Schedule, Notes/Handouts & Videos

Theta positive representations form a class of subgroups of higher rank Lie groups which can be understood as an analogue of holonomies of hyperbolic structures on surfaces. In particular suitably chosen associated length functions can be computed with the aid of geodesic currents. I will discuss joint work with Burger-Iozzi-Parreau in which we extend this link with geodesic currents at infinity of the character variety, and discuss properties of limiting currents.

Carrière proved in 1989 that every compact flat Lorentzian manifold (i.e. pseudo-Riemannian of signature (n,1)) is complete. Beyond this Lorentzian case, no completeness is known in higher signature. In collaboration with Farid Diaf and Malek Hanounah, we prove the completeness of compact flat Kleinian pseudo-Riemannian manifolds of signature (2,2). This result falls into the context of the study of completeness of compact affine manifolds with a parallel volume form (Markus' conjecture). I will present a sketch of the proof of completeness in signature (2,2), and then focus on proving a key lemma, which is a reduction result for certain closed kleinian affine manifolds. 

Ratner’s theorem implies strong topological rigidity for immersed  totally geodesic submanifolds in finite-volume locally symmetric spaces. In infinite-volume settings, however, the topological behavior of totally geodesic submanifolds is much less understood: while it has been studied in certain rank-one cases, the higher-rank setting has remained unexplored.

In this talk, I will describe a construction of the first higher-rank examples exhibiting a failure of such rigidity using floating geodesic planes, joint work with Hee Oh. More precisely, we construct  a Zariski-dense Hitchin surface group inside $\mathrm{SL}_3(\mathbb{Z})$ whose associated locally symmetric space contains floating geodesic planes whose closures have non-integer Hausdorff dimension. 

In recent years, there has been significant progress in establishing effective versions of equidistribution theorems in homogeneous dynamics. In this talk, we will present new results in this direction and discuss their applications to number theory.

In this talk I want to report on recent advances on spectral theory for Anosov representations. Motivated by the theory of Laplace and Ruelle resonances for convex cocompact hyperbolic surfaces I will discuss the corresponding higher rank analogs, i.e. the joint spectrum of the Algebra of invariant differential operators and Weyl chamber flows. In particular I want to explain how the spectral theoretic results imply new results for the Anosov subgroups, such as bounds on growth rates of meromorphic Poincaré series.

We study the limit cone of a positive representation of a surface group into a real-split, semi-simple Lie group. We focus on the problem of identifying which curves and geodesic currents are able to find the boundary. We show that for a typical boundary face, the curves and currents mapping to that face are supported on a sub-flow of low complexity that we call a quasi-lamination. It is analogous to the maximally stretched lamination appearing in the story of Thurston’s asymmetric metric. Joint work with Fran\c{c}ois Gu\’eritaud and Fanny Kassel.

For Kleinian groups there is a very rich study of examples of geometric limits which are different than the algebraic limit, inspired by fundamental examples described by Jorgensen and Kerckhoff-Thurston. I will discuss joint work with Beatrice Pozzetti, where we generalize these examples and study geometric limits "à la Jorgensen" for cyclic and free subgroups of rank-1 Lie groups. I will also describe our work in progress where we generalize this theory to higher rank Lie groups and to surface group representations. 

In this talk, we study the distribution of determinant values taken by lattice points in the space of d by d real matrices. Unless the lattice has an additional multiplicative structure like the lattice of integer matrices, it turns out that the determinant values are dense in the real line, as a consequence of Ratner's theorem. A natural question is how these determinant values are distributed in the real line. We give a complete answer to this question for d=3. For d=2 our approach gives an alternative proof for the quantitative version of the Oppenheim conjecture for quadratic forms of signature (2,2), obtained by Eskin-Margulis-Mozes (2005). This is joint work with Hee Oh.

We know that certain types of locally symmetric spaces are (very) rigid. What about Riemannian manifolds that look locally symmetric 99% of the time?

Rigidity phenomena for these types of spaces are related to higher property (T), representation stability and the existence of a non-sofic group. Not much is known, but I will describe one result in this direction: a rigidity theorem for branched covers of octonionic hyperbolic manifolds with small branching locus. Based on a joint work with Ben Lowe. 

We study the variation of complex length with respect to bending deformations of quasifuchsian groups. As one application, we exhibit an explicit open neighborhood of the Fuchsian locus in quasifuchsian space so that the only critical points of the entropy function in this neighborhood lie on the Fuchsian locus. As a second application, we exhibit an explicit open neighborhood of the Fuchsian locus so that the adjoint of every representation which is not Fuchsian is the linear part of a proper affine action on the Lie algebra of SL(2,C).