# Comparison geometry and applications

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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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