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

摘要：欧几里得空间、光滑流形和度量空间之间的Sobolev映射理论在偏微分方程、变分法、非线性弹性、微分几何、度量空间分析、几何类型和代数拓扑等领域的当代发展中起着重要作用。该项目侧重于上述领域的广泛问题。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)在映射的正则性不足以保证雅可比矩阵可积的情况下流形间弱可微映射的度与同伦理论。流形拓扑结构的连接。(3)非线性弹性中出现的映射类。(4)度量空间和度量空间间Sobolev映射的Lipschitz近似。特别地，从欧几里得空间到海森堡群的映射的近似。(5)度量空间上的连续、Sobolev和光滑抛射。构造可微的peano型映射。特别是卡诺群之间的光滑射波的存在性。20世纪初，随着经典的可微函数概念的确立，偏微分方程理论和变分学进入了死胡同。只有对导数的概念进行适当的推广，才能进一步发展。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的领域越来越多，在数学、工程和物理的不同领域之间建立桥梁的可能性也越来越大。该项目的目标之一是在分析、变分演算、几何和拓扑学等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。这将加强科研机构之间的合作，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，而且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、研讨会和学校中展示。PI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.