Author Archive

Dunkl Operators

December 15, 2007

This post is basically a write-up of notes for a talk I gave for the Olivetti club, the weekly Cornell grad student talk (of course, the post has pretty much everything I wanted to say, while my talk unsurprisingly did not). In 1989 Dunkl introduced a commuting family of operators which is a deformation of the family of directional derivative operators. More specifically, given a finite reflection group W acting on a vector space V, the choice of one complex parameter for each conjugacy class in W determines a family of commuting linear operators on $\mathbb{C}[V]$, and if all the parameters are chosen to be 0 then this family is just the family of directional derivative operators. For almost all parameter choices, this family is surprisingly well-behaved, and many constructions involving directional derivatives can be extended, including the Fourier transform, the heat equation, and the wave equation. As far as I can tell, these operators are pretty mysterious. From looking at the formula, it’s not even obvious they should commute, and further properties are even more surprising. In this post I’ll tell you some of the surprising things about them, but I unfortunately won’t be able to say much about why they exist.
(more…)

G-equivariant embeddings of manifolds

October 24, 2007

This is my first post, and I plan on sporadically writing some in the future. I’m Peter, a third year grad student at Cornell, and I talk to Greg pretty often, so I thought I’d write down some of the things I say. This first post won’t be long or deep, but it’s kind of cute, and the trick behind it is useful in many other situations, so I decided to share it. Let’s say we have a finite group $G$ acting on a compact manifold $M$. The Whitney embedding theorem says that we can embed $M$ into $\mathbb{R}^k$ for sufficiently large $k$, and what I want to show in this post is that you can do this in a $G$-equivariant way, i.e. there is an embedding $\phi:M \to \mathbb{R}^k$ and an injective homomorphism $f:G \to O(\mathbb{R}^k)$ such that $\phi(g\cdot x) = f(g)\cdot \phi(x)$. I guess the moral of the story is that compact manifolds are really just “nice” subsets of Euclidean space, and a compact manifold with a finite group action is really nothing but a “nice” subset of Euclidean space that is preserved by the action of a finite group of permutation matrices. Also, if one were to summarize the moral of the trick used, one might say “averaging over the elements of a group makes things equivariant,” and this idea comes up almost uncountably many times in many areas of mathematics (of course, averaging takes on different meanings in different situations).