Geometric measure theory

Until now I have been thinking about various questions including the following.

• Regularity of sets, currents and varifolds. I am interested in interior and boundary regularity theories developed in geomertic measure theory. I have showed that the first stage (Lipschitz graph approximation) in the classical scheme of proving a regularity theorem (for instance for stationary varifolds (Allard) or minimal sets (Almgren)) depends only upon lower density ratio bounds and almost monotonicity of the density ratios. I am working on the next stages as well as on other questions related to the regularity and the structure of singularities of monotone measures and stationary varifolds.
• Existence for Plateau type problems (there is a beautiful gallery of images produced by John M. Sullivan at Urbana-Champaign, check the soap films section there). The powerful existence theory of H. Federer and W.H. Fleming, developed in 1960's for mass minimizing integral and rectifiable currents, doesn't yield all the surfaces that arise as soap films. In particular, singularities observed in soap films don't show in mass minimizing currents. Currents however (in the sense of De Rham) provide a useful boundary operator. So far, working in collaboration with R. Hardt has yield existence results of currents minimizing functionals somewhere between mass and size (introduced by Almgren) in case of smooth compact boundaries. This also yield us to use rectifiable scans to describe limits of minimzing sequences with unbounded mass. We are now studying whether or not size minimizers do exist. I am also sharing some dreams with G. David about proving existence of area minimizing sets in some classes of constraints modeling a boundary condition (we refer to the work of R.E. Reifenberg for homological conditions).
• Points of differentiablity of Lipschitz functions. Rademacher's theorem states that each Lipschitz function from Rⁿ to R^m is differentiable almost everywhere. This is not optimal: indeed David Preiss constructed a negligible set S in R² such that every Lipschitz function R² → R is differentiable in at least one point of S. He asked P. Huovinen and myself whether this result extends to general dimensions. The method certainly would possibly work only if m < n. In that case we showed that every Lipschitz function has a point of ε differentiability in a universal countable union of m dimensional affine subspaces. These questions are also heavily related to affine approximation schemes of Lipschitz functions defined on infinite dimensional spaces.
• Non absolutely convergent integrals. Lebesgue's integral doesn't integrate all derivates F' of one variable. There are several integrals in dimension one which extend Lebesgue's and do integrate derivatives. Riemann type definitions are due to R. Henstock and J. Kurzweil. Finding a Riemannian definition of a multi-dimensional integral for which the divergence theorem holds true under minimal hypotheses has been more difficult. The class of Henstock-Kurzweil integrable functions in several variables is not invariant under rigid motions, for instance. Also, the right domains of integration are now the sets of finite perimiter in the sense of De Giorgi. Successful integration theories were introduced by W.F. Pfeffer. I had the opportunity to work with Z. Buczolich and W.F. Pfeffer on determining the so-called multipliers (to give an analogy, L^∞ is the space of multipliers for L¹): we have proved that these are bounded BV functions (in the sense of De Giorgi). I also studied the dual of the space of BV functions with no Cantor part in the gradient, when endowed with its BV norm. It turned out that it is undecidable with ZFC whether or not there is an integral representation of the action on the jump part as the flux of a vectorfield through the jump set.