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Bill Allard : The Boundary Finder

(This abstract is in TeX source code. Sorry!) Fix a small positive number $h$. Let $$G=h\mathbb{Z}^2=\{(ih,jh):i,j\in\mathbb{Z}\};$$ thus $G$ is a rectangular grid of points in $\mathbb{R}^2$. Let $\Omega$ be an bounded open subset of $\mathbb{R}^2$ with $C^1$ boundary and let $E=\{x\in G:x\in\Omega\}$. {\bf Question One.} Given $E$ can one determine the length of $\partial\Omega$ to within $O(h)$? The answer to this question is ``yes'', provided $\Omega$ satisfies a certain natural ``thickness'' condition; without this additional assumption the answer may be ``no''. {\bf Question Two.} Is there a fast algorithm for determining the length of $\partial\Omega$. The answer to this question also ``yes''. In this talk I will describe the proof that the answer to Question One is ``yes'' and I will describe the fast algorithm whose existence is implied in the answer to Question Two. If time permits, I will describe some applications.

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