31–35 of 35 posts
Still bothers me that we didn't get a black bar. Can someone explain what the criterion is?
Likewise. A Fields medalist deserves better. In fact, the first female Fields medalist.
But then, we can see your tantrum over the matter above..
Earlier quoted context omitted.
Likewise. A Fields medalist deserves better. In fact, the first female Fields medalist.
If that's the only reason, then it's not good enough. But then, we can see your tantrum over the matter above..
That said, I understand how HN is not really a Math/CS forum, it's more about industry.
"this is about as good motivation as we'll ever have to keep plugging along." So, other people dying of cancer not as good motivation?
- Random wanderer leaping to the defence of a fellow human.
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…
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.
Earlier quoted context omitted.
If that's the only reason, then it's not good enough. But then, we can see your tantrum over the matter above..
I don't agree with gp's inflammatory tone, but do you not think a Fields medalist is not worth a black bar? I can buy not agreeing with tokenism, but her merit stands on its own imo. We lost a great mind; nobody's saying we should be in sack cloth and ashes but I think it's worth recognizing in some manner. That said, I understand how HN is not really a Math/CS forum, it's more about industry.