Research
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.
Publications
Preprints
Epstein Surfaces, W-Volume, and the Osgood-Stowe Differential
co-author: Kenneth Bromberg
Preprint 2024
Download: Pdf
Universal Liouville action as a renormalized volume and its gradient flow
co-author: Kenneth Bromberg, Franco Vargas Pallete, Yilin Wang
Preprint 2023
Download: Pdf
Ghost polygons, Poisson bracket and convexity
co-author: Francois Labourie
Preprint 2023
Download: 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
Download: Pdf
Published
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
Download: 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
Download: 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
Download: Pdf
The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores
co-authors: Jeffrey Brock, Kenneth Bromberg
Geometry & Topology, Volume 27, Issue 8, pp. 3183–3228, 2023
DOI: https://doi.org/10.2140/gt.2023.27.3183
Download: 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
Download: 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
Download: 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
Download: 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
Download: 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
Download: Pdf
Simple length rigidity for Hitchin representations,
Advances in Mathematics, Volume 360, January 2020
co-authors: Richard Canary, François Labourie
Download: 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
Download: 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
Download: Pdf
Simple root flows for Hitchin representations,
Geometrica Dedicata, Volume 192, Issue 1, pp. 57–86, 2018
co-authors: Richard Canary, Francois Labourie, Andres Sambarino,
Download: Pdf
Renormalized volume and volume of the convex core,
Annales de l’Institut Fourier, Volume 67, Issue 5, pp. 2083-2098, 2017
co-author: Richard Canary
Download: Pdf
Simple length rigidity for Kleinian surface groups and applications,
Commentarii Mathematici Helvetici, Volume 92, Issue 4, pp. 715–750, 2017
co-author: Richard Canary
Download: Pdf
An improved bound for Sullivan's convex hull theorem,
Proceedings of the London Math. Society, January 2016
co-authors: Richard Canary, Andrew Yarmola
Download: Pdf
Identities on Hyperbolic Manifolds,
The Handbook of Teichmuller Theory, Vol. 5, pp. 19-53, EMS Publishing, January 2016
co-author: Ser Peow Tan
Download: Pdf
The pressure metric for Anosov representations,
Geometric and Functional Analysis, Vol. 25, No. 4, 2015
co-authors: Richard Canary, Francois Labourie, Andres Sambarino
Download: 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
Download: 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
Download: Pdf
Orthospectra of geodesic laminations and dilogarithm identities on moduli space,
Geometry and Topology, Vol. 15, No. 2, 2011
Download: 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
Download: Pdf
Hyperbolic Volume of n-manifolds with geodesic boundary and orthospectra,
Geometric and Functional Analysis, Vol. 20, No. 5, 2010
co-author: Jeremy Kahn
Download: Pdf
Hausdorff dimension and the Weil-Petersson extension to quasifuchsian space,
Geometry and Topology, Vol. 14, No. 2, 2010
Download: Pdf
An extension of the Weil-Petersson metric to quasi-fuchsian space,
Mathematische Annalen, Vol. 341, No. 4, 2008
co-author: Edward Taylor
Download: 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
Download: 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
Download: Pdf
Patterson-Sullivan measures and quasi-conformal deformations,
Communications in Analysis and Geometry, Vol. 13, No. 3, 2005
co-author: Edward Taylor
Download: Pdf
Bounding the bending of a hyperbolic three-manifold,
Pacific Journal of Math, Vol. 218, No. 2, 2005
co-author: Richard Canary
Download: 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
Download: Pdf
Random geodesics,
In the tradition of Ahlfors and Bers, III, AMS Contemporary Math Series, Vol. 355, 2004
Download: Pdf
Bounds on the average bending of the convex hull of a Kleinian group,
Michigan Math Journal, Vol. 51, No. 2, 2003
Download: Pdf
From the boundary of the convex core to the conformal boundary,
Geometrica Dedicata, Vol. 96, No. 1, 2003
co-author: Richard Canary
Download: Pdf
Length distortion and the hausdorff dimension of limit sets,
American Journal of Mathematics, Vol. 122, 2000
co-author: Edward Taylor
Download: Pdf
Average bending of convex pleated planes in H3,
inventiones mathematicae, Vol. 132, 1998
Download: Pdf
Average curvature of convex curves in H2,
Proceedings of the American Mathematical Society, Vol. 126, No. 1, 1998
Download: Pdf
Bounds on Volume increase under dehn drilling operations,
Proceedings of the London Mathematical Society, Vol. 77, No. 3, 1998
Download: Pdf
The structure and enumeration of link projections,
Transactions of the American Mathematical Society, Vol. 348, No. 6, 1996
Download: Pdf
Notes and Preprints
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.
Lectures Online
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