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How does Perelman proof involve elliptic curves? I thought it was about properties of the solutions to the Ricci flow equation, hence more of a difficult proble
by pathsjs 9y ago
How does Perelman proof involve elliptic curves? I thought it was about properties of the solutions to the Ricci flow equation, hence more of a difficult problem in PDE. I have no clue where elliptic curves would arise
- wayn3 9y ago"Hamilton created a list of possible singularities that could form but he was concerned that some singularities might lead to difficulties. He wanted to cut the manifold at the singularities and paste in caps, and then run the Ricci flow again, so he needed to understand the singularities and show that certain kinds of singularities do not occur. Perelman discovered the singularities were all very simple: essentially three-dimensional cylinders made out of spheres stretched out along a line. An ordinary cylinder is made by taking circles stretched along a line. Perelman proved this using something called the "Reduced Volume" which is closely related to an eigenvalue of a certain elliptic equation." "an eigenvalue of a certain elliptic equation" is probably an elliptic curve. I know very little about number theory. There are many techniques involved. What I know about the Poincare Conjecture comes from a story a professor told during one of his classes.
- pathsjs 9y agoI think it is an elliptic equation as in elliptic partial differential equation https://en.wikipedia.org/wiki/Elliptic_partial_differential_equation https://en.wikipedia.org/wiki/Elliptic_partial_differential_... (the simplest one being the Laplace equation, which is satisfied by harmonic functions). As far as I know, there is no direct link between elliptic PDEs and elliptic curves. Elliptic PDEs are called like that because the coefficient in the second order term look like those of the equation of an ellipse (as opposed to parabolic or hyperbolic PDEs) Elliptic curves take their name because they were first studied in the context of elliptic functions, which are certain integrals that arise in computing the arc length of an ellipse. Even then, the kind of elliptic curves used by number theorists are generalizations where the algebraic form of the curve is the same, but the field of coefficients is a finite extension of the rationals (while the first appearance was with complex numbers). There is a relation between the two, but in general the study of the curve over the complex field only tells part of the story. I am not aware of any more direct relations, but it would be fascinating to learn that there is something that relates them!