On global rigidity of Anosov G-actions

• Danijela Damjanovic (KTH)

Room: Amphithéâtre Hermite / Darboux I will present some recent progress on the global rigidity for Anosov actions of semisimple Lie groups of higher rank. This is joint work with Spatzier, Vinhage and Xu.

Constructing super-expanders from actions of higher rank lattices

• Tim De Laat (University of Münster)

Room: Amphithéâtre Hermite / Darboux Super-expanders are sequences of finite, d-regular graphs that satisfy some nonlinear form of spectral gap with respect to all uniformly convex Banach spaces. This notion vastly strengthens the classical notion of expander. In this talk I will explain some recent constructions of super-expanders, coming from actions of higher rank lattices on Banach spaces and on manifolds. I will also review some recent constructions of (usual) expanders, for which we do not know whether they are super-expanders.

Arithmeticity for Smooth Maximal Rank Positive Entropy Actions of R^k

• Alp Uzman (University of Utah)

Room: Amphithéâtre Hermite / Darboux We prove an arithmeticity theorem in the context of nonuniform measure rigidity. Adapting machinery developed by A. Katok and F. Rodriguez Hertz [J. Mod. Dyn. 10 (2016), 135–172; MR3503686] for Z^k systems to R^k systems, we show that any maximal rank positive entropy system on a manifold generated by k>=2 commuting vector fields of regularity C^r for r>1 is measure theoretically isomorphic to a constant time change of the suspension of some action of Z^k on the (k+1)-torus or the (k+1)-torus modulo {id,-id} by affine automorphisms with linear parts hyperbolic. Further, the constructed conjugacy has certain smoothness properties. This in particular answers a problem and a conjecture from a prequel paper of Katok and Rodriguez Hertz, joint with B. Kalinin [Ann. of Math. (2) 174 (2011), no. 1, 361–400; MR2811602].

Zimmer's embedding theorem for tractor solutions and conformal actions of simple groups on closed pseudo-Riemannian manifolds

• Karin Melnick (University of Luxembourg)

Room: Amphithéâtre Hermite / Darboux Zimmer's embedding theorem concerns actions of connected Lie groups by automorphisms of differential-geometric structures and has yielded important restrictions on which groups can act on a manifold with a given structure. It has a useful version for Cartan geometries which generalizes rather easily to tractor solutions on parabolic-type geometries. Tractor solutions are parallel sections of associated vector bundles for connections which can encode a very wide array of geometric PDEs. An application for the conformal-to-Einstein tractor connection is a rigidity theorem for conformal actions of SU(p',q') on closed (p,q)-pseudo-Riemannian manifolds in the real-analytic setting: 2p' <= p+1 and if 2p'=p+1, then the metric is conformally flat. This is work in progress with K. Neusser.

Higher rank lattice actions with positive entropy

• Homin Lee (Northwestern University)

Room: Amphithéâtre Hermite / Darboux In this talk, I will discuss about smooth higher rank lattice actions on manifolds with positive entropy. From dynamical information, we can detect information on groups and manifolds. For instance, when lattices in SL(n,R) act on an n-dimensional manifold with positive entropy, we can see that the lattice is abstractly commensurable with SL(n,Z). This is joint work with Aaron Brown.

Continuity of a measurable conjugacy between linear cocycles

• Victoria Sadovskaya (PennState University)

Room: Amphithéâtre Hermite / Darboux We consider Holder continuous GL(d,R)-valued cocycles over hyperbolic and partially hyperbolic diffeomorphisms. We discuss results on continuity of a measurable conjugacy between two cocycles. We focus on perturbations of constant cocycles and on cocycles with one Lyapunov exponent. We also mention related results on continuity of measurable invariant geometric structures. As an application, we bootstrap regularity of a conjugacy between an Anosov toral automorphism and its perturbation.

Higher-rank lattices actions on conformal structures

• Vincent Pecastaing (Université Côte d'Azur)

Room: Amphithéâtre Hermite / Darboux In this talk I will discuss both Lie groups and lattices actions by conformal transformation of a pseudo-Riemannian manifold, related to the Lorentzian Lichnerowicz' conjecture. I will first discuss dynamics of SL(2,R)-actions on closed Lorentzian manifolds, and then I will detail recent advances for higher-rank lattices conformal actions, and ongoing works with Thierry Barbot in the specific Lorentzian case.

Systems with rank one factors

• Kurt Vinhage (University of Utah)

Room: Amphithéâtre Hermite / Darboux Lie group actions with rank one factors have natural families of perturbations arising from modifying each factor independently. I will explain ways to obtain semi-rigid settings, in particular, various characterizations of product systems. Partially based on work with R. Spatzier, and work in-progress with A. Uzman.

Non tame cocycle rigidity above affine unipotent abelian actions on the torus

• Bassam Fayad (University of Maryland)

Room: Amphithéâtre Hermite / Darboux Cocycle rigidity with tame solutions is a crucial ingredient in KAM theory. We are interested in cocycle rigidity above affine unipotent abelian actions on the torus with Diophantine translation data. We consider _unlocked_ actions whose rank one factors are non vanishing translations (the locked actions do not have any kind of stability). It follows from Katok and Robinson's observations that when one generator of the action is of step less or equal to 2 then cocycle rigidity with tame solutions holds. Moreover, Damjanovic, Fayad and Saprykina proved that in this case _almost_ cocycles also have _almost_ solutions (with a low regularity control on the error), and from there concluded KAM-rigidity of these actions. In a joint work with S. Durham, we find examples of affine $\mathbb Z^2$-actions on the torus above which smooth cocycle rigidity holds but is not tame. The linear part of the action is generated by unipotent matrices of step 3. Our examples show that KAM-rigidity for higher rank actions by affine unipotent toral actions does not hold in general when no element of the actions is of step less or equal to 2.

