Upcoming/recent talks.

• Iberoamerican Congress on Geometry, Pucón, Chile, 10-13 December 2010
• Special session in Geometric group theory, AMS sectional meeting,
  Albuquerque, New Mexico, 17-18 April 2010.
• Grad student Topology and Geometry conference (open problems session),
  Ann Arbor, 10-11 April 2010.
• Special session on Spectral problems on compact Riemannian manifolds, Joint Meetings, San Francisco, January 2010.
• New directions in geometric group theory, Brisbane, Australia, 14-18 December 2009.
• Topology seminar, Michigan State, 20 November 2009.
• Topology seminar, Louisiana State, 13 November 2009.
• Midwest Dynamics Seminar, University of Illinois at Chicago, 6-8 November 2009.
• Joint Dynamics/Geometry seminar, University of Maryland, 8 October 2009.
• XXIst Rolf Nevanlinna Colloquium, Kyoto, Japan, 7-11 September 2009.
• Hebrew University, Jerusalem, 30 June 2009; the Technion, Haifa, 2 July 2009.
• "Tour de France", Spring 2009: Rennes (11 June), Orleans (2 June), Lille (29 May), Marseille (18 May), Strasbourg (6 May), Orsay (10 April)

Papers, preprints, projects.

• Thin triangles and a multiplicative ergodic theorem for Teichmuller geometry, submitted. (math.GT/0508046)

  We show that the Teichmuller metric satisfies a four-point curvature condition which provides a version of large-scale negative curvature. Using this, we derive a multiplicative ergodic theorem, or "ray approximation," for the random action of the mapping class group: we show that almost every sample path is tracked sublinearly by a geodesic.

• Curvature, stretchiness, and dynamics, In the Tradition of Ahlfors and Bers IV, Contemp. Math. 432 (2007).

  Here, we present the definition of stretchiness, a four-point curvature condition used in a previous work for the study of Teichmuller space, and elaborate some of its properties and applications. We provide computations in some Hilbert geometries and in certain Banach spaces to illustrate that stretchiness is well-suited to the study of non-Riemannian metric spaces.

• Divergence of geodesics in Teichmuller space and the mapping class group (with Kasra Rafi), GAFA, Volume 19, Issue 3 (2009), 722--742. (PDF-- newer than ArXiv)

  We show that both Teichmuller space (with the Teichmuller metric) and the mapping class group (with a word metric) have geodesic divergence that is intermediate between the linear rate of flat spaces and the exponential rate of hyperbolic spaces. For every two geodesic rays in Teichmuller space, we find that their divergence is at most quadratic. Furthermore, this estimate is shown to be sharp via examples of pairs of rays with exactly quadratic divergence. The same statements are true for geodesic rays in the mapping class group. We explicitly describe efficient paths "near infinity" in both spaces.

• Stars at infinity in Teichmuller space (with Joseph Maher), in deep freeze.

  We study Karlsson's conjecture relating the intersection patterns of metric half-spaces in Teichmuller space to the intersection numbers of foliations in PMF.

• The length spectrum of a flat metric (with Chris Leininger and Kasra Rafi), submitted.
  (math.GT/0907.2082)

  We consider the class of flat metrics on surfaces (those induced by quadratic differentials) with respect to the marked length spectrum of simple closed curves. We show that a set of curves is "spectrally rigid" if and only if the set is dense in PMF. In particular, knowing the lengths of all simple closed curves is enough to determine a flat metric, but no finite set of curves suffices. We construct an embedding of Flat(S) into the space of geodesic currents on S and build a boundary for Flat(S) consisting of "mixed structures": part flat metric, part foliation.

• Higher divergence functions (with Aaron Abrams, Noel Brady, Pallavi Dani, Anne Thomas, and Robert Young), in preparation.
• Sprawl in groups and metric spaces (with Christopher Mooney and Ralf Spatzier),
  in progress. (appendix by Duchin-Lelievre)

Other research activity.