24-28 Jun 2019 Sète (France)

Titles and abstracts

Martin Bobb

Decomposition of convex divisible domains

Abstract: The following analogue of Benoist's decomposition theorem for 3-dimensional convex projective domains is true: properly embedded codimension-1 simplices in a convex divisible sets in dimension n > 2 form a discrete set. The image of this set in the quotient is a finite collection of virtual tori, and the constituents of the complement of this collection of tori (if it is non-empty) are cusped convex projective manifolds with type n cusps.

Pierre-Louis Blayac

Recurrence properties of the geodesic flow on convex projective manifolds

Abstract: Convex projective geometry provides interesting examples of geodesic flows. These share many features with geodesic flows on hyperbolic manifolds: geodesics are straight lines in the projective space, isometries are projective automorphisms, and when the universal cover of the manifold is strictly convex and smooth the flow exhibits some good hyperbolicity properties, as noticed by Benoist, Crampon and Marquis. However, in general the universal cover is not homogeneous, the metric is not Riemannian, and the flow is not hyperbolic in the absence of strict convexity. We will discuss when and how one can extend basic recurrence properties of the geodesic flow on hyperbolic manifolds to convex projective manifolds.

Harrison Bray

Bowen-Margulis measure for nonstrictly convex Hilbert geometries on 3-manifolds

Abstract: Hilbert geometries are a class of projectively invariant Finsler metrics on properly convex domains in projective space. The geometry lacks smoothness, but exhibits metric behavior resembling nonpositive curvature and benefits from the geometric data of the domains. Bowen-Margulis measure is an explicitly constructed measure of maximal entropy for the associated Hilbert geodesic flow on compact quotients of these domains under discrete groups of projective transformations. We discuss the construction of this measure for a particularly interesting class of quotients of nonstrictly convex domains discovered by Benoist and prove ergodicity.

Jeffrey Danciger (mini course)

Convex cocompact subgroups in real projective geometry

Abstract: This series of lectures is about a theory of convex cocompactness for discrete subgroups of the projective general linear group acting on real projective space. These groups display geometric and dynamical behavior similar to convex cocompact groups in rank one Lie groups (hyperbolic geometry) with some interesting differences. The tentative plan is as follows:

  • The first lecture will introduce these groups and their basic properties.
  • The second lecture will explain a connection with Anosov representations.
  • The third lecture will focus on constructing examples.

This lecture series is based on joint work with François Guéritaud and Fanny Kassel.

Antonin Guilloux

Hilbert metric in non convex situations

Abstract: With E. Falbel and P. Will, we propose a definition of a metric, reminiscent of and generalizing the Hilbert metric
for convex bodies, which makes sense in unusual settings. I will present in particular its meaning in the case of uniformized CR-spherical manifolds, i.e. manifolds obtained as the quotient by a discrete group of an open subset of the sphere at infinity of the complex hyperbolic space.

Inkang Kim

Convex real projective structures on manifolds

Abstract:I overview some recent results on convex real projective surfaces in terms of affine connections, Kahler metrics, dynamical point of view and harmonic maps. I try to discuss the interplay between geometry, ergodic theory, Riemannian geometry and bundle theory.

Gye-Seon Lee (mini course)

Coxeter groups in real projective geometry

Abstract: Coxeter groups are finitely generated groups that resemble the groups generated by reflections. They play important roles in the various areas of mathematics. In particular there are many connections between Coxeter groups and geometry. This lecture focuses on how one can use Coxeter groups to construct interesting examples of reflection groups in convex real projective geometry.

Arielle Leitner (mini course)

Generalized Cusps on Convex Projective Manifolds

Abstract: Convex projective manifolds are a generalization of hyperbolic manifolds. Generalized cusps are certain ends of convex projective manifolds. They may contain both hyperbolic and parabolic elements.  We will describe their classification (due to Ballas-Cooper-Leitner), and explain how generalized cusps turn out to be deformations of cusps of hyperbolic manifolds.

