Nonsmooth Structures and Geometric Function Theory

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Abstract Heinonen The PI will search for conditions that help recognizing when a given metric space can be parametrized by a homeomorphism that changes distances only in a controlled manner; that is, we ask for (local) bi-Lipschitz parametrizations by Euclidean space. This question is not suficiently understood even for two dimensional surfaces lying in Euclidean three space. There is a direct link from the parametrization problem to the problem of understanding what measurable structures in Euclidean space are locally standard in that they arise as pullbacks of the standard structure by a homeomorphism. This latter question can be asked both in bi-Lipschitz and quasiconformal categories. To that end, the PI propose new integrability conditions for overdetermined systems that may be solvable in a nontraditional sense by geometric methods. Closely related also is the nonlinear problem of recognizing Jacobian determinants of quasiconformal transformations in Euclidean space. Finally, the PI will discuss to what extend certain nonsmoothable four manifolds could be brought to bear some first order differential analysis; while this cannot be accomplished via traditional charts, it could be possible to exhibit metric structures that allow for such analysis. The main intellectual merit of this proposal lies in the synthesis and the common geometric point of view for seemingly separate problems. To that end, nontraditional and venturesome approaches and solutions are proposed. The broader impacts resulting from the proposed activity constitute of bringing together different fields of mathematics, as well as mathematicians of different training and expertise. Students and postdoctoral assistants will be trained as well as learned from, and a diverse group of visitors are brought in for consultation.