Jul 11, 2022
Monday

09:30 AM  10:30 AM


Lecture & MiniCourse 1: "Geometric Inequalities: Homotopies, Fillings andĀ Geodesics"
Regina Rotman (University of Toronto)

 Location
 
 Video

 Abstract
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 2sphere (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 curvaturefree 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 noncompact 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.
 Supplements




Jul 12, 2022
Tuesday

09:30 AM  10:30 AM


Lecture & MiniCourse 1: "Geometric Inequalities: Homotopies, Fillings andĀ Geodesics"
Regina Rotman (University of Toronto)

 Location
 
 Video

 Abstract
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 2sphere (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 curvaturefree 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 noncompact 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.
 Supplements




Jul 13, 2022
Wednesday

09:30 AM  10:30 AM


Lecture & MiniCourse 1: "Geometric Inequalities: Homotopies, Fillings and Geodesics"
Regina Rotman (University of Toronto)

 Location
 
 Video

 Abstract
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 2sphere (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 curvaturefree 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 noncompact 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.
 Supplements




Jul 14, 2022
Thursday

09:30 AM  10:30 AM


Lecture & MiniCourse 1: "Geometric Inequalities: Homotopies, Fillings and Geodesics"
Regina Rotman (University of Toronto)

 Location
 
 Video

 Abstract
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 2sphere (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 curvaturefree 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 noncompact 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.
 Supplements




Jul 15, 2022
Friday

09:30 AM  10:30 AM


Lecture & MiniCourse 1: "Geometric Inequalities: Homotopies, Fillings and Geodesics"
Regina Rotman (University of Toronto)

 Location
 
 Video

 Abstract
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 2sphere (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 curvaturefree 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 noncompact 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.
 Supplements



