Lectures/Mini courses and preparatory reading lists

Metric Geometry and Geometric Analysis Summer School 2022

LECTURES/MINI COURSES

Week 1

Regina Rotman

Course Title: 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, such 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. Serre) and the existence of three simple
closed geodesics on a 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.

Urs Lang

Course Title: Isoperimetric filling inequalities in CAT(0) spaces

The minicourse will start with a quick introduction, essentially from scratch, to currents

in metric spaces in the sense of Ambrosio-Kirchheim. This will be followed by a proof

of the isoperimetric filling inequality of Euclidean type for cycles in CAT(0) spaces.

This important inequality is due to Federer-Fleming for Euclidean space and to Gromov

and Wenger in the general case. Some applications will be discussed. If time permits,

an improvement of the isoperimetric inequality for cycles of dimension greater than or

equal to the asymptotic rank of the underlying CAT(0) space, also due to Wenger,

will be sketched. This pertains to notions of higher-rank hyperbolicity studied recently

in work of Kleiner, the lecturer, and others.

Recommended preparatory reading: Parts I.1, I.2, I.3, I.8, II.1, and pp. 175-179

of II.2 in “Metric Spaces of Non-Positive Curvature” by Bridson-Haefliger

Week 2

Bruce Kleiner

Course Title: Metric geometry and analysis on boundaries of Gromov hyperbolic spaces, and applications

Abstract: The minicourse will cover some aspects of metric and analytical structure on boundaries of Gromov hyperbolic spaces, applications to rigidity, and open problem.