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Sobolev Mappings Between Manifolds and Metric Spaces
Quasiconformal mappings and geometric variational problems like the theory of harmonic mappings lead to study of Sobolev mappings between manifolds and even between metric spaces. One of the main questions in this context is the question about density of smooth mappings in the Sobolev class (in the case of mappings between metric spaces we ask about density of Lipschitz mappings). I will discuss new connections between the problem of the density and the algebraic topology of manifolds and polyhedra.
Although bi-Lipschitz equivalent metric spaces are usually regarded in analysis as equivalent, I will show that the density of Lipschitz mappings can be lost if we modify the target metric spaces in a bi-Lipschitz way.
Finally I will discuss the Hardy space regularity of the pullback of the volume form. This is a nontrivial extension of recent important results on the borderline of harmonic analysis and partial differential equations.
Thursday, January 29, 2004, 445 Altgeld Hall, 2:00 p.m.
Mathematics Colloquia homepage