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.

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.

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.