Nonsmooth Structures and Geometric Function Theory

PI将搜索有助于识别给定度量空间何时可以通过仅以受控方式改变距离的同胚来参数化的条件；即，我们要求欧氏空间的(局部)双Lipschitz参数化。即使对于位于欧几里得三维空间中的二维曲面，这个问题也不能很好地理解。从参数化问题到了解欧几里得空间中什么可测结构是局部标准的问题有一个直接的联系，因为它们是作为标准结构的同胚拉回而产生的。后一个问题可以在双Lipschitz范畴和拟共形范畴中提出。为此，PI为超定系统提出了新的可积性条件，这些条件在非传统意义上可能是用几何方法可解的。与此密切相关的还有识别欧氏空间中拟共形变换的雅可比行列式的非线性问题。最后，PI将讨论某些非光滑的四个流形可以在多大程度上承担一些一阶微分分析；虽然这不能通过传统的图表来完成，但可以展示允许这种分析的度量结构。这一建议的主要智力价值在于对看似独立的问题的综合和共同的几何观点。为此，提出了非传统和冒险的方法和解决办法。拟议的活动产生的更广泛的影响是将不同数学领域以及不同培训和专长的数学家聚集在一起。学生和博士后助理将接受培训，并向他们学习，并邀请不同群体的参观者进行咨询。

AbstractHeinonenThe 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.