The Stochastic Analysis Seminar is devoted to an informal seminar course on contemporary topics in probability or stochastic analysis that are not currently taught in the curriculum. The current and previous topics are detailed below. Talks by outside speakers can be found in the Probability Seminar.

For more information on this seminar, please contact Ramon van Handel.

## Spring 2016: Optimal transportation

*Announcements and schedule changes will be posted on the stochastic analysis seminar blog.*

Optimal transportation is the problem of coupling probability measures in an optimal manner (in the sense that a certain expected cost functional is minimized.) The rich theory that arises from this problem has had far-reaching consequences in probability theory, geometry, PDEs, interacting particle systems, etc. The aim of these informal lectures is to introduce some of the basic ingredients of this theory and to develop some probabilistic applications. Potential topics include: Wasserstein distances and Kantorovich duality; the Brenier map; the Monge-Ampere equation; displacement interpolation; Caffarelli's theorem; applications to functional inequalities, measure concentration, dissipative PDEs, and/or interacting particle systems.

**Time and location:** Thursdays, 4:30-6:00 PM, Bendheim Center classroom 103.

The first lecture will be on February 4. **There will be no lecture on February 25.**

**References:**

- C. Villani, "Topics in Optimal Transportation", AMS (2003).
- L. Ambrosio and N. Gigli, "A User's Guide to Optimal Transport", lecture notes.
- K. Ball, "An Elementary Introduction to Monotone Transportation", lecture notes.
- D. Cordero-Erausquin, "Some Applications of Mass Transport to Gaussian-Type Inequalities", Arch. Rational Mech. Anal. 161 (2002) 257–269.
- D. Bakry, I. Gentil, and M. Ledoux, "Analysis and Geometry of Markov Diffusion Operators", Springer (2014), Chapter 9.

## Fall 2014: Roots of polynomials and probabilistic applications

*We will again experiment with the stochastic analysis seminar blog this semester.
Notes from the lectures, announcements and schedule changes will be posted there.*

Understanding the roots of polynomials seems far from a probabilistic issue, yet has recently appeared as an important technique in various unexpected problems in probability as well as in theoretical computer science. As an illustration of the power of such methods, these informal lectures will work through two settings where significant recent progress was enabled using this idea. The first is the proof of the Kadison-Singer conjecture by using roots of polynomials to study the norm of certain random matrices. The second is the proof that determinantal processes, which arise widely in probability theory, exhibit concentration of measure properties. No prior knowledge of these topics will be assumed.

**Time and location:** Thursdays, 4:30-6:00, Sherrerd Hall 101.

The first lecture will be on September 18.

**References:**

- A. W. Marcus, D. A. Spielman, N. Srivastava, "Interlacing families I/II"
- Notes by T. Tao
- Notes by N. K. Vishnoi
- J. Borcea, P. Brändén, T. M. Liggett, "Negative dependence and the geometry of polynomials"
- R. Pemantle and Y. Peres, "Concentration of Lipschitz functionals of determinantal and other strong Rayleigh measures"

## Fall 2013: Information theoretic methods in probability

*We will again experiment with the stochastic analysis seminar blog this semester.
Notes from the lectures, announcements and schedule changes will be posted there.*

While information theory has traditionally been based on probabilistic methods, ideas and methods from information theory have recently played an increasingly important role in various areas of probability theory itself (as well as in statistics, combinatorics, and other areas of mathematics). The goal of these informal lectures is to introduce some topics and tools at the intersection of probability theory and information theory. No prior knowledge of information theory will be assumed. Potential topics include: entropic central limit theorems, entropic Poisson approximation, and related information-theoretic and probabilistic inequalities; connections to logarithmic Sobolev inequalities and Stein's method; entropic inequalities for additive, matroidal, and tree structures, and their applications; transportation cost-information inequalities and their relation to concentration of measure; basic large deviations theory.

**Prerequisites:** Probability at the level of ORF 526 is assumed.

**Time and location:** Thursdays, 4:30-6:00, Bendheim Center classroom 103.

The first lecture will be on September 19. Schedule changes will be posted to the blog.

**References:**

- N. Gozlan and C. Leonard, "Transport Inequalities. A Survey"
- O. Johnson, "Information Theory and the Central Limit Theorem"
- M. Ledoux, "The Concentration of Measure Phenomenon"
- C. Villani, "Topics in Optimal Transportation"

## Spring 2013: Random graphs

*New feature this semester: stochastic analysis seminar blog.
Notes from the lectures and any announcements will be posted there.*

Complex graphs and networks are ubiquitous throughout science and engineering, and possess beautiful mathematical properties. The goal of these informal lectures is to provide an introduction to some of the most basic results in the theory of (Erdos-Renyi) random graphs from the probabilistic viewpoint: chromatic number and clique number, and the size of the largest component in the subcritical, supercritical, and critical regimes. If time permits, we will discuss Brownian limits.

