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May 22, Tuesday
12:00 – 13:00

In 1934 H. Whitney posed the following problem: Let $f$ be a function defined on a closed subset $E$ of $R^n$. How can we tell whether $f$ extends to a $C^m$-smooth function defined on all of $R^n$? We discuss different aspects of this classical problem including its interesting connections with Convex Geometry (Helly's theorem), Lipschitz selections of set-valued functions and Analysis on Riemannian manifolds. The main part of the talk will be addressed to the "finiteness principal" which states that the Whitney extension problem can be reduced to the same kind of the problem, but for finite sets with prescribed numbers of points. We will present several constructive criteria for restrictions of $C^2$-functions and Sobolev $W^1_p$ and $W^2_p$-functions to arbitrary closed subsets of $R^2$.