My main areas of research are hyperbolic geometry, Kleinian groups and the geometry of representation varieties. I am particularly interested in Teichmüller geometry, geometric/spectral identities, quasifuchsian groups, convex hulls of Kleinian groups, Hausdorff dimension of limit sets, renormalized volume, Patterson-Sullivan theory and generalizations of the Weil-Petersson metric to representation varieties.
Research is partially supported by NSF grant DMS-2405291
Bounds on bending in terms of the Schwartzian derivative and Teichmüller distance, co-author: Ming Hong Tee, Preprint 2025, Pdf
Epstein Surfaces, W-Volume, and the Osgood-Stowe Differential, co-author: Kenneth Bromberg, Preprint 2024, Pdf
Ghost polygons, Poisson bracket and convexity, co-author: Francois Labourie, Preprint 2023. Pdf
A bound on the L^2 norm of a projective structure by the length of the bending lamination co-author: Kenneth Bromberg, Preprint 2022. Pdf
Universal Liouville action as a renormalized volume and its gradient flow, co-author: Kenneth Bromberg, Franco Vargas Pallete, Yilin Wang, Duke Math Journal, accepted January 2025. Pdf
L^2-bounds for drilling short geodesics in convex co-compact hyperbolic 3-manifolds
co-author: Kenneth Bromberg, Advances in Mathematics, Volume 451, August 2024, DOI: https://doi.org/10.1016/j.aim.2024.109804. Pdf
Variation of holonomy for projective structures and an application to drilling hyperbolic 3-manifolds co-author: Kenneth Bromberg, Geometriae Dedicata special edition, Groups, Geometry, and Dynamics: a celebration of Francois Labourie's 60th birthday, Volume 218, article 62, (2024), DOI: https://doi.org/10.1007/s10711-024-00908-0. Pdf
Convergence of the gradient flow of renormalized volume to convex cores with totally geodesic boundary, co-authors: Kenneth Bromberg, Franco Vargas Pallete, Compositio Mathematica, Volume 159, Issue 4, pp. 830-859, April 2023, DOI: https://doi.org/10.1112/S0010437X2300708X. Pdf
The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores, co-authors: Jeffrey Brock, Kenneth Bromberg, Geometry and Topology, Volume 27, Issue 8, pp. 3183–3228, 2023, DOI: https://doi.org/10.2140/gt.2023.27.3183. Pdf
Hessian of Hausdorff dimension on purely imaginary directions, co-authors: Beatrice Pozzetti, Andrés Sambarino, Anna Wienhard, Bulletin of the London Math Society, Volume 54, Issue 3, pp. 1027-1050, June 2022, DOI: https://doi.org/10.1112/blms.12612. Pdf
Strata separation for the Weil-Petersson completion and gradient estimates for length functions, co-author: Kenneth Bromberg, Journal of Topology and Analysis, Vol. 16, No. 02, pp. 227-259, (2024), DOI: https://doi.org/10.1142/S1793525321500667. Pdf
Lower bounds for volumes and orthospectra of hyperbolic manifolds with geodesic boundary, co-author: Mikhail Belolipetsky, Algebraic and Geometric Topology, Volume 22, pp. 1255–1272, 2022, DOI: https://doi.org/10.2140/agt.2022.22.1255. Pdf
Uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures, co-author: Yunhui Wu, Crelle, Volume 2021, Issue 770, pp. 159-181, January 2021. Pdf
Dilogarithm identities for solutions to Pell's equation in terms of continued fraction convergents, The Ramanujan Journal, Volume 55, Issue 1, pp. 141-161, 2021. Pdf
Simple length rigidity for Hitchin representations, Advances in Mathematics, Volume 360, January 2020, co-authors: Richard Canary, François Labourie. Pdf
Schwarzian derivatives, projective structures, and the Weil-Petersson gradient flow for renormalized volume, Duke Math Journal, Volume 168, Number 5, pp. 867-896, April 2019, co-authors: Jeffrey Brock, Kenneth Bromberg. Pdf
An introduction to pressure metrics on higher Teichmüller spaces, Ergodic Theory and Dynamical Systems, Volume 38, Issue 6, pp. 2001-2035, September 2018, co-authors: Richard Canary, Andres Sambarino. Pdf
Simple root flows for Hitchin representations, co-authors: Richard Canary, Francois Labourie, Andres Sambarino, Geometrica Dedicata, Volume 192, Issue 1, pp. 57–86, 2018. Pdf
Renormalized volume and volume of the convex core, co-author: Richard Canary, Annales de l’Institut Fourier, Volume 67, Issue 5, pp. 2083-2098, 2017. Pdf
Simple length rigidity for Kleinian surface groups and applications, co-author: Richard Canary, Commentarii Mathematici Helvetici, Volume 92, Issue 4, pp. 715–750, 2017. Pdf
An improved bound for Sullivan's convex hull theorem, Proceedings of the London Math. Society, January 2016, co-authors: Richard Canary, Andrew Yarmola. Pdf
Identities on Hyperbolic Manifolds, The Handbook of Teichmuller Theory, Vol. 5, pp. 19-53, EMS Publishing, January 2016 , co-author: Ser Peow Tan. Pdf
The pressure metric for Anosov representations, Geometric and Functional Analysis, Vol. 25, No. 4, 2015, co-authors: Richard Canary, Francois Labourie, Andres Sambarino. Pdf
Higher derivatives of length functions along earthquake deformations, Michigan Math Journal, Vol. 64, 2015, Download: Pdf
Moments of the boundary hitting function for the geodesic flow on a hyperbolic manifold, Geometry and Topology, Vol. 18, No. 1, 2014, co-author: Ser Peow Tan. Pdf
