This is an elective course for second-year M.Math students.
The following syllabus (prone to minor changes and not strictly same as the official syllabus) is only indicative of the topics to be covered.
Class timings : Tuesdays : 2 - 3 PM and 4.30 - 6 PM. Thursdays : 4.30 - 6 PM.
Lectures :
Diagrammatic summary of the course (excluding Chapter 4 of [vH]).
Week 1. Basics of Gaussian random variables ; see Mathew's notes for proof of Gaussian integration by parts ; See Section 2.5 of [5] for Marcinkiewicz's theorem.
Week 2. Exercise from Week 1, Overview of vH's notes (Chapter 1), Slepian's inequality and Sudakov-Fernique inequality
See Theorem 2.2.5 of [4] for a quantitative version of Sudakov-Fernique and on why Slepian's is non-obvious see footnote below Theorem 2.2.1 of [4]. Strassen's theorem for stochastic domination - Lindvall's article
Week 3 : Tensorization and bounded differences, Markov semi-groups. (Section 2.1 and 2.2 of vH) ; Poincare' inequalities and applications (Section 2.3 of vH).
See Appendix A of [7] for techical details relating to Markov semigroups and in particular, the issues about domains.
Week 4: Variance identities and exponential ergodicity (Section 2.4 of vH) ; Problems from Chapter 2 of vH.
Week 5 : Entropy method and Modified Log Sobolev inequalities, (Section 3.3 and 3.4 of vH). Concentration in metric spaces (Section 4.1 of vH).
For more on the isoperimetric method to prove concentration inequality, see Problem 4.2 and Chapter 6 of [11]. A simpler exposition and applications may be found in Chapter 6 of [12]. Also see Problem 4.8.
The Herbst argument (the unpublished 1975 letter of I. Herbst to L. Gross – private communication of Professor L. Gross)
Week 6 : Discussion of Problems from Chapter 2 and Chapter 3 (Sections 3.1 and 3.2).
Week 7 : Discussion of Problems from Chapter 3 (Sections 3.3 and 3.4). Transportation inequalities and tensorization (Sections 4.2of vH) and problems.
For more on Bakry-Emery criterion and local Poincare inequalities (Exercises 2.13 and 3.19 of [vH]), see Section 4.7 of [7].
Week 8 : Mid-Sem Break.
Week 9 : Talagrand's concentration inequality, Dimension-free concentration and T2 inequality (Sections 4.3 and 4.4 of vH) and related problems.
Week 10 : Section 4.4 of vH, Sharp transitions to concentration, hypercontractivity and log-Sobolev inequalities (Sections 8.1 and 8.2 of vH).
Weeks 11 and 12 : NO CLASSES
Week 13: Problems from Chapter 4. Talagrand's L1-L2 inequality (Section 8.3 of vH)
Week 14 : Talagrand's L1-L2 inequality (Section 8.3 of vH). Problems from Section 8.1. Lindeberg method (Section 9.1 of vH)
Week 15 : Stein's method and Second-order Poincare inequality (Sections 9.2 and 9.3 of vH).
( For a massive amount of information on Stein's method, see this website).
Week 16 : Problems from Chapters 8 and 9.
We shall have presentations on the following topics
May 13 - 11.30 AM Yogesh : Superconcentration and Chaos (Chapter 3 of [10]).
May 13 - 1.45 PM Parthanil : Berman comparison lemma and extremes of stationary Gaussian sequences.(Notes (there might be errors) )
May 13 - 3.15 PM Sarvesh : Lyapunov Function method and Local Poincare inequalities (Sections 4.6 and 4.7 of [7]).
May 15 - 1.45 PM Prabhanka : Modified log-Sobolev inequalities for strong Rayleigh measures ([9]).
May 15 - 3.15 PM Tejaswi : Generic Chaining (from Chapters 5 and 6 of [vH])).
Class Schedule :
Tuesdays and Fridays : 2- 3 PM. Tuesdays and Thursdays : 4.30 - 5.30 PM.
Marking Scheme :
End-Sem : 50
Class presentation / assignments : 50.
References :
1. [vH] Ramon van Handel : Probability in high dimensions : Notes
2. Manjunath Krishnapur's course on Gaussian processes : Notes
3. R. J. Adler : Introduction to continuity, extrema and related topics for general Gaussian processes.
4. R.J. Adler and J. E. Taylor : Random fields and Geometry.
5. M. Talagrand : Upper and lower bounds for stochastic processes.
6. W. Bryc : Normal distribution: characterizations with applications. (Link)
7. D. Bakry, I. Gentiland M. Ledoux: Analysis and Geometry of Markov Diffusion Operators.
8. S. Chatterjee : A general method for lower bounds on fluctuations of random variables.
9. J. Salez and J. Hermon : Modified log-Sobolev inequalities for strong-Rayleigh measures (see also Modified log-Sobolev inequalities for log-concave measures ).
10. S. Chatterjee : Superconcentration and Related Topics
11. J. M. Steele : Probability Theory and Combinatorial Optimization.
12. J. E. Yukich : Probability Theory of Classical Euclidean Optimization problems.
13. Yuan Liu : The Poincaré inequality and quadratic transportation-variance inequalities (see the references to Ledoux's and Jourdain's papers)
For mid and End-sem exam schedule, see here.