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

1900年，大卫·希尔伯特提出了一份著名的清单，列出了对20世纪数学产生根本影响的23个开放问题。希尔伯特的第20个问题是关于偏微分方程组和变分问题的广义解的存在性。这是一个至关重要的问题，因为这个理论走进了死胡同：经典的可微函数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。没有索波列夫空间，非线性偏微分方程的进一步发展(无论是理论上还是实践上)都是不可能的。然而，近几十年来，Sobolev空间的应用范围远远超出了偏微分方程组的理论；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的分析，仅举几例。这是当代数学中一个非常活跃的领域，有许多悬而未决的问题和新兴的研究领域。本研究项目涉及Soblev函数和映射的解析、几何和拓扑性质，以及具有低可微正则性的其他相关映射类。主要研究人员的重点是在分析、几何和拓扑学领域中看似不相关的方面之间找到新的桥梁。研究生和年轻的研究人员将通过参与该项目的研究进行培训。更详细地说，首席研究员计划探索以下主题。(1)凸函数的逼近(它的二阶分布导数是Radon测度，因此从可微的观点看正则性很低)。(2)Sobolev扩环的一般理论。(3)平移不变算子的存在性(这个问题不涉及弱可微函数)。(4)有限偏差Orlicz-Sobolev映射的连续性。(5)Sobolev等距浸没的正则性。(6)Sobolev同胚雅可比的符号。(7)Sard定理的拓扑非平凡Kaufman型反例(当Sard定理处理充分光滑映射时，主要研究者将研究低正则映射的情况)。(8)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)Heisenberg群的Lipschitz同伦群。(11)到Heisenberg群的接触映射的Whitney扩张定理。(12)微分形式演算和Hölder连续映射及其在Heisenberg群上的应用。(13)准备一本关于海森伯格团队的专著。这一奖项反映了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。

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.