Uniformly perfect domains and convex hulls: improved bounds in a generalization of a theorem of Sullivan, Pure and Applied Mathematics Quarterly, Vol. 9, No. 1, 2013, co-author: Richard Canary. Pdf
Orthospectra of geodesic laminations and dilogarithm identities on moduli space, Geometry and Topology, Vol. 15, No. 2, 2011. Pdf
The Thurston Metric on Hyperbolic Domains and Boundaries of Convex Hulls, Geometric and Functional Analysis, Vol. 20, No. 6, 2010, co-author: Richard Canary. Pdf
Hyperbolic Volume of n-manifolds with geodesic boundary and orthospectra, Geometric and Functional Analysis, Vol. 20, No. 5, 2010, co-author: Jeremy Kahn. Pdf
Hausdorff dimension and the Weil-Petersson extension to quasifuchsian space, Geometry and Topology, Vol. 14, No. 2, 2010. Pdf
An extension of the Weil-Petersson metric to quasi-fuchsian space, Mathematische Annalen, Vol. 341, No. 4, 2008, co-author: Edward Taylor. Pdf
Quasiconformal homogeneity of hyperbolic surfaces with fixed-point full automorphisms, Proceedings of the Cambridge Philosophical Society, Vol. 143, No. 1, 2007, co-authors: Petra Bonfert-Taylor, Richard Canary, Edward Taylor. Pdf
Distribution of intersection lengths of a random geodesic with a geodesic lamination, Ergodic Theory and Dynamical Systems, Vol. 27, No. 4, 2007, co-author: David Dumas. Pdf
Patterson-Sullivan measures and quasi-conformal deformations, Communications in Analysis and Geometry, Vol. 13, No. 3, 2005, co-author: Edward Taylor. Pdf
Bounding the bending of a hyperbolic three-manifold, Pacific Journal of Math, Vol. 218, No. 2, 2005, co-author: Richard Canary. Pdf
Distortion of the exponent of convergence in space, Annales Academicae Scientiarum Fennicae Mathematica, Vol. 29, 2004, co-authors: Petra Bonfert-Taylor, Edward C. Taylor. Pdf
Random geodesics, In the tradition of Ahlfors and Bers, III, AMS Contemporary Math Series, Vol. 355, 2004. Pdf
Bounds on the average bending of the convex hull of a Kleinian group, Michigan Math Journal, Vol. 51, No. 2, 2003. Pdf
From the boundary of the convex core to the conformal boundary, Geometrica Dedicata, Vol. 96, No. 1, 2003, co-author: Richard Canary. Pdf
Length distortion and the hausdorff dimension of limit sets, American Journal of Mathematics, Vol. 122, 2000, co-author: Edward Taylor. Pdf
Average bending of convex pleated planes in H3, inventiones mathematicae, Vol. 132, 1998. Pdf
Average curvature of convex curves in H2, Proceedings of the American Mathematical Society, Vol. 126, No. 1, 1998. Pdf
Bounds on Volume increase under dehn drilling operations, Proceedings of the London Mathematical Society, Vol. 77, No. 3, 1998. Pdf
The structure and enumeration of link projections, Transactions of the American Mathematical Society, Vol. 348, No. 6, 1996. Pdf
Wolpert formulae for the Poisson bracket of Hitchin Length Functions
In this preprint, we use Goldman's formula for the Poisson bracket of invariant functions to derive formulae for the Poisson bracket of length functions on Hitchin space. In particular we derive the formulae for the first and second derivatives. These generalize formulae of Wolpert given in the Teichmuller case for the Weil-Petersson Poisson bracket. We use the formula to prove convexity results for spectral length functions along the associated Hamiltonian vector fields. We also derive a formula for the Poincare dual of the Hamiltonian of the Poisson bracket of two length functions.
Goldman's formula for the Poisson bracket of the Atiyah– Bott–Goldman symplectic form on surface group representation varieties
These notes contain a brief survey of Goldman's paper "The symplectic nature of fundamental groups of surfaces".
Lecture Notes on Renormalized Volume
These are notes from lectures on the work of Krasnov and Schlenker on Renormalized volume. The lectures were given in Fall 2014 as part of Curt McMullen's Informal seminar at Harvard.
Epstein Surfaces, W-Volume, and the Osgood-Stowe Differential
This preprint further develops the link between the W-Volume, Epstein surfaces and the generalized Schwarzian defined by Osgood and Stowe.
Progress in low-dimensional topology: Teichmüller theory and 3-manifold groups
Centre for Quantum Geometry of Moduli Spaces, Aarhus, 2012
A Weil-Petersson metric on the Hitchin component
QGM Masterclass (20 lectures, joint with R. Canary and A. Sambarino)
Centre for Quantum Geometry of Moduli Spaces, Aarhus, Denmark, 2013
Bonahon 60th Conference
University of Paris-Sud, Orsay, France, 2015
Simple Length Rigidity and the Isometry group of the Quasifuchsian Intersection Pairing
Teichmüller Theory, Hyperbolicity and Dynamics
IMPA, Rio de Janeiro, Brazil, 2019
Fields Institute Minicourse (2 lectures, joint with K. Bromberg)
Fields institute, Toronto, Canada, 2024
Renormalized volume in finite and infinite dimensions
Hyperbolic Manifolds, Their Submanifolds and Fundamental Groups
IMPA, Rio de Janeiro, Brazil, 2025
Loewner Energy, Renormalized Volume and their Weil-Petersson gradient flows
Higher rank geometric structures, Higgs bundles and physics
Institut Henri Poincaré, Sorbonne Université, Paris, 2025
Ghost polygons, Poisson bracket and convexity