I am a professor of
Mathematics, Statistics, and Computer Science
the University of Illinois at Chicago
My primary research areas are differential geometry, geometric topology, and complex analysis.
Most of my research projects focus on moduli spaces of locally homogeneous geometric structures on manifolds, often in cases where complex-analytic techniques can be applied.
I use computer experiments in my research, and I pursue mathematical visualization projects for both research and expository purposes.
Titles link to PDF preprints, DOI links go to published versions.
- Uniformization of compact complex manifolds by Anosov homomorphisms (with Andrew Sanders). To appear in Geometric and Functional Analysis.
- Opers and nonabelian Hodge: numerical studies (with Andrew Neitzke). Preprint, 2020.
- Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations (with Andrew Sanders). Geometry & Topology 24 (2020), no. 4, 1615-1693.
- Coarse and fine geometry of the Thurston metric (with Anna Lenzhen, Kasra Rafi, and Jing Tao).
Forum of Mathematics, Sigma 8 (2020), e28.
- Asymptotics of Hitchin's metric on the Hitchin section (with Andrew Neitzke). Communications in Mathematical Physics 367 (2019), 127-150.
- 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
Mathematical Visualization Projects
Lecture videos and slides
Most of my research presentations are chalk/whiteboard talks; these are some exceptions:
- Math at UIC (2008-)
- Math at Brown (2005-2008, Tamarkin Assistant Professor and NSF Postdoc)
- Math at Rice (2004-2005, NSF Postdoc)
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.