Abstract: We will discuss a method of carrying out integration on moduli spaces defined from Fredholm systems. The main point is that we do away with the regularity assumptions that is generally used in such a setup to obtain smooth manifolds (or orbifolds). We will also discuss some applications for such an integration. Most of the constructions will be elementary and will be motivated from finite dimensional examples. This is based on works of Chen-Li-Tian and works in progress with Chen and Hyvrier.

Abstract: Iterating a map on the plane can produce complicated self-similar patterns resulting in fractal pictures. The map that produces the do-it-yourself fractal operates just on points with integer coordinates and is simple enough to be worked out by hand. I will use it to explain how self-similarity arises naturally in fractal pictures.

When our simple map is extended to operate on the whole plane, things become much more complicated. A bit of Calculus, a bit of Number Theory, and plenty of Linear Algebra will be needed to look at the question, "What Happens Eventually?"