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

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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.