Concentration of Measure Seminar
Welcome to the Student Analysis Seminar at UC Santa Barbara!
In Summer 2025, we’ll explore the phenomenon of measure concentration and the applications of it to functional inequalities, PDE’s, geometry, probability, and statistics. The main references we’ll refer to are
- (N) Asaf Naor’s lecture notes
- (L) Michel Ledoux’s book The Concentration of Measure Phenomenon
- (DP) Concentration of Measure for the Analysis of Randomised Algorithms by Devdatt P. Dubhashi & Alessandro Panconesi
Location & Time
Update: We will meet in South Hall 6617 at 11:30AM on Tuesdays. After meetings, any lecture notes from the presenter will be linked and posted below.
Schedule
| Date | Topic | Presenter | References | Lecture Notes |
|---|---|---|---|---|
| 2025-07-01 | Introduction to Concentration of Measure | Connor Marrs | L Ch. 1, 2 | TBD |
| 2025-07-08 | More Examples of Concentration Inequalities | Claire Murphy | L Ch. 1, N Pg. 4-19 | Download the PDF |
| 2025-07-15 | Isoperimetry & Functional Inequalities | Jack Pfaffinger | L Ch. 2 | TBD |
| 2025-07-22 | Skip for OT Conference | N/A | NA | NA |
| 2025-07-29 | Concentration in Product Spaces | Christian Hong | L Ch. 4, N Pg. 22-30 | TBD |
| 2025-08-05 | Entropy & Log-Sobolev | Connor Marrs | L Ch. 5 | TBD |
| 2025-08-12 | Log-Sobolev \& Langevin Dynamics | Claire Murphy | L Ch. 5 | Download the PDF |
| 2025-08-19 | Transportation Cost Inequalities | Connor Marrs | L Ch. 6 | TBD |
| 2025-08-26 | Gaussian Processes | Jack Pfaffinger | L Ch. 7 | TBD |
Abstracts
Week 1: Introduction to Concentration Phenomena (Connor)
Abstract: At first glance, if I have a sum of random variables, it may seem that the variance of this sum will be quite large. In the worst case scenario, this is true, but provided we sum up enough terms and the random variables are sufficiently uncorrelated, the sum will have almost no variance (i.e. it is close to constant). This is one instance of a general phenomenon in which random quantities that depend “nicely” on sufficiently uncorrelated inputs turn out to be nearly constant. This leads to so-called concentration inequalities that can be used to study the isoperimetric problem, functional inequalities (e.g. Sobolev & Poincare), and high dimensional geometry. As a result, we can develop new techniques to study geometry, PDE’s, stochastic processes, and statistics. I’ll introduce some motivating examples, prove a simple concentration inequality, and state some key results.
Week 2: More Examples of Concentration Inequalities (Claire)
Abstract: TBD
Week 3: Isoperimetry & Functional Inequalities (Jack)
Abstract: TBD
Week 4: Concentration in Product Spaces (Christian)
Abstract: TBD
Week 5: Log-Sobolev Inequalities (Connor)
Abstract: TBD
Week 6: Log-Sobolev Inequalities and Stochastic Dynamics (Claire)
Abstract: TBD
References
- Michel Ledoux, The Concentration of Measure Phenomenon (Mathematical Surveys and Monographs, Vol. 89)
- Asaf Naor’s lecture notes on Concentration of Measure available here
- Dubnashi’s & Panconesi’s notes available here