My research is in so-called Geometric Analysis with an emphasis on Geometric Function Theory and especially QuasiConformal Mappings and QuasiHyperbolic Geometry. Very recently I have jumped on the band wagon of "doing analysis in the general metric space setting''. I follow a program of applying geometric methods, using techniques from geometric measure theory, differential geometry and geometric topology, to investigate various problems in quasiconformal function theory. Of particular interest to me is the analysis of certain classes of spaces and mappings which preserve their structure. For example, I have studied uniform spaces for many years. I also investigate conformal metrics (such as the hyperbolic and quasihyperbolic ones, but others too) especially with an eye towards finding geometric estimates for such metrics. Much of my work involves well-known (quasi)conformal (quasi)invariants such as modulus of curve families and variational or conformal capacity.
In June 2003 I organized a workshop, Future Trends in Geometric Function Theory, which was held at the University of Jyväskylä in Finland immediately following the XIX Rolf Nevanlinna Colloquium. Here is a partial reading list for our RNC workshop.
My E-Talks
I just recently started giving these using beamer; here are my first four.
Metric Space Inversions and Quasihyperbolic Geometry presented July 14, 2006 in the Conference on Geometric Analysis and Applications at the University of Illinois, Urbana-Champaign, IL.
Metric Space Inversions presented September 29, 2006 in a colloquium at the Virginia Polytechnic University in Blacksburg, VA.
Quasiconvex Plane Domains presented October 21, 2006 in an AMS Special Session at the University of Cincinnati.
Euclidean Quasiconvexity presented March 16, 2007 in an AMS Special Session at Miami University.
QuasiHyperbolic Universal Convexity presented November 8, 2014 at Texas Tech University.
My Recent Papers
Below you can download some of my papers. Please note that these are preprint versions and may differ slightly from the published versions! These are a mix of postscript and pdf; if you'd prefer something different, please just ask! If you really want to, you can see my complete publication list. I would be happy to send you reprints of any of my papers.
Fractal inner chordarc disks (with Terry Sullivan), J.d'Analyse Math. 84 (2001) 173-205
Quasiextremal distance sets (with Shanshuang Yang), Analysis 23 (2003) 27-50
Mappings of finite distortion: gauge dimension of generalized quasicircles (with Pekka Koskela), Illinois Math. J., 47 No.4 (2003) 1243-1259
A Möbius invariant metric for regions on the Riemann sphere (with William Ma and David Minda), in Future Trends in Geometric Function Theory, Report 92, Univ. Jyvöskylä Dept. Math. Stat. (2003) 101-118
Conformal deformations of uniform Loewner spaces, Math. Proc. Cambridge Phil. Soc., 136 (2004) 325-360
Quasiconformal deformations and volume growth , Proc. London Math. Soc., (3) Vol. 92 (2006) 161-199.
Estimates for conformal metric ratios (with William Ma and David Minda), Computational Methods and Function Theory, Vol. 5, No. 2 (2005) 323-345.
Uniform domains and capacity (with Stephen Buckley), Israel Journal of Mathematical, accepted October 2005, 21 pages.
Uniform spaces and Gromov hyperbolicity, Proceedings International Workshop on Quasiconformal Mappings and their Applications, Forum d'Analyste, accepted January 2006, 39 pages.
Geodesics and curvature of Möbius invariant metrics (with Zair Ibragimov and David Minda), Rocky Mountain Journal of Mathematics, accepted February 2006, 20 pages.