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So I am trying to undesrstand what her work was about. It seems fields medal was for "the dynamics and geometry of Riemann surfaces and their moduli spaces" (

So I am trying to undesrstand what her work was about. It seems fields medal was for "the dynamics and geometry of Riemann surfaces and their moduli spaces" (my emphasis). What is dynamics to a geometer? Clearly it is important -- because it is important in physics. But I want to understand what mathematicians are getting excited about. We can talk about a family of geometrical objects foo(t) where the parameter t is a single real value, which stands in for time. But -- other than its relation to the physical world -- why is that $t$ parametrisation important? Why not complex parameters, or multiple parameters, or something completely different?

stonesixone 9y ago
My memory may be a bit fuzzy on this. But I believe one way the term "dynamics" is used in her work is in reference to the space of Riemann surfaces (or equivalently, the space of hyperbolic geometries on a surface). This space is called moduli space. Certain topological operations on the surface (e.g. twisting a "handle") give rise to a corresponding transformation on moduli space. So one of the things mathematicians in this area can study is the dynamics of such operations on moduli space. This means, for example, studying the behavior of these operations on moduli space, in the limit. Also, to a mathematician, "dynamics" can often simply mean studying the behavior of iterations of a single function from a space to itself. So the parameter "t" in this case is over the positive integers (or all integers if the function has an inverse).
sn9 9y ago
The Quanta piece linked early on in the OP gives a deeper description of her work and life. It's from the time of her Fields win.
dbranes 9y ago
Not gonna describe what dynamics is because that's a broad question best answered by some googling. Will comment on the more restricted question "why do we like 1-parameter families". 1-parameter families of objects usually arise from differential equations in geometric settings. Some diffrential geometric examples that one should be able to appreciate without needing motivation from physics: - Integral curves and 1-parameter flows generated by integrable vector fields. A important special case in dynamics is Hamiltonian diffeomorphisms on symplectic manifolds. - Ricci flow in Riemannian manifolds - Geodesics and Jacobi fields on Riemannian manifolds One plausible reason for why we like 1-parameter families is that they're just easier, than, say, 2-parameter families. This is true in more than one way - 1-parameter families tend to arise from easier differential equations. For example, compare a geodesic problem to a minimal surface problem. Conversely it's easier to write down natural PDE systems when you're only trying to evolve 1 time variable. - The domain is easier to understand. For simplicity let's say we're parametrizing things continuously with a connected, compact domain. Topologically there's just two such 1-manifolds, the circle and the closed interval. When we consider a family of object parametrized by, say, 2 parameters, the domain is a surface (which we at least know a classification). If you then consider 3 parameters, the domain is a 3-manifold (which we don't know how to classify). As an example, a simple "family of geometric objects" you can parametrize is just a family of points on a manifold. In dimension 1, this amounts to some classification of curves. Things that come from this are, for example, the fundamental group, rank 1 (say, singular) homology, morse-smale theory which counts curves between critical points, geodesics which are solutions to some length functional. All these things are fairly easy to understand given some standard education in differential/algebraic topology. Now if you consider the same problem in dimension 2, the first step is to understand the ways you can embed your domain into your target manifold. This is more or less a starting point for the theory of moduli spaces which is already non-trivial even in the genus 0 case. For example, under certain technical assumptions this can be done with Gromov-Witten theory. These are much more sophisticated issues.

stonesixone 9y ago

My memory may be a bit fuzzy on this. But I believe one way the term "dynamics" is used in her work is in reference to the space of Riemann surfaces (or equivalently, the space of hyperbolic geometries on a surface). This space is called moduli space. Certain topological operations on the surface (e.g. twisting a "handle") give rise to a corresponding transformation on moduli space. So one of the things mathematicians in this area can study is the dynamics of such operations on moduli space. This means, for example, studying the behavior of these operations on moduli space, in the limit. Also, to a mathematician, "dynamics" can often simply mean studying the behavior of iterations of a single function from a space to itself. So the parameter "t" in this case is over the positive integers (or all integers if the function has an inverse).

sn9 9y ago

The Quanta piece linked early on in the OP gives a deeper description of her work and life. It's from the time of her Fields win.

dbranes 9y ago

Not gonna describe what dynamics is because that's a broad question best answered by some googling. Will comment on the more restricted question "why do we like 1-parameter families". 1-parameter families of objects usually arise from differential equations in geometric settings. Some diffrential geometric examples that one should be able to appreciate without needing motivation from physics: - Integral curves and 1-parameter flows generated by integrable vector fields. A important special case in dynamics is Hamiltonian diffeomorphisms on symplectic manifolds. - Ricci flow in Riemannian manifolds - Geodesics and Jacobi fields on Riemannian manifolds One plausible reason for why we like 1-parameter families is that they're just easier, than, say, 2-parameter families. This is true in more than one way - 1-parameter families tend to arise from easier differential equations. For example, compare a geodesic problem to a minimal surface problem. Conversely it's easier to write down natural PDE systems when you're only trying to evolve 1 time variable. - The domain is easier to understand. For simplicity let's say we're parametrizing things continuously with a connected, compact domain. Topologically there's just two such 1-manifolds, the circle and the closed interval. When we consider a family of object parametrized by, say, 2 parameters, the domain is a surface (which we at least know a classification). If you then consider 3 parameters, the domain is a 3-manifold (which we don't know how to classify). As an example, a simple "family of geometric objects" you can parametrize is just a family of points on a manifold. In dimension 1, this amounts to some classification of curves. Things that come from this are, for example, the fundamental group, rank 1 (say, singular) homology, morse-smale theory which counts curves between critical points, geodesics which are solutions to some length functional. All these things are fairly easy to understand given some standard education in differential/algebraic topology. Now if you consider the same problem in dimension 2, the first step is to understand the ways you can embed your domain into your target manifold. This is more or less a starting point for the theory of moduli spaces which is already non-trivial even in the genus 0 case. For example, under certain technical assumptions this can be done with Gromov-Witten theory. These are much more sophisticated issues.