Internet Analysis Seminar

How one can discern order in a seemingly very disordered set? If the set in question has some self-similarity then a dynamical systems approach can be of use. But suppose there is no a priori structure. We consider one such situations when the full a priori knowledge about the set is the following: 1) its Hausdorff dimension is given, and we know that the Hausdorff measure in this dimension is (positive) and finite, 2) the set is a singularity set of a non-constant Lipschitz function satisfying some (fractional) Laplace equation. Or, instead of 2) one can say that singular integrals from a small collection (e.g., Riesz transforms) are bounded in $L^2$ with respect to Hausdorff measure. Or, instead of 2) one can say that a certain Calderón-Zygmund capacity of the set is positive. Then what geometry, if any, is imposed on the set by these conditions? It turns out (or conjectured) that automatically we can "connect" points from a non-trivial part of the set by a smooth manifold. In other words, the points of such a set should "feel" each others presence in a very quantitative and geometric way. This multi-dimensional analytic traveling salesman problem is the subject of the lectures. This is because several (but not all) such problems were recently solved, and they turned out to be entangling PDE, Harmonic Analysis and Geometric Measure Theory into one knot.

First, there is a particular (but very interesting and important) family of problems on the plane. These are problems posed by Painlevé, Denjoy, Ahlfors, Vitushkin, and were solved in the last 12 years by the efforts of a large group of mathematicians. Sets of finite Hausdorff measure $\mathcal{H}^1$ and positive analytic capacity $\gamma$ on the plane must contain a subset of positive $\mathcal{H}^1$-measure of a rectifiable curve. This "analysis-to-geometry" statement was known as "Denjoy's problem" and was solved almost simultaneously, and by different methods by David-Mattila-Léger and Nazarov-Treil-Volberg.

The higher dimensional analogue of this question is a very interesting area of current research, and is the following: Is it true that the sets of finite Hausdorff measure $\mathcal{H}^m$, $1\le m\le d$, $m$ an integer, and positive $\gamma(m, d)$-capacity must contain a non-trivial $m$-rectifiable subset? This is known as the David-Semmes problem and is completely analogous to Denjoy's problem in dimension greater than $2$. Unfortunately, in higher dimensions the main geometric tool, called Menger's curvature, is "cruelly missing". The topic of the Internet Analysis Seminar this year will focus on the machinery necessary to understand the David-Semmes problem and the recent work of Nazarov, Tolsa and Volberg in the co-dimension one case. The lectures will touch upon the themes connecting analysis (singular integral operators, operator capacity) with geometry (geometric measure theory).

During Spring 2014 the lectures for the Internet Analysis Seminar will be prepared by Professor Alexander Volberg.