Koszul showed that the set of holonomies of convex projective structures on a compact manifold is open in the representation variety. We will describe an extension of this result to convex projective manifolds whose ends are generalized cusps,  due to Cooper-Long-Tillmann. If time permits we will discuss current work
on the moduli space of generalized cusps (joint with Ballas and Cooper).

Louis Merlin

Volume growth in Hilbert geometries of low regularity.

Abstract:The volume entropy of a metric measure space is the exponential growth rate of volumes of balls. A recent result of N. Tholozan shows that, in the context of Hilbert geometries, entropy can never exceed the hyperbolic entropy (n-1 in dimension n). This result is absolutely not rigid and in fact, maximal entropy is achieved as soon as the boundary is sufficiently regular. This leads to the general question on the relation between the regularity of the convex set and the value of the volume entropy. The aim of this talk is to present two results which state that the relation does exist. This is a joint work with Jan Cristina.

Xin Nie

Holomorphic cubic differentials and convex domains in the real projective plane.

Abstract: The Cheng-Yau affine sphere in R^3 projectivizing to a given proper convex domain in RP^2 endows the domain with a canonical holomorphic cubic differential. While Labourie and Loftin deduced from this construcution a holomorphic parametrization of the SL(3,R)-Hitchin component of a closed surface, I will explain how a flat infinite sector of the cubic differential gives rise to a line segment on the boundary of the convex domain, which implies a holomorphic parametrizations of the space of certain convex projective structures on an open surface, generalising the result of Dumas and Wolf. I will also discuss problems arising from the analogy between this construction and harmonic maps to the hyperbolic plane.  

Anne Parreau

Degeneration of convex projective surfaces and geodesic currents.

Abstract: Geodesic currents were introduced by Bonahon to study Thurston's compactification of the Teichmuller space. For higher Teichmuller spaces, this compactification can be generalized replacing the length function by the Jordan projection, seen as a Weyl chamber valued length function.  I will explain, focusing on the case of the space of convex real projective structures on a surface, how to associate geodesic currents to degenerations of such structures and
how to use them to construct a natural non-empty open domain of discontinuity for the action of the mapping class group on the boundary. This is based on joint work with Marc Burger, Alessandra Iozzi, and Beatrice Pozzetti.

Andres Sambarino

Hyperconvex representations and Hausdorff dimension

Abstract: The purpose of the talk is to study the Hausdorff dimension of limit sets of certain discrete subgroups $\Gamma$ of ${\mathsf{SL}}_d(\mathbb{R})$ (or ${\mathsf{SL}}_d(\mathbb{C})$) that verify a property that we call \emph{Hyperconvex}. This is an open property in the space $\textrm{hom}(\Gamma,{\mathsf{SL}}_d)$ that implies some form of conformality of the group action when restricted to its limit set.
This is joint work with B. Pozzetti and A. Wienhard

Anna Schilling

Horofunction and Satake compactifications of symmetric spaces

Abstract: There are various ways to compacting a Riemannian symmetric space $X=G/K$ of non-compact type. The horofunction compactification embeds $X$ into the space of real valued functions using the metric on the space and takes the closure there. The (generalized) Satake compactification is associated to an (irreducible) faithful projective representation of $G$. In this talk I will explain these two compactifications and show that any (generalized) Satake compactification can be realized as a horofunction compactification with respect to a special Finsler norm on $X$. This is joint work with Thomas Haettel, Cormac Walsh and Anna Wienhard.

Cormac Walsh

Approximability of convex bodies and volume growth in Hilbert geometries.

Abstract :The approximability of a convex body measures how quickly the complexity of an approximating polytope must increase as one approximates the body more and more closely. I will discuss how this quantity is related to another one: the exponential rate of growth of the volume of a ball in the Hilbert metric on the convex body. This is joint work with Constantin Vernicos.

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