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The soul theorem of Riemannian geometry largely reduced the study of non-compact complete Riemannian manifolds with non-negative curvature to the study of compact submanifolds. The soul theorem was published in 1972 by Cheeger and Gromoll, and in 1994 Perelman proved a significant conjecture based on the soul theorem. In this talk we will briefly outline the proof of the soul theorem, mainly focusing on the existence of the soul (a submanifold) and its properties. The proof is constructive, hence has great applicational value.
This talk will introduce a PDE that describes the porosity of a given medium. It will cover existence theory for a simplified PDE, which starts with the L2 case and can be extended to L1 by functional analytic arguments.
Citing from the Department of Physics Website:
The geometry of special relativity can be neatly described using
hyperbolic trigonometry. The geometry of general relativity can be
similarly described using differentials and differential forms.
This talk presents an excursion through both special and general
relativity, emphasizing geometric structure and using (mostly)
elementary concepts from trigonometry, linear algebra, and vector
calculus. You may never view those topics the same way again...