**Prerequisites:** Probability at the level of ORF 526 is assumed.

**Time and location:** Thursdays, 4:50-6:20, Bendheim Center classroom 103.

The first lecture will be on February 28. **There will be no lecture on March 28** (colloquium).

**References:**

- N. Alon and J. Spencer, "The Probabilistic Method"
- R. Durrett, "Random Graph Dynamics"
- A. Nachmias and Y. Peres, "The critical random graph, with martingales", Israel J. Math. 176 (2010), p. 29-41
- D. Aldous, "Brownian excursions, critical random graphs and the multiplicative coalescent", Ann. Probab. 25 (1997), p. 812-854
- G. Grimmett, "Probability on Graphs"
- R. van der Hofstad, "Random Graphs and Complex Networks" (lecture notes)
- B. Bollobas, "Random Graphs"
- S. Janson, T. Luczak, A. Rucinski, "Random Graphs"

## Fall 2012: Random matrices

Both the theory and applications of random matrices have attracted significant attention in recent years. In particular, much effort has gone into understanding the spectral properties of large random matrices, with applications ranging from statistical physics to signal processing to high-dimensional statistical inference.

The goal of these informal lectures is to provide a basic introduction to some selected results in random matrix theory. Potential topics include: Wigner matrices and the semicircle distribution; Wishart matrices and the Marcenko-Pastur distribution; asymptotics of the top and bottom singular values; fluctuations of linear eigenvalue statistics and of the top eigenvalue; Dyson Brownian motions and related topics. If time permits (which is unlikely) we will introduce some free probability.

**Prerequisites:** Probability at the level of ORF 526 is assumed.

**Time and location:** Thursdays, 4:30-6:00, Bendheim Center classroom 103.

The first lecture is on September 20. **There will be no lecture on October 25 and November 1.**

**References:**

- G. W. Anderson, A. Guionnet, O. Zeitouni, "An Introduction to Random Matrices"
- Z. Bai and J. W. Silverstein, "Spectral Analysis of Large Dimensional Random Matrices"
- T. Tao, "Topics in Random Matrix Theory"
- R. Vershynin, "Introduction to the non-asymptotic analysis of random matrices" (lecture notes)

## Fall 2011: Concentration of measure

The law of large numbers states that the average of many independent random variables is close to its expectation. It turns out that this simple fact is a special case of a much more general phenomenon that could be informally phrased as follows: "a function of many independent random variables, that does not depend too much on any one of them, is nearly constant". This idea, called concentration of measure, appears in many different areas of probability and its applications, and there exists a powerful set of tools to establish that such properties hold in a precise quantitative sense (in the form of sharp nonasymptotic estimates on the deviation of the random variable of interest from its expectation or median).

The goal of these informal lectures is to provide a basic introduction to such methods. Potential topics include Chernoff bounds, Hoeffding, Bernstein and Azuma inequalities, bounded differences, Gaussian concentration, isoperimetry, log-Sobolev inequalities, transportation cost inequalities, Talagrand's concentration inequalities, and some random applications.

**Prerequisites:** Probability at the level of ORF 526 is assumed.

**Time and location:** Thursdays 4:30-5:30 PM in Bendheim Center classroom 103.

The first lecture is on September 15. **There will be no lecture on October 6 and 27, and December 15.**

**References:** The following references (ordered alphabetically) discuss theory and applications of concentration of measure from different perspectives. We may use parts of some of these references, but will not follow one particular reference throughout.

- N. Alon and J. Spencer, "The Probabilistic Method"
- N. Berestycki and R. Nickl, "Concentration of Measure" (lecture notes)
- D. Dubhashi and A. Panconesi, "Concentration of Measure for the Analysis of Randomised Algorithms"
- M. Ledoux, "The Concentration of Measure Phenomenon"
- G. Lugosi, "Concentration-of-measure inequalities" (lecture notes)
- P. Massart, "Concentration Inequalities and Model Selection" (St Flour lecture notes)
- C. McDiarmid, "Concentration"

C. McDiarmid, "On the method of bounded differences" - G. Pisier, "Probabilistic methods in the geometry of Banach spaces"
- J. M. Steele, "Probability Theory and Combinatorial Optimization"
- M. Talagrand, "Concentration of measure and isoperimetric inequalities in product spaces"

M. Talagrand, "New concentration inequalities in product spaces"

M. Talagrand, "A new look at independence"