Analysis and Geometric Measure Theory

The workshop will be dedicated to problems where there is strong interplay between analysis (in particular harmonic analysis and complex analysis) and geometric measure theory (in particular rectifiability and variational methods).

Topics to be covered include:

1. Analytic capacity and rectifiability
   The classical Painleve problem consists in finding a geometric characterization for compact sets of the complex plane which are removable for bounded analytic functions. The methods used to study this problem come from complex analysis (analytic capacity), harmonic analysis (Cauchy singular integral operator) and geometric measure theory (rectifiability). In 1998, G. David solved the Vitushkin conjecture which provides an answer to Painleve's question for sets with finite $1$-dimensional Hausdorff measure. His work relied on the ideas of many mathematicians among others M. Christ, P. Jones, P. Mattila, M. Melnikov and J. Verdera.
   Recently, X. Tolsa proposed a solution for the Painleve problem in terms of Menger curvature.

   Problems to be discussed during the workshop include:

   - Discussion of Tolsa's conditions;

   - Bilipschitz invariance of the class of removable sets for bounded analytic functions in the complex plane;

   - Relationship between analytic capacity and Favard length;

   - Harmonic analysis in nonhomogeneous spaces;

   - The higher dimensional case, namely the study of removable sets for Lipschitz harmonic functions in R^n (the main problem is that although there exist analogs of the Menger curvature for n-1 -dimensional sets, they are not adapted to the study of the Riesz transforms. Hence, only a few basic things about this problem are known).

2. Analysis and rectifiability in singular metric spaces
   Partially motivated by questions arising in classical differential geometry, several authors have begun developing theories of analysis and rectifiability in metric spaces. To this effect basic tools of geometric function theory, for example Poincare inequalities or quasi-conformal mappings, have been introduced and studied in general metric spaces. Counterparts to the classical theorems in Euclidean spaces have been proved in metric spaces with bounded geometry. For instance, in 1999 J. Cheeger proved a version of Rademacher's theorem on the differentiability of Lipschitz functions on metric spaces where Poincare inequalities hold. The tools from non-smooth analysis play a crucial role in understanding limiting phenomena arising from smooth geometry.

   Problems to be discussed during the workshop include:

   - Geometric analysis (Poincare inequalities, Sobolev spaces, ..) and applications to PDE and geometry;

   - Basic tools of geometric measure theory (Sets of finite perimeter, area and co-area formulas, ..) in metric spaces;

   - The Kakeya problem;

   - Definitions of rectifiability in metric spaces (for instance, in Carnot groups).

3. Mumford-Shah functional
   This functional was introduced in connection with image segmentation. Let Omega be a bounded domain in the plane and let g be a bounded function on Omega. The Mumford-Shah functional is given by

   $$J(u,K) = \int \int_{\Omega \setminus K} |u - g|^{2} + \int \int_{\Omega \setminus K} |\nabla u|^{2} + H^{1}(K).$$

   The existence of minimizers (u,K) (in a reasonable sense) is known, but the main problem consists in studying the geometric properties of the set of singularities K. The Mumford-Shah conjecture states that K should be the finite union of C^1 arcs. Recent progress have been made by G. David, A. Bonnet, L. Ambrosio, S. Solimini, N. Fusco among others, but the conjecture is still open.
   The study of the Mumford-Shah functional in higher dimensions is a vibrant new question which seems to be related to the theory of minimal surfaces.

   Problems to be discussed during the workshop include:

   - Complete classification of the global minimizers of the 2-dimensional Mumford-Shah functional;

   - Study of cracktips (C^1 regularity, calibration,... ) for the 2-dimensional Mumford-Shah functional;

   - Study of the 3-dimensional Mumford-Shah functional, in particular connexions with minimal surfaces, complete classification of global minimizers.

   Confirmed Participants

   Programme (PDF)

   Videos

   Final Report (PDF)