Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.

Quasiconformal Maps

Ok, time to get back to Riemann surfaces! We’ve been all about hyperbolic surfaces, and so first let’s compare the two of them: every oriented hyperbolic surface is a Riemann surface, and the conformal class of a complex structure contains … Continue reading

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We’re going to get to quasi-conformal maps soon, but first, we’re going to want to build some new objects on our Riemann surfaces.  Differential forms are a fairly standard thing, and asserting that they’re holomorphic isn’t exactly a revolution.  On … Continue reading

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Mapping Class Group Elements

Let’s get into the mapping class group and talk a bit about its elements and its structure.  I’m going to omit proofs, because I can’t beat Minsky’s exposition, and this is just some flavor and definitions, most of which won’t … Continue reading

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The Mapping Class Group

Last time, we touched on the mapping class group, so now, we’re going to dig in. Now, we’re not going to dig too deeply, there’s a LOT here (see the wonderful book by Farb and Margalit for a hint at … Continue reading

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Fenchel-Nielsen Coordinates

Welcome back, and hope all you readers had a good 2014 and particularly good holidays and new year’s celebrations, if you do those things.  Today, we’re going to keep on the road to producing the moduli space of curves, by … Continue reading

Hyperbolic Surfaces

Ok, with the hyperbolic plane and its metric and geodesics out of the way, we can start getting into some surface theory.

The hyperbolic plane

So, I know I usually talk about strictly algebraic geometry stuff, but the moduli of curves lives in an interesting place.  It’s both an algebraic and an analytic object.  So we’re going to start by talking a bit about hyperbolic … Continue reading

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New Series: Moduli of Riemann Surfaces

Though I’m not quite ready to start (next week!) I feel that, in the spirit of Jim getting back to the blogging and my continued promises, I’d announce my series now.  I’m going to start a detailed series on the … Continue reading