Category Archives: Complex analysis

Slightly elevated Teichmuller theory

Last week at my invitation, David Dumas spoke in the U Chicago geometry seminar and gave a wonderful introductory talk on the theory of convex real projective structures on surfaces. This is the first step on the road to what is colloquially known … Continue reading

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Roots, Schottky semigroups, and Bandt’s Conjecture

It has been a busy quarter. Since August, I have made 10 trips, to conferences or to give colloquia. On 8 out of the 10 trips, I talked about a recent joint project with Sarah Koch and Alden Walker, on a topic in … Continue reading

Posted in Complex analysis, Dynamics, Hyperbolic geometry, Number theory | Tagged , , , , , , , , , , , , , | 1 Comment

Dipoles and Pixie Dust

The purpose of this blog post is to give a short, constructive, computation-free proof of the following theorem: Theorem: Every compact subset of the Riemann sphere can be arbitrarily closely approximated (in the Hausdorff metric) by the Julia set of … Continue reading

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Groups quasi-isometric to planes

I was saddened to hear the news that Geoff Mess recently passed away, just a few days short of his 54th birthday. I first met Geoff as a beginning graduate student at Berkeley, in 1995; in fact, I believe he gave the … Continue reading

Posted in 3-manifolds, Complex analysis, Groups, Hyperbolic geometry, Uncategorized | Tagged , , , , , | 2 Comments

Kähler manifolds and groups, part 2

In this post I hope to start talking in a bit more depth about the global geometry of compact Kähler manifolds and their covers. Basic references for much of this post are the book Fundamental groups of compact Kähler manifolds by … Continue reading

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Kähler manifolds and groups, part 1

One of the nice things about living in Hyde Park is the proximity to the University of Chicago. Consequently, over the summer I came in to the department from time to time to work in my office, where I have … Continue reading

Posted in Algebraic Geometry, Complex analysis, Geometric structures, Lie groups | Tagged , , , , | 8 Comments

Liouville illiouminated

A couple of weeks ago, my student Yan Mary He presented a nice proof of Liouville’s theorem to me during our weekly meeting. The proof was the one from Benedetti-Petronio’s Lectures on Hyperbolic Geometry, which in my book gets lots … Continue reading

Posted in Complex analysis, Euclidean Geometry, Rigidity | Tagged , , , | 7 Comments