The goal of the summer school is for the students to first become familiar with the concept of concentration of measure in different settings (Euclidean, Riemannian and discrete), and the main open problems surrounding it. The students will later become familiar with the proof techniques that involve the different types of localization and obtain expertise on the ways to apply the localization techniques. After attending the graduate school, the students are expected to have the necessary background that would give them a chance to both conduct research around open problems in concentration of measure, find new applications to existing localization techniques and perhaps also develop new localization techniques.

The topics of the graduate school are suited for any student with an undergraduate degree in mathematics. It will require no expertise in any particular field and will be completely self contained. The required background (as described below) is usually obtained in any program of undergraduate studies. Each lecture series will begin with a ”preliminaries” lecture that would make sure that all the students are up to speed in terms of mathematical background.

Prerequisites

The students are required to have the following prerequisites.

• Familiarity with basic concepts in measure theory, probability theory stochastic processes and martingales. Two good textbooks are: A. Chapters 1 and 4 in Probability: Theory and Examples (alternate) / R. Durrett. B. Probability with Martingales / Williams, Part A + Chapter 10.
• Basic familiarity with differentiable manifolds (e.g., Lecture Notes on Elementary Topology and Geometry / Singer and Thorpe, especially Chapter 5)
• Some basic familiarity with stochastic calculus will be helpful (but not mandatory). Probability: Theory and Examples (alternate) / R. Durrett, Chapter 7