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Wolfe's Theorem in R^n

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Title: Wolfe's Theorem in R^n
Author(s): Pim, Jonathan
Contributor: University of Helsinki, Faculty of Science
Degree program: Master 's Programme in Mathematics and Statistics
Specialisation: Mathematics
Language: English
Acceptance year: 2022
Abstract:
Wolfe’s Theorem states that there is an isometric isomorphism between the space of flat k-cochains and the flat differential k-forms in R^n . The flat forms are the space of essentially bounded differential forms with an essentially bounded weak exterior derivative. The flat cochains are the dual space of the flat chains which are geometric objects based on finite linear combinations of k-simplices. In this sense, Wolfe’s Theorem connects geometry and analysis. After proving Wolfe’s Theorem, we give two corollaries: that the isomorphism from Wolfe’s Theorem can be concretely approximated by convolution with smooth mollifiers, and a version of Stokes’ Theorem for flat chains. Our method for proving Wolfe’s Theorem involves isometrically embedding the flat chains, as well as a predual of the flat forms, into the space of flat currents. By way of some approximation theorems in the space of flat currents, the images of these two embeddings coincide. Thus, the flat chains are isomorphic to that predual. This isomorphism lifts to their dual spaces giving Wolfe’s Theorem.
Keyword(s): Wolfe’s Theorem Flat chains Flat forms Currents Differential k-forms k-vector fields


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