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Local Structure of Sets, Measures and Currents

Final Report Summary - LOCALSTRUCTURE (Local Structure of Sets, Measures and Currents)

The focus of the project has been on studying local behaviour of sets, functions and measure-theoretical objects with the view of establishing, or showing impossibility, of modern developments in the line of research started by the crucial results from the dawn of modern analysis such as Lebesgue's Density Theorem, Rademacher's Theorem on differentiability of Lipschitz functions almost everywhere and many similar results. The highlights include:

-- The breakthrough discovery of the structure of non-differentiability sets of Lipschitz self-mappings of Euclidean spaces, and the result of Csornyei and Jones that every Lebesgue null set admits this structure. These difficult results are being prepared for publication, but the special measure-theoretic case of the first result already appeared in GAFA, and in this case the second result may be obtained from the work of De Philippis and Rindler.

-- The breakthrough result (published in the Inventiones) that Rademacher's theorem is not sharp for Lipschitz mappings of Euclidean spaces to Euclidean spaces of lower dimensions. Research along the route indicated here was continued by Magnani, Speight and Pinnamonti who extended non-sharpness differentiability results and related results on structure of porous sets to the Heisenberg group and more general Carnot groups.

-- Uniqueness of tangent behaviour everywhere for any 1-dimensional stationary varifold on any Riemannian manifold, completing Kolar's result that in higher dimensional cases even rectifiable stationary varifolds may have non-unique tangent behaviour somewhere.

-- First results towards a sharp version of Rademacher's theorem for real-valued Lipschitz functions.

-- Results on one-dimensional calculus of variations move us close towards a complete understanding of the failure of regularity of minimal objects in this situation.

-- In the Banach spaces setting the main achievements include sharp results on Gateaux differentiability of cone-monotone functions, and further results on Gateaux differentiability, existence of extensions of vector-valued functions with preservation of points of continuity and derivatives, and recovering the compact space from the structure of the space of continuous functions on it.

-- Construction of a Gaussian measure in a Hilbert space for which the Density Theorem fails uniformly.

-- A number of further contributions to several directions of modern mathematical analysis.