Friday, February 16, 2007

a tale of two spaces

Well, in some ways this week was spent studying three spaces, but two of them made their way into the title of the conference: outer space and Teichmüller space. Teichmüller space has a long history in the study of Riemann surfaces; outer space is much more recent, and the conference was organized in part to celebrate the 20th “birthday” of outer space: in 1986 the definition and fundamental properties of outer space were published by Culler and Vogtmann (the latter of whom is a professor at Cornell). At Thursday night’s bouillabaisse dinner, the two of them blew out the candles on a birthday cake. The third kind of space mentioned this week was symmetric spaces; the study of these goes back even further than Teichmüller space. Several talks described the fruitfulness of thinking of these three kinds of spaces jointly, to see what analogues and differences there are.

Part of the idea in the study of these three spaces is to link geometric and algebraic objects so that each can cast light on the other. Generally, the algebra comes from looking at symmetries in the geometry. Theorems arise which describe the restrictions one side puts on the other: that is, if your geometric object looks the same in a lot of places, then there are a lot of ways to move the points of the object around on itself while keeping the same structure, and vice versa. (Think of how a sphere can be rotated around any axis through its center by any amount to give the same sphere, because it looks exactly the same at every point. A cube, while it has lots of pretty symmetries, is less symmetric than the sphere, which you can see by the severe restrictions on which axes you can rotate around and by how much to leave the cube looking as it did before. A highly irregular object looks different no matter how much you rotate it in any direction; this is called having a trivial symmetry group, which is an algebraic way of describing a geometric property.) The basic geometric objects in Teichmüller theory are surfaces with metrics, i.e., ways of measuring distances on the surfaces. The distances can make little patches on the surface look either flat or curvy (for an explanation of what “curvy” means here, see Tim’s entry about bugs on a hotplate). The symmetries of such a surface can be described roughly (almost always) as ways to cut the surface up and glue it back together so that it looks the same. Outer space looks at graphs, which in this context means not plotting a function or the year’s earnings, but taking points and connecting them with edges of various lengths. (A common example here is a single point with several curves that leave and come back to it; this is called a rose.) The symmetries come from ways of exchanging pieces of the graphs. But whereas Teichmüller space interests itself mainly with the geometry of the surfaces, outer space was created specifically to study the ways of moving the graph around—that is, the focus is on the algebraic aspect. This means the study of outer space falls under “geometric group theory,” using the geometry of an object to glean algebraic information about the associated symmetries. (Indeed, it is called “outer space” because the ways of moving things around form the outer automorphism group, an algebraic object that was studied long before the introduction of outer space.)

This was a fantastic week to be in Marseille (or more properly, in Luminy where the conference center is). For those interested in the conference webpage itself, you can find it (containing a description of the week, a list of participants, and summaries of the talks) here. I could throw around the names of some of the top-notch mathematicians who were here, but everyone who cares probably already knows. I made and strengthened a number of good contacts, and got some good references and discussion for the projects I’m currently working on. Congratulations and thanks to the organizers, and to all the speakers, and to my colleagues with whom I met. I look forward to continuing in this field, at the intersection of so many interesting parts of mathematics and drawing on the skills of so many generous and talented mathematicians, for many years.

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