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, Patterson-Sullivan theory and generalizations of the Weil-Petersson metric to representation varieties.

Publications

  • Uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures

co-author: Yunhui Wu

Preprint 2019

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  • Dilogarithm identities for solutions to Pell's equation in terms of continued fraction convergents

Preprint 2019

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  • Simple length rigidity for Hitchin Representations,

Advances in Mathematics, Volume 360, January 2020

co-authors: Richard Canary, François Labourie

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  • 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

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  • 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

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  • Simple root flows for Hitchin representations,

Geometrica Dedicata, Volume 192, Issue 1, pp. 57–86, 2018

co-authors: Richard Canary, Francois Labourie, Andres Sambarino,

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  • 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

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  • Simple length rigidity for Kleinian surface groups and applications,

Commentarii Mathematici Helvetici, Volume 92, Issue 4, pp. 715–750, 2017

co-author: Richard Canary

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  • An improved bound for Sullivan's convex hull theorem,

Proceedings of the London Math. Society, January 2016

co-authors: Richard Canary, Andrew Yarmola

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  • Identities on Hyperbolic Manifolds,

The Handbook of Teichmuller Theory, Vol. 5, pp. 19-53, EMS Publishing, January 2016

co-author: Ser Peow Tan

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  • The pressure metric for Anosov representations,

Geometric and Functional Analysis, Vol. 25, No. 4, 2015

co-authors: Richard Canary, Francois Labourie, Andres Sambarino

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  • Higher derivatives of length functions along earthquake deformations,

Michigan Math Journal, Vol. 64, 2015

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  • 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

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  • 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

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  • Orthospectra of geodesic laminations and dilogarithm identities on moduli space,

Geometry and Topology, Vol. 15, No. 2, 2011

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  • The Thurston Metric on Hyperbolic Domains and Boundaries of Convex Hulls,

Geometric and Functional Analysis, Vol. 20, No. 6, 2010

co-author: Richard Canary

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  • Hyperbolic Volume of n-manifolds with geodesic boundary and orthospectra,

Geometric and Functional Analysis, Vol. 20, No. 5, 2010

co-author: Jeremy Kahn

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  • Hausdorff dimension and the Weil-Petersson extension to quasifuchsian space,

Geometry and Topology, Vol. 14, No. 2, 2010

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  • An extension of the Weil-Petersson metric to quasi-fuchsian space,

Mathematische Annalen, Vol. 341, No. 4, 2008

co-author: Edward Taylor

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  • 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

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  • 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

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  • Patterson-Sullivan measures and quasi-conformal deformations,

Communications in Analysis and Geometry, Vol. 13, No. 3, 2005

co-author: Edward Taylor

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  • Bounding the bending of a hyperbolic three-manifold,

Pacific Journal of Math, Vol. 218, No. 2, 2005

co-author: Richard Canary

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  • Distortion of the exponent of convergence in space,

Annales Academicae Scientiarum Fennicae Mathematica, Vol. 29, 2004

co-authors: Petra Bonfert-Taylor, Edward C. Taylor

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  • Random geodesics,

In the tradition of Ahlfors and Bers, III, AMS Contemporary Math Series, Vol. 355, 2004

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  • Bounds on the average bending of the convex hull of a Kleinian group,

Michigan Math Journal, Vol. 51, No. 2, 2003

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  • From the boundary of the convex core to the conformal boundary,

Geometrica Dedicata, Vol. 96, No. 1, 2003

co-author: Richard Canary

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  • Length distortion and the hausdorff dimension of limit sets,

American Journal of Mathematics, Vol. 122, 2000

co-author: Edward Taylor

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  • Average bending of convex pleated planes in H3,

inventiones mathematicae, Vol. 132, 1998

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  • Average curvature of convex curves in H2,

Proceedings of the American Mathematical Society, Vol. 126, No. 1, 1998

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  • Bounds on Vol. increase under dehn drilling operations,

Proceedings of the London Mathematical Society, Vol. 77, No. 3, 1998

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  • The structure and enumeration of link projections,

Transactions of the American Mathematical Society, Vol. 348, No. 6, 1996

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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.

Pdf

Older version Pdf


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".

Pdf

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.

Lecture 1

Lecture 2

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.

Pdf

Effective bounds for Wolpert’s Strata Separation

In this note, we obtain effective bounds on the Weil-Petersson distance between strata in the Weil-Petersson completion of Teich(S).

Pdf