This quarter we are having Reading Seimnar in Geometry (A Part of the Geometry Seminar) Graduate Students are strongly encouraged to participate.
We are going to cover two very exciting topics of Riemannian Geometry and Geometrical Analysis, with applications to physics in mind.
Topic 1: Gromov's macroscopic picture of scalar curvature
Scalar curvature is the trace part of the curvature tensor, and is important for understanding conformal geometry, Einstein metrics, general relativity and many other subjects. According to Gromov, manifolds have "macroscopic dimension", which is the dimension you see when you view a manifold from a far distance. Small directions of the manifold are compressed in such view, so you get a macroscopic picture. (So we humans are 0-dimensional in this view!) The theme of Gromov is this: manifolds of positive scalar curvature have macroscopic dimension n-2, i.e. 2 dimensions are compressed when viewed from afar. Many interesting topics such as spectrum pop up in the course of analysing this picture.
Topic 2: Penrose Conjecture in General Relativty and Spin Geometry
Do you know what is the most important global physical quantity associated with an isolated space-time (such as our solar system)? It's the so-called ADM mass. It is a rather mysterious, yet fundamental physical entity. A fundamental conjecture of Penrose gives a lower bound for the ADM mass of a universe in terms of its boundary area (the area of the horrible event horizon). Of course, this can be formulated in terms of geometry (indeed, Riemannian geometry). What have geometers done about this conjecture? They proved it. We are going to present the details of the proof. The tool is spin geometry and PDE. (A secret message: the story of Penrose conjecture is not yet finished. Using the methods, one may do more.)
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