Research

limit
          set

Action of a quasifuchsian group



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

1. The structure and enumeration of link projections,  Transactions of the American Mathematical Society, Vol. 348, No. 6, 1996
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2. Bounds on Vol. increase under dehn drilling operations, Proceedings of the London Mathematical Society , Vol. 77, No. 3, 1998
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3. Average curvature of convex curves in H2, Proceedings of the American Mathematical Society , Vol. 126, No. 1, 1998
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4. Average bending of convex pleated planes in H3, inventiones mathematicae, Vol. 132,  1998
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5. Length distortion and the hausdorff dimension of limit sets, American Journal of Mathematics, Vol. 122, 2000
co-author: Edward Taylor
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6. 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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7. 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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8. Random geodesics, In the tradition of Ahlfors  and Bers, III, AMS Contemporary Math Series, Vol. 355, 2004
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9. 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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10. 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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11. 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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12. 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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13. 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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14. 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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15. Hausdorff dimension and the Weil-Petersson extension to quasifuchsian space, Geometry and Topology,  Vol. 14,  No. 2, 2010
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16. 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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17. 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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18. Orthospectra of geodesic laminations and dilogarithm identities on moduli space, Geometry and Topology,  Vol. 15, No. 2, 2011
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19. 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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20.  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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21. Higher derivatives of length functions along earthquake deformations, Michigan Math Journal,  Vol. 64, 2015
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22. The pressure metric for Anosov representations, Geometric and Functional Analysis, June, 2015
co-authors: Richard Canary, Francois Labourie, Andres Sambarino
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23. Identities on Hyperbolic Manifolds, The Handbook of Teichmuller Theory, Vol. 5, to appear
co-author: Ser Peow Tan
Preprint 2013
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24. An improved bound for Sullivan's convex hull theorem,
co-authors: Richard Canary, Andrew Yarmola
Preprint 2014
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25. Renormalized volume and volume of the convex core

co-author: Richard Canary
Preprint 2015
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26. An introduction to pressure metrics on higher Teichmüller spaces
co-authors:  Richard Canary, Andrés Sambarino
Preprint 2015
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Preprints and Notes

The Poisson Bracket of Length functions in the Hitchin Component
Wolpert's cosine formula on Teichmüller space gives the Weil-Petersson Poisson bracket {lα,lβ} for geodesic length functions lα,lβ of closed curves α,β as the sum of the cosines of the angle of intersection of the associated geodesics. This was recently generalized to Hitchin representations by Labourie in the case of certain length functions of simple curves with a single point of intersection. In this paper, we give a complete generalization for all length functions and curves using Goldman's formula for the Poisson bracket on representation varieties of surface groups into reductive Lie groups.
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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".
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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

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