Research

My research focuses on the geometry, topology, and deformation theory of locally homogeneous geometric structures on manifolds, a subject with roots in Felix Klein’s 1872 Erlangen program that features a blend of differential geometry, Lie theory, representation theory, and dynamics. I study an array of low-dimensional geometric structures modeled on non-Riemannian geometries including semi-Riemannian, affine, and projective geometries. Most of my work is aimed at understanding and exploiting a phenomenon, known as geometric transition, by which different moduli spaces of geometric manifolds interact with one another.

Here is a short Research Summary.

Recently Jean-Marc Schlenker gave a Séminaire Bourbaki about my joint work with François Guéritaud and Fanny Kassel.

Recent papers

  1. Convex cocompact actions in real projective geometry
    joint with F. Guéritaud and F. Kassel.
  2. Convex cocompactness in pseudo-Riemannian symmetric spaces
    joint with F. Guéritaud and F. Kassel, Geometriae Dedicata, special issue Geometries: A celebration of Bill Goldman's 60th birthday (to appear).
  3. Higher signature Delaunay triangulations
    joint with S. Maloni and J.-M. Schlenker
  4. Convex projective structures on non-hyperbolic three-manifolds
    joint with S. Ballas and G.-S. Lee
  5. Fundamental domains for free groups acting on anti-de Sitter 3-space
    joint with F. Guéritaud and F. Kassel, Math. Res. Lett. 23 (2016), no. 3, pp. 735--770.
  6. Polyhedra inscribed in a quadric
    joint with S. Maloni and J.-M. Schlenker
  7. Limits of geometries
    joint with D. Cooper and A. Wienhard, Trans. Amer. Math. Soc., DOI: https://doi.org/10.1090/tran/7174, 2017.
  8. Margulis spacetimes via the arc complex
    joint with F. Guéritaud and F. Kassel, Invent. Math., 204 (2016), no. 1, pp. 133--193.
  9. Geometry and topology of complete Lorentz spacetimes of constant curvature
    joint with F. Guéritaud and F. Kassel, Ann. Sci. Éc. Norm. Supér. 49 (2016), no. 1, pp/ 1--56.
  10. Ideal triangulations and geometric transitions
    J. Topol. 7 (2014), no. 4, pp. 1118--1154.
  11. A Geometric transition from hyperbolic to anti de Sitter geometry
    Geom. Topol. 17 (2013), no. 5, pp. 3077--3134
  12. Some open questions on anti de Sitter geometry
    joint with T. Barbot, F. Bonsante, W.M. Goldman, F. Guéritaud, F. Kassel, K. Krasnov, J.-M. Schlenker, A. Zeghib

The following works are in preparation. Preliminary drafts may be available upon request.

  1. Margulis spacetimes with parabolic elements
    joint with F. Guéritaud and F. Kassel
    (in preparation)
  2. Proper affine actions of right angled Coxeter groups
    joint with F. Guéritaud and F. Kassel
    (in preparation)
  3. Examples and counter-examples of convex cocompact groups
    joint with F. Guéritaud and F. Kassel
    (in preparation)
  4. Induced metrics on convex hulls of quasi-circles
    joint with Francesco Bonsante , S. Maloni, and J.-M. Schlenker
    (in preparation)

Thesis

Geometric transitions: from hyperbolic to AdS geometry
ph.d. thesis, Stanford University (2011).