Geometric Analysis and Topology Seminar
A few conjectures on intrinsic volumes on Riemannian manifolds and Alexandrov spaces
Speaker: Semyon Alesker, Tel Aviv
Location: Warren Weaver Hall 202
Date: Wednesday, March 6, 2019, 11 a.m.
Synopsis:
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. While there is no such continuity in general, we will describe a new conjectural compactification of the set of isometry classes 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.