Geometric Function Theory in Euclidean and Metric Spaces

微积分在科学上有无数的应用。在19世纪和20世纪，微积分的概念被研究用于定义在称为光滑流形的一般空间上的函数。在过去的30年里，数学出现了新的方向微积分，现在被称为分析，在更一般的空间，称为度量空间，导致非光滑和光滑空间之间的遥远世界的新桥梁。度量空间的分析，连同定量拓扑学，如今是一个活跃而独立的领域，汇集了来自数学光谱不同部分的研究人员。它具有深远的应用。目前的项目旨在研究度量空间和定量拓扑分析中的广泛问题，同时包括分析中的其他相关主题。这项工作强调了如何成功地采用类似的技术来回答不同数学领域中看似无关的问题。该项目还包括研究生的培训。该项目的共同主题是函数和映射理论的分析，几何和拓扑方面的低阶正则性（凸函数，Sobolev函数，Lipschitz和Hölder连续映射，映射是一次连续可微的，等等）。这种映射出现在当代数学的几个领域，该项目试图在分析，几何和拓扑的不同领域之间建立桥梁。更确切地说，研究者将研究以下领域的主题：（1）凸函数的逼近;（2）与拓扑和变分学有关的Sobolev同胚的Jacobian符号;（3）与定量拓扑有关的球面同伦群和导数秩低的映射几何;（4）导数秩低的映射的逼近;（5）与定量拓扑有关的Jacobian符号。（5）度量空间中的面积和余面积公式;（6）隐函数定理在度量空间中的推广;（7）数量隐函数定理和通过树的因子分解;（8）Heisenberg群中Hölder连续映射的分析性质;（9）Heisenberg群的Lipschitz和Hölder同伦群;（10）海森堡群的惠特尼延拓定理及其在接触映射近似中的应用。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Calculus has a myriad of applications in science. In the nineteenth and twentieth centuries the notions of calculus were studied for functions defined on general spaces called smooth manifolds. The last thirty years have witnessed the emergence of new directions in mathematics—calculus, now called analysis, on more general spaces, called metric spaces—leading to new bridges between the distant worlds of non-smooth and smooth spaces. Analysis on metric spaces, together with quantitative topology, is nowadays an active and independent field bringing together researchers from disparate parts of the mathematical spectrum. It has far-reaching applications. The current project aims at investigating a broad spectrum of questions in analysis on metric spaces and quantitative topology while including other related topics in analysis. The work emphasizes how similar techniques can be successfully employed to answer seemingly unrelated questions in different areas of mathematics. The project also includes the training of graduate students.The common themes of the project are the analytic, geometric, and topological aspects of the theory of functions and mappings with low order of regularity (convex functions, Sobolev functions, Lipschitz and Hölder continuous mappings, mappings that are one time continuously differentiable, etc.). Such mappings appear in several areas of contemporary mathematics, and the project attempts to create bridges between different areas of analysis, geometry, and topology. More precisely the investigator will study topics in the following areas: (1) Approximation of convex functions; (2) Sign of the Jacobian of Sobolev homeomorphisms with connections to topology and the calculus of variations; (3) Homotopy groups of spheres and geometry of mappings whose derivatives have low rank, with connections to quantitative topology; (4) Approximation of mappings whose derivatives have low rank; (5) Area and coarea formulas in metric spaces; (6) Generalization of the implicit function theorem to metric spaces; (7) Quantitative implicit function theorem and factorization through trees; (8) Analytic properties of Hölder continuous mappings in the Heisenberg groups; (9) Lipschitz and Hölder homotopy groups of the Heisenberg groups; (10) Whitney extension theorem for the Heisenberg group with applications to approximation of contact mappings.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.