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 Teichm├╝ller 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.


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 goedesic flow on a hyperbolic manifold, Geometry and Topology, Vol. 18, No. 1, 2014
co-author: Ser Peow Tan
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21. The pressure metric for convex representations
co-authors: Richard Canary, Francois Labourie, Andres Sambarino
Preprint 2013
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22. Identities on Hyperbolic Manifolds, The Handbook of Teichmuller Theory, Vol. 5, to appear
co-author: Ser Peow Tan
Preprint 2013
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23. An improved bound for Sullivan's convex hull theorem,
co-authors: Richard Canary, Andrew Yarmola
Preprint 2014
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24. Higher derivatives of length functions along earthquake deformations, Michigan Math Journal, to appear
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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