Speaker: Luis Atzin Franco Reyna
University of Notre Dame
Will give a Graduate Student Geometry Seminar entitled:
Gromov’s Pre-Compactness Theorem
Abstract: The Gromov-Hausdorff distance measures how far away are two metric spaces from being isometric. This introduces a metric structure in the space of all compact metric spaces and allows us to talk about convergence. Similarly to Arzelà-Ascoli’s theorem, Gromov proved a theorem which allows us to show a sequence of compact metric spaces has a converging subsequence. We will give a sketch of the proof of Gromov’s Pre-Compactness Theorem. This theorem tells us when a sequence of compact metric spaces has a converging subsequence. In particular, one can look at sequences of manifolds of a fixed dimension, bounded diameter and bounded Ricci or Sectional curvature. This sequences have converging subsequences and their limit space is a compact metric space. One can quickly see that this limit spaces may fail to be smooth manifolds and if one wishes to hope for “curvature” to be preserved under convergence, then there is a need to generalize bounds in curvature. This leads to the notion of Alexandrov spaces and RCD(k,n) spaces.
Date: 03-01-2024
Time: 2:00 pm
Location: 117 Hayes-Healy Bldg