Weakly Differentiable Mappings and Functions: Analysis, Geometry, and Topology

1900年，大卫·希尔伯特（David Hilbert）提出了一份著名的23个开放问题清单，这些问题对20世纪的数学产生了根本性的影响。希尔伯特第20个问题是关于偏微分方程和变分问题的广义解的存在性问题。这是一个至关重要的问题，因为这个理论陷入了死胡同：经典的可微函数的概念不足以进一步发展。这导致了20世纪最伟大的数学发现之一：索博列夫空间理论。没有Sobolev空间，非线性偏微分方程的进一步发展（理论和实践）是不可能的。然而，最近几十年表明，Sobolev空间的应用范围远远超出了偏微分方程理论；该理论适用于微分几何、几何群论、代数拓扑、次黎曼几何和度量空间分析，仅举几例。这是当代数学中一个非常活跃的领域，有许多尚未解决的问题和新兴的研究领域。本研究项目涉及Sobolev函数和映射的解析、几何和拓扑性质，以及其他具有低可微正则性的相关映射类。首席研究员的重点是在分析、几何和拓扑学领域中看似无关的方面之间寻找新的桥梁。研究生和年轻的研究人员将通过参与该项目的研究而得到培训。更详细地说，首席研究员计划探讨以下主题。(1)凸函数的逼近（其二阶分布导数为Radon测度，因此从可微的角度来看正则性较低）。(2) Sobolev扩展域的一般理论。(3)平移不变算子的存在性（不涉及弱可微函数的问题）。(4)有限畸变Orlicz-Sobolev映射的连续性。(5) Sobolev等距浸没的规律性。(6) Sobolev同胚的雅可比矩阵符号。(7) Sard定理的拓扑非平凡kaufman型反例（虽然Sard定理处理的是足够光滑的映射，但主要研究者将研究具有低规律性的映射的情况）。度量空间上的Sobolev空间。(9)度量空间中Lipschitz映射的隐函数定理。（10）海森堡群的Lipschitz同伦群。（11）接触映射入Heisenberg群的Whitney扩展定理。（12）微分形式的微积分和Hölder连续映射及其在海森堡群中的应用。（13）准备一本关于海森堡群的专著。该奖项反映了美国国家科学基金会的法定使命，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。

In 1900, David Hilbert presented a famous list of 23 open questions that had fundamental influence in the mathematics of the 20th century. Hilbert's 20th problem was about existence of generalized solutions of partial differential equations and variational problems. This was a crucial question since the theory was at a dead end: the classical notion of a differentiable function was not sufficient for further development. This led to one of the greatest discoveries of mathematics of 20th century: the theory of Sobolev spaces. Without Sobolev spaces, further development (both theoretical and practical) of nonlinear partial differential equations would not be possible. Recent decades showed however, that the scope of applications of Sobolev spaces goes far beyond the theory of partial differential equations; the theory applies to differential geometry, geometric group theory, algebraic topology, sub-Riemannian geometry, and analysis on metric spaces, just to name a few. This is a very active area of contemporary mathematics with many unsolved problems and new emerging areas of research. This research project concerns analytic, geometric, and topological properties of Sobolev functions and mappings as well as other related classes of mappings that have low differentiability regularity. The focus of the principal investigator is on finding new bridges between seemingly unrelated aspects of the fields of analysis, geometry, and topology. Graduate students and young researchers will be trained through research involvement in the project.In more detail, the principal investigator plans to explore the following topics. (1) Approximation of convex functions (which have second order distributional derivatives being Radon measures and thus low regularity from the differentiable viewpoint). (2) General theory of Sobolev extension domains. (3) Existence of translation-invariant operators (a question that does not deal with weakly differentiable functions). (4) Continuity of Orlicz-Sobolev mappings of finite distortion. (5) Regularity of Sobolev isometric immersions. (6) Sign of the Jacobian of a Sobolev homeomorphism. (7) Topologically nontrivial Kaufman-type counterexamples to the Sard theorem (while the Sard theorem deals with sufficiently smooth mappings, the principal investigator will investigate the case of mappings with low regularity). (8) Sobolev spaces on metric spaces. (9) Implicit function theorem for Lipschitz mappings into metric spaces. (10) Lipschitz homotopy groups of the Heisenberg groups. (11) Whitney extension theorem for contact mappings into the Heisenberg group. (12) Calculus of differential forms and Hölder continuous mappings with applications to the Heisenberg groups. (13) Preparation of a monograph about the Heisenberg groups.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.