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Quantitative Implicit Function and Extension Theorems for Lipschitz Maps. September 14, 2011 (11:30 AM PDT - 12:30 PM PDT)
Parent Program: --
Location: MSRI: Simons Auditorium
Speaker(s) Jonas Azzam (University of Edinburgh)
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We discuss recent work with Raanan Schul. ``All Lipschitz maps from $R^7$ to $R^3$ are orthogonal projections". This is of course quite false as stated. There is, however, a surprising grain of truth in this statement.

We show that all Lipschitz maps $f:R^{n+m}\rightarrow R^{n}$ can be precomposed with a bi-Lipschitz map $g:R^{n+m}\to R^{n+m}$ such that $f\circ g$ will satisfy, when we write the domain as $R^n\times R^m$ and restrict to a large subset $E$, that $f\circ g$ will be constant in the first coordinate and bi-Lipschitz in the second coordinate.
Geometrically speaking, the map $g$ distorts $R^7$ in a controlled manner, so that the fibers of $f$ are straightened out. Moreover, the target space can be replaced by any metric space.

Our results are quantitative. The size of the set $E$ is an important part of the discussion, and examples such as Kaufman\\'s 1979 construction of a singular map of $[0,1]^3$ onto $[0,1]^2$ are motivation for our estimates.

On route we will discuss an extension theorem which is used to construct the bi-Lipschitz map $g$. We show that for any $f:[0,1]^{n}\rightarrow R^{D}$ whose image has positive content, one may extend $f$ from a large subset of its domain to a global bi-Lipschitz map $F:R^{n}\rightarrow R^{D}$.

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