# Comparison geometry and applications

## Vitali Kapovitch (Toronto University)

created by paolini on 10 Jan 2014
modified by gigli on 17 Jun 2014

Abstract.

1. Definitions and examples of Alexandrov spaces with curvature bounded below, Globalization theorem. Gromov-Hausdorff convergence. 2. First order structure: tangent cones, first variation formula. Volume, Bishop-Gromov comparison. 3. Perelman's theorem on bounding the number of isolated singular points in nonnegatively curved spaces and application to the Erdos problem about discrete isometric actions on $R^n$. 4. Semiconcave functions, gradient flows of semiconcave functions and their Lipschitz properties. 5. Application of the gradient flows: Virtual nilpotency of the action of the fundamental group on higher homotopy groups for almost nonnegatively curved manifolds.

As a reference I'll be using the draft of our book with Anton Petrunin and Stephanie Alexander

http://www.math.psu.edu/petrunin/papers/alexandrov-geometry

and the paper

Nilpotency, almost nonnegative curvature and the gradient push, V. Kapovitch A. Petrunin and W. Tuschmann, Annals of Mathematics, Vol. 171 (2010), No. 1, 343-373

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