Nahum Zobin (W&M)
Whitney Problems: old and new ideas
In 1934 Hassler Whitney published three extremely influential papers all dealing with versions of the following problem: given an arbitrary subset $E\subset \R^n$ and a function $f: E\rightarrow \R$ (or a string of functions $f^\alpha: E \rightarrow \R, \alpha \in \Z_+^n, |\alpha|\le k$) determine whether there exists a function $F\in C^m(\R^n)$ such that $F|_E = f$ (or $\forall\, \alpha \in \Z_+^n, |\alpha|\le k \le m \quad F^{(\alpha)}|_E = f^\alpha$). Whitney proposed ingenious constructions which led to very effective answers in the following cases: (i) $k = m$, (ii) $k = 0, n = 1.$ Also in the case of $E$ being an open bounded domain in $\R^n,$ Whitney proved that the restrictions of functions from $C^m(\R^n)$ to $E$ give the whole space $C^m(E)$ if the domain $E$ is quasi-convex (i.e., the geodesic metric on $E$ is equivalent to the Euclidean metric). There have been a lot of activity in the area, especially since mid-90s. In 1998 I was able to completely resolve an old question about the necessity of quasi-convexity in the last mentioned Whitney result. But the most striking developments came in the early 2000s, when C. Fefferman gave very satisfactory answers to the remaining Whitney problems. In recent years the interest has shifted to Sobolev spaces, and there are some very important developments in this area. I will explain a "quantum" version of the Whitney problems, that I have proposed recently, and discuss related ideas.