The Cauchy Integral, Its Friends and Family: Analysis, Geometry and Combinatorics
by
Alexander Volberg
Michigan State University and University of Edinburgh

In 1733 count de Buffon asked the question: What is the probability for a needle of length L < 1 to intersect a grid of parallel lines on the plane having distance 1 between each other? In 1898 Paul Painlevé asked another question: How can one describe geometrically the compact sets on the plane such that the only functions analytic and bounded in the complement of these sets are constants? At the end of 20th century it became clear that these two questions are closely related. Moreover, they are closely related to a wide variety of problems, from percolation on graphs to electrostatics.

The key words here are "Calderón-Zygmund capacities".

Capacities with positive kernels, even their non-linear counterparts, are well understood. But, recently the focus has been on capacities with signed, complex or vector-valued kernels. It is usually quite difficult to prove that they are even "capacities". In particular, this was the essence of Vitushkin's question and Tolsa's answer about one of them, analytic capacity assigned to the Cauchy kernel on the complex plane. Tolsa's proof does not work in three dimensions, however the corresponding capacity does exist in three dimensions. It is related to the gradient of the fundamental solution for the Lapalace equation, this is an exact counterpart of Cauchy kernel on the plane. We explain the universal approach to proving subadditivity of such new capacities. We will mention also a pretty amazing connection between Calderón-Zygmund capacities and the usual (but non-linear) capacities.

Already Vitushkin explored a very enigmatic connection between analytic capacity and Geometric Measure Theory. He put forward the question about the "equivalence" of analytic capacity and the so-called Buffon needle probability. We will describe the status of this problem today. This will bring us naturally to a seemingly elementary problem of estimating the Buffon needle probability of a finite collection of disks located in a Cantor pattern. We will indicate the relation of this problem with percolation on graphs, tiling, the Besicovitch projection theorem, and the Kakeya problem.

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Date received: January 23, 2008