Explore the intricacies of high-dimensional objects through pathwise analysis in this 45-minute lecture by Ronen Eldan for the International Mathematical Union. Delve into the emergence of simple structures in high-dimensional settings, dimension-free concentration on Gaussian space, and Gaussian-like behavior beyond Gaussian measures. Examine high-dimensional convexity, the KLS conjecture, and pure state decompositions of interacting particle systems. Learn about martingale techniques and their simple illustrations, followed by an in-depth look at the pathwise approach, including analysis of variance and stochastic localization. Discover the basic properties of time-derivatives corresponding to spatial moments, the emergence of Gaussians, and the collapse into localized measures. Conclude with practical applications, focusing on concentration on the discrete hypercube.
Ronen Eldan- Revealing the Simplicity of High-Dimensional Objects via Pathwise Analysis
International Mathematical Union via YouTube
Overview
Syllabus
Intro
Emergence of a simple structure in high-dimensional settings
Dimension-free concentration on Gaussian space
Gaussian-like behavior beyond the Gaussian measure
High dimensional convexity and the KLS conjecture
Pure state decompositions of interacting particle systems
Martingale techniques - a simple illustration
A Pathwise approach
Analysis of variance
Stochastic Localization (discrete version)
Basic properties of
Time-derivatives correspond to spatial moments
Emergence of a Gaussian
Collapsing into localized measures
Some applications
Application #4: Concentration on the discrete hypercube
Taught by
International Mathematical Union
