[cmath] University of Toronto's 21st Annual R. A Blyth Lectures: Assaf Naor - The Hidden Structure of Metric Spaces, March 8-10

[Allison Andres]

University of Toronto's

21St Annual R. A. Blyth Lecture Series

 

Speaker: Assaf Naor

Series Title: The Hidden Structure of Metric Spaces 

 

Schedule:

 

The Ribe program

Tuesday March 8, 4:00PM

 

Discretization

Wednesday March 9, 4:00PM

 

Metric X_p inequalities

Thursday March 10, 4:00PM

 

Abstract: A classical rigidity theorem of Martin Ribe (1975) suggests that
certain important properties of normed spaces are actually metric properties
in disguise, i.e., they can be characterized while using only distances
between points and without any reference to the underlying linear structure
whatsoever. Ribe's theorem inspired a longstanding research program that
aims to uncover an explicit dictionary that translates concepts and
phenomena from the structured linear world of normed spaces to the seemingly
wild and unstructured world of general metric spaces. Any "entry" in such a
dictionary could potentially be used to apply insights from the linear
theory to other remote settings, including graphs, groups, Riemannian
manifolds and various metric spaces that arise as continuous relaxations of
combinatorial optimization problems. Previous advances in the Ribe program
involved the use of diverse mathematical tools, and they led to many
powerful applications to various areas. Nevertheless, many central mysteries
remain, and the hidden dictionary that the Ribe program aims to uncover
still contains many missing entries that are needed for further progress.

 

The purpose of the first lecture is to provide an introduction to the Ribe
program, with an indication of examples of key milestones that have already
been achieved, examples of applications, and some of the many important
problems that remain open. The second lecture will be devoted to a
description of one of the abstract mechanisms that underlie the Ribe
theorem, as discovered by Bourgain (1986), and its relation to analytic
issues such as quantitative versions of differentiation. The third lecture
will be devoted to a description of the recent completion of a step in the
Ribe program, which discovers the last invariant of metric spaces that was
missing from a useful repertoire of invariants that have been discovered
over the past three decades that (among their many applications) certify
when L_q fails to admit a bi-Lipschitz embedding into L_p. This new
invariant yields a metric version of a classical linear theorem of Paley
(1936), and the proof of its validity in L_p relies on modern Fourier
analytic tools.

 

Visit
<http://www.math.utoronto.ca/cms/assets/MathFiles/Seminars/Files/21st-Blyth.
pdf>
http://www.math.utoronto.ca/cms/assets/MathFiles/Seminars/Files/21st-Blyth.p
df for complete details.

 

Allison Andres

Executive Assistant to the Chair

 <mailto:aandres at math.utoronto.ca> aandres at math.utoronto.ca

416-978-3317

 <https://www.math.toronto.edu/> https://www.math.toronto.edu/

 

 

 

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