- John D. MacArthur Professor of Mathematics, Caltech, USA (from 2013 to 2020)
- Sadleirian Professor, University of Cambridge (from 2013 to 2014)
- Professor of Mathematics, Caltech, USA (from 2011 to 2013)
- Professor of Mathematics, University of Warwick (from 2006 to 2011)
- Lecturer/Reader, University of Warwick (from 2002 to 2006)
- Low dimensional geometry, topology and dynamics
- Functional and geometric analysis
- Harmonic Maps and the Schoen Conjecture. Journal of American Mathematical Society 30 (2017), no. 3, 799-817
- The good pants homology and a proof of the Ehrenpreis conjecture. Annals of Mathematics (2) 182, No. 1, 1-72 (2015) (with J. Kahn)
- Harmonic maps between hyperbolic 3-spaces. Inventiones Mathematicae. 199, No. 3, 921-951 (2015)
- Immersing almost geodesic surfaces in a closed hyperbolic 3-manifold. Annals of Mathematics (2) 175 (2012), no. 3, 1127–1190. (with J. Kahn)
- Realization of the Mapping class group by homeomorphisms. Inventiones Mathematicae 168, no. 3, 523–566. (2007)
- Quasisymmetric groups. Journal of American Mathematical Society 19, no. 3, 673–715 (2006)
- The logarithmic spiral: A counterexample to the K=2 conjecture. Annals of Mathematics 161, no. 2, 925–957 (2005) (with D. Epstein)
- Quasiconformal homeomorphisms and the convex hull boundary. Annals of Mathematics 159, no. 1, 305–336 (2004) (with D. Epstein, A. Marden)
- Simons Investigator Award (2016).
- ICM Invited Speaker, Korea (2014).
- Elected Fellow of the Royal Society (2014).
- Research Merit Award, Royal Society (2014).
- Clay Research Award, awarded by Clay Institute (2012).
- Whitehead Prize, awarded by LMS (2004).
- Leverhulme Prize, awarded by Leverhulme Trust (2004).
- Distinguished Lectures: Distinguished Lectures at University of Indiana, 2020; Namboodiri Lectures at University of Chicago, 2019; Ordway Lectures at University of Minnesota, 2013.