this is your brain on drugs

zeroth: dimensions. a point has dimension 0. a line has dimension 1. a plane has dimension 2. the space we move around in has dimension 3. if you throw time into it you get a 4th dimension. you can do this all day. n-dimensions is just a space with n independent linear coordinates. if you think about the line as being all real numbers, and the plane as a pair of real numbers, then real n-space is just an n-tuple of real numbers.

first: a manifold (of dimension n) is built by "gluing" pieces of the standard n-space together in a nice way. (the maps mujst be invertible on the overlaps and the "type" of the manifold depends on how restrictive a condition you place on the overlap functions, also known as the transition functions.) each piece of n-space is called a coordinate patch and the entire collection of patches and transition functions is called an atlas. just like an atlas for the globe or north america.

examples: ignore the functions and just think about imaginary sheets of rubber. if you take an balloon and cut it in half you get two disks tha you can flatten into, well, flat disks. if we glue them back together along the edge niceley we get a balloon back. we can do this more generally. think about a rubber square. if we glue the left edge to the right edge you get a tube (cylinder). if we glue the top and bottom circles of the cylinders together, we get a torus (the surface of a donut). we can do this more generally and get all surfaces: the surface of some number of donuts that have been glued together. the number of donuts you need is galled the genus of the surface. another way to create this is to make an n-holed donut. the surface of this has genus n.

second: calculus. calculus lends the idea of a tangent line to the graph of a function. manifolds have tangent bundles that allow you to compute derivates. they also allow you to do geometry. at each point of a manifold, you have a vector space. these fit together nicely as you move through the manifold. a geometry gives you an additional structure to the vector space. (an inner product on the vector spaces (that changes in a nice way when you change points) gives you a Riemannian geometry.) if your inner products are all skew symmetric then the geometry is Symplectic. Riemannian geometry is rigid. our day to day lives follow such a rigid geometry. Symplectic geometry is flexible. you can wiggle things without changing their "shape." this does mean, though, that they are more useful for topological applications. a Calabi-Yau manifold is a 6-dimensional manifold that has a special symplectic geometry on it. string theory is currently acting on the assumption that the 10-dimensional universe is built on 4-dimensional spacetime and a 6-dimensional Calabi-Yau manifold. a symplectic geometry on a manifold is determined by a closed, nondegenerate 2-form, omega. this 2-form is a skew-symmetric pairing on vector fields, and gives a duality isomorphism from differential forms to vector fields. the derivative of a real-valued function is a differential form and it's symplectic dual is a Hamiltonian vector field. it's not (too) hard to see that the flow along the Hamiltonian preserves the symplectic structure (by the nature of the duality isomorphism).

lastly (for now): a Poisson structure (on a manifold) is simply a bilinear pairing on smooth functions {,} (on a manifold) satisfying a leibnitz property for products: {f,gh}=g{f,h}+h{f,g}. a symplectic structure gives such a Poisson structure by pairing the two Hamiltonians.

i'm also interested in naked people and ice wizards . i'm very very sad that amn and loser's lounge are being cancelled. i may never be happy again.