So we have this thingummy, and we’re trying to find out more about it. What would you do? Shake it a little and see what happens, maybe? That’s one way you might describe the motivation for (or, at least, one motivation for) deformation theory. By figuring out how a geometric structure—a triangulation, say, or a foliation, or a smooth atlas, or a hyperbolic structure—can be perturbed, we learn more about the nature of the structure at hand.
Deformation spaces are not a precisely-defined concept; moduli spaces are slightly more precisely-defined, though not completely so, and in some sense a similar (though perhaps somewhat broader) idea.
One key example of a moduli space is the genus g Teichmüller space Teich(g): this is the space of all marked hyperbolic metrics on a genus g surface (the genus g surface, when we regard it as the unique topological object in its homeomorphism class), modulo homotopy.
Here “marked” indicates that points in Teich(g) correspond not just to a choice of hyperbolic metric on (described mathematically by e.g. an orientation-preserving homeomorphism from a “standard” / fixed surface with a metric which is declared to be an isometry), but also a choice of isomorphism to the fundamental group —not just hyperbolic clothing, but also instructions on how to wear it. The specification of a marking “rigidifies” and helps clarify the effect of automorphisms (i.e. non-trivial mapping classes) on our hyperbolic surface—something which is concern e.g. when we try to build a fine moduli space.
It is one of the great triumphs of 19th century mathematics that Teich(g) is also the space of all marked conformal structures on the genus g surface, modulo biholomorphisms homotopic to the identity. Any conformal structure yields a hyperbolic metric via uniformization; conversely, given a hyperbolic metric, we may produce a conformal structure using isothermal coordinates.
This dual identity allows Teichmüller theory—the theory which describes Teichmüller space and what it can do—to draw on tools from both hyperbolic geometry and complex analysis, and gives the theory much of its richness.
Setting out from the viewpoint of hyperbolic geometry, we may obtain, for instance, the Fenchel-Nielsen coordinates—which start by taking all-right hyperbolic hexagons, gluing these to get pairs of pants with specified cuff lengths, and then gluing pairs of pants with specified twists to get hyperbolic surfaces—and a proof of the Nielsen-Thurston classification, parts of which look at how lengths of curves on the hyperbolic surface change under the action of (various types of) mapping classes.
Setting out from the viewpoint of complex analysis, we obtain a systematic description of the tangent and cotangent spaces to Teichmüller space as spaces of various sorts of differential forms—Beltrami differentials and holomorphic quadratic differentials, respectively—, which in some sense encode representations of infinitesimal deformations of the metric structure—quasiconformal maps and transverse measured foliations, resp.
From there we may obtain results on optimal deformations between the various hyperbolic / conformal structures—this is the content of Teichmüller’s existence and uniqueness theorems, and the basis of Teichmüller geometry, i.e. the study of the geometry of Teich(g) as not just a collection of spaces, but a space in its own right.
I’ve written elsewhere, at some [mild] length, about moduli spaces in general and Teichmüller space in particular, including the motivation for these, so I should not continue on here. I did want, nevertheless, to emphasise the dual nature of the space and the tools involved—something I’ve come to appreciate more recently—, and to highlight the theory’s role as a base case and starting point for some of the subsequent areas I explore next: higher Teichmüller theory, and Weil-Petersson geometry.