Differentiable structures on metric measure spaces, einstein spaces, quantitative behavior of singular sets

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The project is part of an ongoing program involving the study of situations that are in some way singular i.e. the objects may have discontinuities, may not be everywhere smooth, may have subsets on which they become infinite, etc. One theme involves putting constraints on the size or nature of the singularities which can arise in the solutions to certain nonlinear partial differential equations (PDEs) which are of importance in mathematics and physics. The principle investigator and collaborators have introduced methodology which, for a significant class of such PDEs, provides better control on the singularities than was previously available. Another theme is to study spaces which can be quite wild (e.g. they can have fractional dimensions) but are nonetheless well enough behaved so that one can employ the methods of calculus, if differentiation is understood in a sufficiently generalized sense. Apart from their intrinsic interest, these methods have had a surprising application to a basic problem in theoretical computer science, the sparsest cut problem with general demands. The project focuses on three main areas: 1) The analytical and geometric structure of metric measure spaces with Lipschitz differentiable structure. 2) Degeneration of Einstein metrics. 3) Quantitative behavior of singular sets. Regarding 1), a basic question is to study the extent to which Lipschitz differentiability spaces are more general than spaces for which the measure satisfies a doubling condition and a Poincar\'e inequality holds in the sense of Heinonen-Koskela. Recent work with Bate characterizing Lipschitz differentiability spaces in terms of Alberti representations should play an important role. Regarding 2), a challenging question is whether for noncollapsed Gromov-Hausdorff limits of Einstein spaces with bounded Einstein constant, the singular set has Hausdorff codimension 4 (as conjectured by M. Anderson). Regarding 3), a goal is to extend to new cases, the techniques developed by the principle investigator, with A. Naber (and partly with R. Haslhofer), for studying the quantitative behavior of singular sets of certain elliptic and parabolic PDEs.