device, show that the set of those functions is stable under the Poisson bracket and give an application to the convexity of length functions, generalizing the result of Kerckhoff on Teichmüller space. We shall start by giving an introduction to Anosov representations, define precisely what are the functions we consider, and explain the combinatorial device involved. This is a joint work with Martin Bridgeman.