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为定义在一般空间上的函数，称为光滑流形。在过去的三十年里，数学中出现了新的方向--微积分，现在称为分析，在更一般的空间，称为度量空间--导致了非光滑空间和光滑空间这两个遥远世界之间的新桥梁。度量空间分析与定量拓扑学一起，是当今一个活跃而独立的领域，汇集了来自数学光谱中不同部分的研究人员。它有着深远的应用。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，同时在分析中包括其他相关主题。这项工作强调了如何成功地使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论的解析、几何和拓扑方面，以及具有低阶正则性的映射(凸函数、Sobolev函数、Lipschitz和Hölder连续映射、一次连续可微的映射等)。这种映射出现在当代数学的几个领域，该项目试图在分析、几何和拓扑学的不同领域之间建立桥梁。更确切地说，研究者将研究下列领域的课题：(1)凸函数的逼近；(2)与拓扑学和变分有关的Sobolv同胚雅可比的符号；(3)球面的同伦群和其导数具有低秩且与定量拓扑有关的映射的几何；(4)其导数具有低秩性的映射的逼近；(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)定量隐函数定理和树的因式分解；(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)Heisenberg群的惠特尼扩张定理及其应用到接触映射的逼近。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.