Lecture & Mini-Course 1: "Geometric Inequalities: Homotopies, Fillings and Geodesics"

We will discuss various geometric inequalities motivated by famous existence  theorems of various minimal objects in differential geometry proven by  topological methods. Let M be a closed Riemannian manifold. Quantitative  versions of such theorems as the existence of a periodic geodesic on M due to  A. Fet and L. Lusternik, the existence of infinitely many geodesics between an  arbitrary pair of points on M (J. P. Serve) and the existence of three simple  closed geodesics ona Riemannian 2-sphere (L. Lusternik and L. Schnirelmann)  will be presented.

We will begin with a discussion of surfaces, next explore how the results for  surfaces can be generalized to curvature-free estimates on higher dimensional  manifolds. We will next discuss geometric inequalities that involve curvature  bounds. If time permits, we will also talk about the case of non-compact  complete manifolds with some geometric constraints, like finite volume.

In terms of the prerequisites, in addition to Do Carmo's Riemannian Geometry, I  would expect students to know some fundamentals of Algebraic Topology, such as  Homology and Homotopy Groups, which can be found in Hatcher's textbook.

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