Rigidity of lattice actions on boundaries

• Mitul Islam (MPI Leipzig)

Room: Amphithéâtre Hermite / Darboux Lattices in linear semi-simple Lie groups have a natural action on their (generalized) Furstenberg boundaries, e.g. a lattice in SL_n(R) acting on the different flag varieties. For such natural actions, I will discuss the local rigidity question---are these actions ‘stable’ under small deformations in the homeomorphism group of the boundary? We will answer this question for higher rank uniform lattices by showing that any small deformation is semi-conjugate to the natural action. This is joint work with Chris Connell, Thang Nguyen, and Ralf Spatzier. Time permitting, I will discuss how local rigidity fails when we replace the Furstenberg boundary with the visual boundary of the symmetric space.

Poisson–Voronoi tessellations and fixed price in higher rank

• Amanda Wilkens (University of Texas)

Room: Amphithéâtre Hermite / Darboux We define and motivate the Poisson point process, which is, informally, a "maximally random" scattering of points in space. We introduce 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 and Sam Mellick, 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. We give some intuition for the proof. No prior knowledge on Poisson point processes, fixed price, or higher rank will be assumed.

Rigidity of commensurators and its applications

• Wouter Van Limbeek (University of Illinois Chicago)

Room: Amphithéâtre Hermite / Darboux I will discuss a question raised independently by Greenberg and Shalom: Can an infinite discrete subgroup of a simple Lie group have dense commensurator and not be a lattice? I will explain the surprising connections between this question and other long-standing open problems, and discuss recent progress on special cases of the question. This is joint work with (subsets of) Brody, Fisher, and Mj.

Global smooth rigidity for toral automorphisms

• Zhenqi Wang (Michigan State University)

Room: Amphithéâtre Hermite / Darboux Suppose f is a diffeomorphism on torus whose linearization A is weakly irreducible. Let H be a conjugacy between f and A. We prove the following: 1 if A is hyperbolic and H is weakly differentiable 2. if A is partially hyperbolic and H is C^1+holder. Then H is C^\infty. Our result shows that the conjugacy in all local and global rigidity results for irreducible A is $C^\infty$. This is a joint work with B. Kalinin and V Sadovskaya.

Rigidity of some higher rank partially hyperbolic actions

• Sven Sandfeldt (KTH)

Room: Amphithéâtre Hermite / Darboux Smooth rigidity of higher rank abelian and lattice actions with some hyperbolicity has been studied extensively. When the manifold is a nilmanifold, results by Rodriguez Hertz, Wang, and Brown, Rodriguez Hertz, Wang show that: If the action contains an Anosov diffeomorphism then the action is globally rigid. I will discuss rigidity of higher rank partially hyperbolic actions on nilmanifolds. In particular, I will discuss global rigidity of abelian and higher rank lattice actions that contain one fibered partially hyperbolic element.

Strong primeness for equivalence relations arising from Zariski dense subgroups

• Cyril Houdayer (ENS Paris)

Room: Amphithéâtre Hermite / Darboux In this talk, I will describe an ongoing joint work with Daniel Drimbe in which we show that equivalence relations arising from essentially free ergodic probability measure preserving actions of Zariski dense discrete subgroups of simple algebraic groups are strongly prime. As a consequence, we obtain a unique prime factorization result for direct products of such equivalence relations. This extends and strengthens Zimmer's primeness result for equivalence relations arising from actions of lattices in simple Lie groups (1981). The key novelty in our approach relies on a combination of ergodic theory of algebraic group actions and Popa's intertwining theory for equivalence relations.

Lichnerowicz conjecture(s)

• Abdelghani Zeghib (ENS Lyon)

Room: Amphithéâtre Hermite / Darboux A priori, the conformal group of a compact Riemannian manifold has no reason to be compact, since it only preserves angles and not distances. A posteriori, however, it turns out that this group is compact, with a single exception: the round sphere! The Lichnerowicz conjecture refers to similar rigidity statements in the cases of pseudo-Riemannian conformal and projective structures.

Rigidity for boundary actions and classification in low dimensions

• Federico Rodriguez-Hertz (PennState University)

Room: Amphithéâtre Hermite / Darboux The plan of the talk is to describe joint work with A. Brown and Z. Wang on the smooth classification of actions of lattices in SL(n,R) on n-1 dimensional manifolds. The method is also amenable to show rigidity for some boundary actions.

Coarse embeddings and group actions preserving rigid geometric structures

• Charles Frances (University of Strasbourg)

Room: Amphithéâtre Hermite / Darboux We will see how the notion of coarse embeddings allows to better understand discrete group actions preserving a rigid geometric structure. The focus will mainly be on isometric and conformal actions. In particular, we will discuss a Tits alternative for isometry groups of compact Lorentzian manifolds.