Geometry and topology of weakly differentiable mappings into Euclidean spaces, manifolds and metric spaces

Piotr Hajlasz：NSF提案DMS-0900871摘要：Sobolev映射到欧氏空间、光滑流形和度量空间之间的理论在偏微分方程组、变分、非线性弹性、微分几何、度量空间分析、几何类型学和代数拓扑学的当代发展中起着重要的作用。特别是，PI计划研究：(1)流形之间映射的变分问题的正则性理论，特别强调临界增长(n-调和映射和H-曲面系统)的非线性问题。(2)当映射的正则性不足以保证雅可比的可积性时，流形间弱可微映射的度与同伦理论。到流形的拓扑结构的连接。(3)出现在非线性弹性中的映象类。(4)度量空间和度量空间之间的Sobolev映射的Lipschitz逼近。特别地，讨论了从欧氏空间到Heisenberg群的映射的逼近问题。(5)度量空间上的连续、Sobolev和光滑满射。可微Peano-型映射的构造尤其是卡诺群之间的光滑满射的存在。在21世纪初，由于经典的可微函数的概念，偏微分方程组和变分法理论走到了死胡同。只有对导数的概念进行适当的推广，才有可能进一步发展。这导致了索博列夫空间的发现。随着索博列夫空间理论应用于越来越多的领域，在数学、工程和物理的不同领域之间建立桥梁有了新的可能性。该项目的目标之一是在分析、变分、几何和拓扑学领域之间建立这种联系。与来自不同机构和国家的研究人员合作是该项目的重要组成部分。这将加强科研机构的合作，并为参与该项目的研究生提供独特的机会，不仅可以研究当代数学前沿的开放问题，还可以与来自世界各地的研究人员建立联系。该项目的成果将在会议、讲习班和学校中介绍。国际和平协会已经参与组织了几次专门讨论类似主题的会议和研讨会。

Piotr Hajlasz: NSF Proposal DMS-0900871Abstract:The theory of Sobolev mappings into Euclidean spaces, between smooth manifolds and between metric spaces plays an important role in the contemporary development of partial differential equations, calculus of variations, nonlinear elasticity, differential geometry, analysis on metric spaces, geometric typology and algebraic topology.The project focuses on a wide range of problems in the areas mentioned above. In particular the PI plans to investigate: (1) Regularity theory for variational problems for mappings between manifolds with a particular emphasis on problems with nonlinearity of the critical growth (n-harmonic mappings and H-surface system). (2) Degree and homotopy theory for weakly differentiable mappings between manifolds in the case in which regularity of mappings is not enough to guarantee integrability of the Jacobian. Connections to the topological structure of manifolds. (3) Classes of mappings that arise in the nonlinear elasticity. (4) Lipschitz approximation of Sobolev mappings into metric spaces and between metric spaces. In particular, approximation of mappings from the Euclidean space into the Heisenberg group. (5) Continuous, Sobolev and smooth surjections onto metric spaces. Construction of differentiable Peano-type mappings. In particular existence of smooth surjections between Carnot groups.At the beginning of XXth century, with a classical notion of differentiable functions, the theory of partial differential equations and calculus of variations came to a dead end. The further development was only possible with a suitable generalization of notion of the derivative. This led to the discovery of the Sobolev spaces. With the increasing variety of areas to which the theory of Sobolev spaces applies there are new possibilities to built bridges between different areas of mathematics, engineering and physics. One of the aims of the project is to develop such connections between fields of analysis, calculus of variations, geometry and topology. Collaboration with researchers from different institutions and countries is an essential part of the project. This will strengthen the cooperation of scientific institutions and give unique opportunity for the graduate students involved in the project not only to work in open problems at the frontiers of the contemporary mathematics, but also to establish contacts with researchers from all over the world. The outcomes of the project will be presented on conferences, workshops and schools. The PI has already been involved in organizations of several conferences and seminars devoted to similar topics.