##### Department of Mathematics,

University of California San Diego

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### Geometric Analysis Seminar

## Raanan Schul

#### UCLA

## Bi-Lipschitz decomposition of Lipschitz functions into a metric space.

##### Abstract:

We will outline the proof of a quantitative version of the following Sard type theorem. Given a Lipschitz function $f$ from the $k-$dimensional unit cube into a general metric space, one can decomposed $f$ into a finite number of Bi-Lipschitz functions $f|_{F_i}$ so that the $k-$Hausdorff content of $f([0, 1]^k \smallsetminus \cup F_i$) is small. The case where the metric space is $\mathbb{R}^d$ is a theorem of P. Jones (1988). This positively answers problem 11.13 in ``Fractured Fractals and Broken Dreams" by G. David and S. Semmes, or equivalently, question 9 from ``Thirty-three yes or no questions about mappings, measures, and metrics" by J. Heinonen and S. Semmes.

Sponsor: Kate Okikiolu

### February 3, 2009

### 1:00 PM

### AP&M 6402

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