Hodge Theory on Metric Spaces

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Passbani, Hossein | 2012

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 43355 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Bahraini, Alireza
Abstract:
Hodge theory is a beautiful synthesis of geometry, topology, and anal- ysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the math- ematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We believe that this constitutes a step towards understanding the geometry of vision. We extend the theory of the Laplacian on domains of Euclidean space or on a manifold and next to metric space. Finally we give a beautiful application in web researcher engin.
Keywords:
Metric Space ; Cohomology Group ; Harmonic Form ; Scale Cohomology