Her work also has some deep connections to theoretical physics.
So math gaf what does this mean?
Wonderful contribution. Give this guy a Pulitzer.
Yeah and the worst thing is that you basically can't become a scientist without being good at math.
This is why I don't think I could ever study pure mathematics at the graduate level. I really don't think my imagination is up to it. (And not seeing many other women do it plays into it. I hope Mirzakhani inspires young girls.) So I respect the hell out of people who do this work.
A surface is something that locally looks like a plane, but may have a different topology on a larger scale. For instance, both a sphere and a plane are surfaces. In both cases, when you look at a small patch of the surface, it is topologically equivalent to a plane. However, the sphere is different from the plane on the global scale (if you'll pardon the pun), because the local neighborhoods are "stitched together" in a different way. This image posted earlier:
ftfy
A surface is something that locally looks like a plane, but may have a different topology on a larger scale. For instance, both a sphere and a plane are surfaces. In both cases, when you look at a small patch of the surface, it is topologically equivalent to a plane. However, the sphere is different from the plane on the global scale (if you'll pardon the pun), because the local neighborhoods are "stitched together" in a different way. This image posted earlier:
is another example of a [Riemann] surface. This particular example most naturally lives in four-dimensional space; for rendering purposes, it has been projected down into our normal 3-dimensional space, which creates the appearance of self-intersection. However, the actual surface does not intersect itself. For if it did, then the neighborhood around the intersection would not look like a plane, and thus the object could not be a surface.
A Riemann surface is a complex manifold, which means that it is a surface equipped with additional algebraic structure that allows us to define holomorphic functions on it. These functions are important because, to put it briefly, we can do complex analysis ("calculus") on them.
A surface is something that locally looks like a plane, but may have a different topology on a larger scale. For instance, both a sphere and a plane are surfaces. In both cases, when you look at a small patch of the surface, it is topologically equivalent to a plane. However, the sphere is different from the plane on the global scale (if you'll pardon the pun), because the local neighborhoods are "stitched together" in a different way. This image posted earlier:
is another example of a [Riemann] surface. This particular example most naturally lives in four-dimensional space; for rendering purposes, it has been projected down into our normal 3-dimensional space, which creates the appearance of self-intersection. However, the actual surface does not intersect itself. For if it did, then the neighborhood around the intersection would not look like a plane, and thus the object could not be a surface.
A Riemann surface is a complex manifold, which means that it is a surface equipped with additional algebraic structure that allows us to define holomorphic functions on it. These functions are important because, to put it briefly, we can do complex analysis ("calculus") on them.