Nonsmooth Structures and Geometric Function Theory

PI将寻找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参数化的条件；也就是说，我们要求（局部）bi-Lipschitz参数化由欧几里德空间。即使在欧几里得三维空间中的二维曲面上，这个问题也没有得到充分的理解。从参数化问题到理解欧几里德空间中哪些可测量结构是局部标准的问题有一个直接的联系，因为它们是标准结构通过同胚的回调而产生的。后一个问题可以在双lipschitz和拟共形范畴中提出。为此，PI提出了可以用非传统几何方法求解的超定系统的新的可积性条件。与此密切相关的还有识别欧几里德空间中拟共形变换的雅可比行列式的非线性问题。最后，讨论了非光滑四流形在何种程度上可以进行一阶微分分析；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。这个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，提出了非传统的、大胆的方法和解决方案。拟议的活动所产生的更广泛的影响包括将不同的数学领域以及不同训练和专业知识的数学家聚集在一起。学生和博士后助理将接受培训和学习，并带来不同群体的访客进行咨询。

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.