Geometry/Topology Seminar

from Tuesday
January 01, 2019 to Friday
May 31, 2019
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Tuesday
February 12, 2019

4:00 PM - 5:30 PM
Math Tower P-131
Semyon Alesker, Tel Aviv University, Tel Aviv, Israel
Few conjectures on intrinsic volumes on Riemannian manifolds and Alexandrov spaces

The celebrated Hadwiger's theorem says that linear combinations of intrinsic volumes on convex sets are the only isometry invariant continuous valuations (i.e. finitely additive measures). On the other hand H. Weyl has extended intrinsic volumes beyond convexity, to Riemannian manifolds. We try to understand the continuity properties of this extension under the Gromov-Hausdorff convergence (literally, there is no such continuity in general). First, we describe a new conjectural compactification of the set of all closed Riemannian manifolds with given upper bounds on dimension and diameter and lower bound on sectional curvature. Points of this compactification are pairs: an Alexandrov space and a constructible (in the Perelman-Petrunin sense) function on it. Second, conjecturally all intrinsic volumes extend by continuity to this compactification. No preliminary knowledge of Alexandrov spaces will be assumed, though it will be useful.


Tuesday
March 12, 2019

4:00 PM - 5:00 PM
Math Tower P-131
Fedor Manin, Ohio State University
Rational homotopy and topological isoperimetry

Soon after Sullivan introduced his model of rational homotopy theory in the 1970's, Gromov noted that the theory had some metric consequences for maps between compact manifolds or simplicial complexes. I will present a systematic view of this relationship which gives a powerful tool for, among other things, resolving the following type of question, asked by Gromov twenty years later:

Given two $L$-Lipschitz maps $f, g: X → Y$, where $X$ and $Y$ are nice compact metric spaces, what is the optimal Lipschitz constant of a homotopy between them?

I will also try to explain why this question is fundamental to quantitative topology.


Tuesday
April 02, 2019

4:00 PM - 5:30 PM
Math Tower P-131
Simons Lectures, Stony Brook University
no Geometry/Topology seminar


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