I am a professor in the
Mathematics, Statistics, and Computer Science
the University of Illinois at
My research concerns the geometry of low-dimensional manifolds, the deformation spaces of locally homogeneous geometric structures on such manifolds, and the use of analytic techniques in these areas.
Computer experiments and visualizations are an important component of
I am on sabbatical for the academic year 2019-2020.
Links go to PDF preprints. The published versions may be available online if your institution or organization has a subscription to the journal.
- Asymptotics of Hitchin's metric on the Hitchin section (with Andrew Neitzke). Communications in Mathematical Physics, 2018. doi:10.1007/s00220-018-3216-7
- Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations (with Andrew Sanders). To appear in Geometry & Topology. Status updated Oct 2019.
- Coarse and fine geometry of the Thurston metric (with Anna Lenzhen, Kasra Rafi, and Jing Tao). To appear in Forum of Mathematics, Sigma. Status updated Jan 2020.
- Holonomy limits of complex projective structures.
Advances in Mathematics 315 (2017), 427-473.
- Polynomial cubic differentials and convex polygons in the projective plane (with Michael Wolf).
Geometric and Functional Analysis 25 (2015), no. 6, 1734-1798.
- Skinning maps are finite-to-one.
Acta Mathematica 215 (2015), no. 1, 55-126.
- Grafting lines fellow travel Teichmüller geodesics (with Young-Eun Choi and Kasra Rafi).
International Mathematics Research Notices 2012 (2012), 2445-2492.
- Bers slices are Zariski dense (with Autumn Kent).
Journal of Topology 2 (2009), no. 2, 373-379.
- Survey: Complex projective structures.
In Handbook of Teichmüller Theory, Volume II. Ed. Athanase Papadopoulos. EMS, 2009.
- Slicing, skinning, and grafting (with Autumn Kent).
American Journal of Mathematics 131 (2009), no. 5, 1419-1429.
- Projective structures, grafting, and measured laminations (with Michael Wolf).
Geometry & Topology 12 (2008), no. 1, 351-386.
- Distribution of intersection lengths of a random geodesic with a geodesic lamination (with Martin Bridgeman).
Ergodic Theory and Dynamical Systems 27 (2007), no. 4, 1055-1072.
- The Schwarzian derivative and measured laminations on Riemann surfaces.
Duke Mathematical Journal 140 (2007), no. 2, 203-243.
- Grafting, pruning, and the antipodal Map on Measured Laminations.
The PDF linked here is an updated version of the paper published in J. Diff. Geom. 74 (2006), 93-118. It includes the correction that is the subject of this erratum.
Journal link: article / erratum
- Precision studies of the nonabelian Hodge and Riemann-Hilbert correspondences (with Andrew Neitzke).
- Anosov representations, locally homogeneous complex manifolds, and deformation theory (with Andrew Sanders).
- Rays of conformally thick convex real projective surfaces (with Michael Wolf).
- Experiments with skinning maps (with Autumn Kent).
Upcoming conferences and other events that I am helping to organize:
- Many years ago, I spent a lot of time making digital electronic circuits with 555 timers and other ICs of a similar vintage. If you chain together two missing pulse detectors (from Forrest Mims' classic "Engineer's Mini-Notebook: 555 Timer IC Circuits") you get a simple PFM decoder!
This material is based upon work supported by the National Science
Foundation. Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the author and do not
necessarily reflect the views of the National Science Foundation.