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, geometric groups theory, dynamics, and topology. I study an array of low-dimensional geometric structures modeled on non-Riemannian geometries including semi-Riemannian, affine, and projective geometries. Of particular interest to me is a phenomenon known as geometric transition, by which different moduli spaces of geometric manifolds interact with one another.

Here is my Research Statement (from 2025).

Slides

Here are slides from two versions of a talk I have been giving recently about joint work in progress with Fanny Kassel and François Guéritaud. The project concerns understanding the shape of the Benoist limit cone for positive representations of free groups and surface groups. For those with some experience in hyperbolic surface geoemtry, here are slides from my lecture, Thurston's asymmetric metric on Teichmuller space revisited, at the Todd Drumm memorial conference, Brin Center, University of Maryland, April 2025. For those having, additionally, experience in Higher Teichmuller-Thurston theory (higher rank semi-simple Lie groups, Anosov representations, positive representations), see the slides from my lecture, Record breakers, slack calculus, and the Benoist limit cone at the Joint Meetings of the AMS and UNI in Palermo, Italy, July 2024.

Here are slides from my lecture, Exotic real projective Dehn surgery space at the conference Geometric Structures in Strasbourg 2022. This is joint work with Sam Ballas, Gye-Seon Lee, and Ludovic Marquis that finds convex real projective structures on infinitely many closed hyperbolic 3-manifolds which are not a continuous deformation of the hyperbolic structure. These are found via a convex real projective analogue of Thurston's hyperbolic Dehn filling theorem.

In 2015, Jean-Marc Schlenker gave a Séminaire Bourbaki about my joint work with François Guéritaud and Fanny Kassel on realizing Margulis spacetimes as limits of anti de Sitter spacetimes.

Papers

1. Combination theorems in convex real projective geometry
   joint with F. Guéritaud, and F. Kassel, arXiv July 2024.
   
2. Convex cocompactness for Coxeter groups
   joint with F. Guéritaud, F. Kassel , G.-S. Lee, and L. Marquis, Journal of the European Mathematical Society, 27 (1) 119--181 (2025).
   
3. Convex cocompact actions in real projective geometry
   joint with F. Guéritaud and F. Kassel, Annales Scientifiques de l'École Normale Supérieure (to appear)
   
4. Proper actions of discrete groups of affine transformations
   joint with T. Drumm, W. Goldman, I. Smilga.
   Dynamics, Geometry, Number Theory: the Impact of Margulis on Modern Mathematics (D. Fisher, D. Kleinbock, G. Soifer, eds.), University of Chicago Press, February 2022.
5. Quasicircles and width of Jordan curves in CP^1
   joint with Francesco Bonsante , S. Maloni, and J.-M. Schlenker, Bulletin of the London Mathematical Society, 53 (2) 507-523 (2021)
6. The induced metric on the boundary of the convex hull of a quasicircle in hyprebolic and anti de Sitter geometry
   joint with Francesco Bonsante , S. Maloni, and J.-M. Schlenker, Geometry and Topology 25, 2827--2911
   
7. Affine actions with Hitchin linear part
   joint with T. Zhang, Geometric and Functional Analysis, 29, 1369-1439 (2019).
   
8. Proper affine actions of right-angled Coxeter groups
   joint with F. Guéritaud and F. Kassel, Duke Mathematical Journal, 169(12): 2231-2280 (2020).
   
9. 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., 192, Issue 1, pp. 87--126, 2018.
   
10. Convex projective structures on non-hyperbolic three-manifolds
    joint with S. Ballas and G.-S. Lee, Geometry and Topology, 22 (2018), pp 1593--1646.
    
11. 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.
    
12. Polyhedra inscribed in a quadric
    joint with S. Maloni and J.-M. Schlenker, Invent. Math., 221 (2020), 237-300.
    
13. Limits of geometries
    joint with D. Cooper and A. Wienhard, Trans. Amer. Math. Soc., 370 (2018), 6585--6627.
    
14. Margulis spacetimes via the arc complex
    joint with F. Guéritaud and F. Kassel, Invent. Math., 204 (2016), no. 1, pp. 133--193.
    
15. 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.
    
16. Ideal triangulations and geometric transitions
    J. Topol. 7 (2014), no. 4, pp. 1118--1154.
17. A Geometric transition from hyperbolic to anti de Sitter geometry
    Geom. Topol. 17 (2013), no. 5, pp. 3077--3134

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

1. Record breakers, slack calculus, and the Benoist limit cone
   joint with F. Guéritaud and F. Kassel
   (in preparation)
2. The shape of the limit cone for (PSL₂(R))^d
   joint with F. Guéritaud and F. Kassel
   (in preparation)
3. Exotic real projective Dehn surgery space
   joint with S. Ballas, G.-S. Lee, and L. Marquis. (in preparation)
4. Margulis spacetimes with parabolic elements
   joint with F. Guéritaud and F. Kassel
   (in preparation. thank you for your patience everyone, we promise to release this soon!)

Thesis

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