Proper actions on group manifolds

• Sourav Ghosh (Ashoka University)

Room: Amphitheater Darboux In this talk, I will introduce group manifolds and discuss proper actions of a word hyperbolic group on group manifolds in terms of Anosov representations. Moreover, I will show how to use this relation to construct new examples of proper actions of free groups on group manifolds. This is a joint work with Toshiyuki Kobayashi.

Combination theorems, Bers slices, and holomorphic correspondences

• Mahan Mj (Tata Institute of Fundamental Research, Mumbai)

Room: Amphitheater Darboux Our starting points consist of the simultaneous uniformization theorem for surface groups and the mating construction for polynomials. We shall describe a hybrid construction that simultaneously uniformizes a polynomial and a surface. We provide two constructions for some genus zero orbifolds and polynomials lying in the principal hyperbolic component: 1) For punctured spheres with possibly order 2 orbifold points using orbit equivalence 2) Generalizing (1) to orbifolds that have, in addition, an orbifold point of order > 2. This uses a factor dynamical system. We characterize the combinations of polynomials and Fuchsian genus zero orbifold groups as explicit algebraic functions. This allows us to embed the 'product' of Teichmüller spaces of genus zero orbifolds and parameter spaces of polynomials in a larger ambient space of algebraic correspondences. We conclude by describing the analog of the Bers slice in this context. and establish the pre-compactness of the Bers slice. This is the analog of the Bers' compactness theorem in classical Teichmüller theory. Joint work with Yusheng Luo and Sabyasachi Mukherjee.

Research seminar at IHES https://www.ihes.fr/~/kassel/seminGGD.html

Growth of $k$-dimensional systoles in congruence coverings

• Mikhail Belolipetsky (IMPA)

Room: Amphitheater Darboux We study growth of absolute and homological $k$-dimensional systoles of arithmetic $n$-manifolds along congruence coverings. Our main interest is in the growth of systoles of manifolds whose real rank $r>1$. We observe, in particular, that in some cases for $k=r$ the growth function tends to oscillate between a power of a logarithm and a power function of the degree of the covering which is a new phenomenon. We also prove the expected polylogarithmic and constant power bounds for small and large $k$, respectively. In the talk I will discuss the general picture and proofs of some results. It is based on joint work with S. Weinberger and work in progress with A. Erdnigor and S. Weinberger.

Junior Seminar Room: Amphitheater Darboux

Moduli of spherical surfaces and their representation spaces

• Gabriele Mondello (Sapienza Università di Roma)

Room: Amphitheater Darboux In this talk we discuss certain topological properties of the moduli space $\mathcal{MSph}_{g,n}(\vartheta)$ of spherical surfaces, namely surfaces of genus $g$ endowed with a metric of curvature $1$ with $n$ conical singularities of angles $2\pi\vartheta_1,...,2\pi\vartheta_n,$ and highlight how different they are from moduli spaces of surfaces of curvature $-1$. We show that their local structure can be studied through certain decorated representation spaces, which are also object of our investigation. Concerning the global topological properties of $\mathcal{MSph}_{g,n}(\vartheta),$ we show that these moduli spaces are homotopy equivalent to finite cell complexes and that their connected components are non-compact (with very few exceptions). Time permitting, we will describe some explicit example. This is joint work with Dmitri Panov (KCL).

Junior reading group Room: Amphitheater Darboux

Transverse spheres in flag manifolds

• Max Riestenberg (MPI MiS Leipzig)

Room: Amphitheater Darboux Recent results provide many flag manifolds whose transverse subsets cannot properly contain circles, which then restricts the corresponding class of Anosov subgroups. We complement those results by providing transverse 2-spheres (or larger) in all remaining cases of partial flag manifolds of split real Lie groups. We also find arbitrarily large maximally transverse spheres in certain full flag manifolds of split groups. This is joint work with Parker Evans.

Conférence Carte Blanche "Trace vives" à la maison Poincaré

• Hugo Parlier (University of Luxembourg)

Room: Maison Poincaré Participation is subject to obtaining a free ticket at https://billetterie-maison-poincare.ihp.fr/evenements/conferences-cartes-blanche/traces-vives ----------------------------------------------------------------------------------------------------------------- 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).

Research seminar at IHES

Junior reading group Room: Amphitheater Darboux

Zariski-closures of Vinberg representations

• Jacques Audibert (Max Planck Institute, Leipzig)

Room: Amphitheater Darboux In this talk we will discuss Coxeter groups and their "Vinberg representations". These are discrete and faithful representations that are of special geometric significance. As we will show, the Zariski-closure of such representations is very restricted. This allows us to construct new examples of Zariski-dense subgroups of $\mathrm{SL}(n,\mathbb Z)$. In particular, we will prove that, for all $n$ at least $3$, $\mathrm{SL}(n,\mathbb Z)$ contains a Zariski-dense subgroup isomorphic to the fundamental group of a closed surface. This is joint work with Sami Douba, Gye-Seon Lee and Ludovic Marquis.

Asymptotic counting of surfaces in negatively curved three manifolds

• Sébastien Alvarez (Universidad de la Republica)

Room: Amphitheater Darboux In this talk, we present a two dimensional analogue of the geodesic flow, namely the space of surfaces of constant curvature inside a closed negatively curved $3$-manifold. The dynamical properties of this space are described in terms of actions of $\mathrm{PSL}(2,\mathbb R)$ and the equidistribution properties allow to study the geometric rigidity associated to the counting of such surfaces according to their area. This is joint work with Graham Smith (PUC RIo de Janeiro) and Ben Lowe (University of Chicago).