# The 2018-2019 Chern Lectures

**The 2018-2019 Chern Lectures** will be given by Assaf Naor of Princeton University, on March 12, 13, 14 and 15, 2019. The title of the series is "**Quantitative embeddability, obstructions, and applications**".

**Abstract of the series:**

Representing a metric space X as a subset of another metric space Y, while preserving up to a controlled error certain geometric features of X, is a powerful paradigm in metric geometry, with a wide variety of decisive applications. This endeavor teaches us about X if certain questions have already been answered about Y, and it also allows us to understand the geometry of Y through those spaces X that can or cannot be faithfully embedded into it. While metric embeddings have been studied extensively from multifaceted perspectives for almost a century, many of the most basic questions remain open. The purpose of these lectures is to give examples of recent progress, often relying on unexpected connections between, and applications to, seemingly disparate mathematical disciplines. The organizing principle that underlies much of the advances over the past four decades is the Ribe program, which is a network of conjectures and analogies about the quest to reveal, and use, a hidden dictionary for translating between linear and nonlinear phenomena. While aspects of this theme will be dispersed throughout these lectures, the talks will not rely on each other and could be understood independently. Our goal is to present self-contained examples of modern developments, and to indicate the rich web of mysteries that remain to be understood. We will not rely on any prerequisites beyond an undergraduate degree in mathematics, and all of the relevant background will be introduced and explained.

**Lecture 1:** The (in)compatibility of 3 and 5 dimensional Heisenberg geometry with Lebesgue spaces

Tuesday, March 12, 2019

4:00 pm - 5:00 pm, Simons Institute Auditorium, Calvin Lab

We will discuss the longstanding bi-Lipschitz embedding problem in ℝkRk, and how over the years it became intertwined with the embeddability properties of the Heisenberg groups into Lp(μ)Lp(μ) spaces. We will explain a recent completion of this project, which exhibits unexpected twists, decisive applications to longstanding open questions in algorithms and metric geometry, and connections to subtle structural issues in analysis.

**Lecture 2:** Extension, discretization, and quantitative differentiation

Wednesday, March 13, 2019

4:00 pm - 5:00 pm, Simons Institute Auditorium, Calvin Lab

We will discuss questions about the relation between discrete phenomena and their continuous counterparts. This relates to extension of partially defined functions, Bourgain’s work on discretization and almost extension for a quantitative version of Ribe’s rigidity theorem, and differentiation questions that are well understood as infinitesimal phenomena but their macroscopic counterparts remain basic mysteries.

**Lecture 3:** An average John theorem (Math Department Colloquium)

Thursday, March 14, 2019

4:00 pm - 5:00 pm, Room 60, Evans Hall

We will prove a sharp average-case variant of a classical embedding theorem of John through the theory of nonlinear spectral gaps. We will use this theorem to provide a new answer to questions of Johnson and Lindenstrauss (1983) and Bourgain (1985) on metric dimension reduction, and explain how it leads to unexpected algorithms for approximate nearest neighbor search.

**Lecture 4:** Nonpositive curvature is not coarsely universal

Friday, March 15, 2019

4:00 pm - 5:00 pm, Simons Institute Auditorium, Calvin Lab

We will discuss coarse embeddings into Alexandrov spaces of nonpositive or nonnegative curvature. By studying subtle invariants that initially arose within the Ribe program and discretization questions, we will answer a question of Gromov (1993) about the coarse universality of Hadamard spaces. Connections to important questions such as the existence of super-expanders will be explained.

**Biography of Speaker:**

**Assaf Naor** obtained his B.Sc. (1996), M.Sc. (1998) and Ph.D. (2002) from the Hebrew University in Jerusalem. From 2002 through 2006 he was a member of the Theory Group of Microsoft Research. From 2006 to 2014 he was a professor at the Courant Institute of Mathematical Sciences of New York University. Since 2014 he is a professor at the Department of Mathematics of Princeton University. He is the director of the Algorithms and Geometry (A&G) Think Tank at the headquarters of the Simons Foundation in NYC.