Some Unnested, Sprinkled Donuts

Great talk on Tuesday by Professor Anil Hirani on harmonic differential forms and generally bringing back things like geometry, topology, and algebra back into numerical analysis. Here's a sampling of what was talked about.

Harmonic differential forms are the basic topological obstructions to finding of solutions of Poisson's equation (∆u=f). Solutions to Poisson's equation represent steady-state phenomena (the scalar version representing heat, and the vector or differential form versions representing flows). On a closed surface, if we have a source term f, it must balance out (sum to zero over the whole surface), so that as much material is being created as it is being destroyed. Otherwise, of course, we could not have a steady state (more and more stuff just keeps getting added!). However, if the domain has some kind of topological nontriviality, such as holes (which could be represented as obstacles of some sort), different kinds of steady-state solutions can exist without sources. This is because stuff can flow "around" the holes, our "out of a cavity" (even in the idealized case that it's a single point and infintesimally small). But if the holes are not there, then it is often true that zero is the only field that satisfies the (reasonable) demand that the flow be continuous at that point. On other things besides donuts, we have planar domains with holes (lotus root slices), whose harmonic fields do vanish somewhere.

And here is an example of flow out of a cavity: 

We call this example the Koosh [Thanks to fellow math homie Adam for locating the slide with this... I didn't get a pic of it during the talk. He'll be one of our guest bloggers from time to time!]

The topological object that describes these fields more precisely is the de Rham cohomology of the space, equal to the "locally constant fields modulo the trivially constant ones" as described by Bott and Tu in their nice book Differential Forms in Algebraic Topology (Prof. Hirani recommends that if there's any class that one should take in graduate school, it's Algebraic Topology. I have my reservations about that, but he also said that we might not know what he is talking about when he says that!). In any case, there is a notion of closed form, which is a form which when integrated over a surface of the appropriate dimension, it is unchanged under small perturbations of the surface, holding its boundary fixed. Too large of a perturbation may cause the surface to cross over one of these topological nontrivialities, which will in fact cause a jump in the value of the integral. On the other hand, if the form is exact, it has what is known as a potential, which and the integral is unchanged regardless of the surface, so long as it has the same boundary, by virtue of Stokes's Theorem.

As abstract as this all seems, it has some concrete interpretations, for example, rankings of sports teams (or the Netflix Problem), where everybody ranks things by comparing two things, and we want to figure out some sort of global, overall ranking. Cohomology tells us the level of inconsistency one may achieve---one can get into situations where A is better than B is better than C is better than A. This also has applications in economics. In order to take advantage of all of this for applications, we obviously have to have a method of computing these things. Geometry allows us to use a very common, bread-and-butter tool of scientists everywhere: least squares. Finding a harmonic form is equivalent to finding a form within the same cohomology class of minimal norm. It is quite beautiful how all these techniques work well together!