interesting thing I learned today

Posted by Haelfix on September 16, 2003 at 22:21:16:

This is probably old hat to a few of you, but today I learned a neat little mathematical fact.

We were studying a case of a group that is both algebraic and lie.

Turns out that there are things known as nonsingular cubic curves. Elliptic curves are closely related, (they are defined over the same field modulo points that are not in the 'model')

In the case of real numbers. The cool thing is, by drawing lines through two points, intersecting with the third.. and drawing the line that connects that point to an arbitrary one, one gets a group action. Turns out its identical for all fields.

Now, in complex space. One can build an isomorphism from doubly periodic elliptical functions and trace out into a pentagon. Thats where the lie connection comes in.

By identifying sides of the Pentagon, we have a manifold, the torus of dimension (2n).

Hence the family of eliptic curves has a morphism into toruses. This is actually important in string theory and other modern physics, as one has an algebraic way of dealing with manifolds that are closely related to toruses.

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