Free Probability and Combinatorics: Math 689
Fall 2012, MWF 11:30–12:20 p.m. in Blocker 148
Professor: Michael Anshelevich, Milner 326, manshel@math.tamu.edu.
Office hours: MW 2–3 p.m., T 11–12 a.m., or by appointment.
Textbook: Alexandru Nica and Roland Speicher, L ECTURES ON THE C OMBINATORICS
P ROBABILITY, Cambridge University Press, 978-0521858526.
OF
F REE
Prerequisites: Math 608 or permission of instructor. All students should be familiar with the
notions of a Hilbert space, linear operator, and linear functional. Analysis students are expected to
have taken Math 607-608 or equivalent. Combinatorics students are expected to have taken Math
630.
Learning Objectives: Free Probability Theory studies non-commuting objects using probabilistic
language and techniques. This course is an introduction to Free Probability with an emphasis on the
combinatorial aspects of the theory. At the conclusion of the course students should be ready to start
research in free probability. For students interested in pursuing other research directions, the course
will provide connections with numerous other fields, notably Operator Algebras, Combinatorics,
and the study of Random Matrices.
Exams: None.
Homework: Weekly.
Final project: The course requirements will include an optional final project, written on a topic
of student’s interest, as approved by the instructor. Deadlines: September 17 (decide whether to
do a project), October 1 (choose a topic), November 12 (first draft due), December 3 (project due),
December 12 (presentations).
Grading: Each student will choose one of the following grading schemes: (A) Homework 100%;
(B) Top half homework assignments 50%, final project 50%. A total score of 90% or more guarantees an A, a score of 80% or more a B, 70% or more a C, 60% or more a D.
Scholastic Dishonesty: Cheating of any form is not acceptable and it will be dealt with harshly.
In particular, copying work done by others, either in-class or out of class, is an act of scholastic
dishonesty and it will be prosecuted to the full extent allowed by university policy. Collaboration
on homework assignments is permitted for this course, however each student must write up their
own solutions. For more information on university policies regarding scholastic dishonesty, see the
University Student Rules.
Aggie Honor Code: “An Aggie does not lie, cheat, or steal or tolerate those who do.”
Students with disabilities: Come talk to me no later than the first week of classes. “The Americans
with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive
civil rights protection for persons with disabilities. Among other things, this legislation requires
that all students with disabilities be guaranteed a learning environment that provides for reasonable
accommodation of their disabilities. If you believe that you have a disability requiring an accommodation, please contact the Department of Student Life, Services for Students with Disabilities, in
Room 126 of the Koldus Building or call 845–1637.”
Attendance: According to the University Student Rules, absence for three or more class days
requires a University-approved excuse and documentation.
1
Preliminary course outline (subject to modification):
• Noncommutative probability spaces and distributions.
– Non-commutative probability spaces. Examples. Group algebras.
– Distributions. Weak law of large numbers.
• Functional analysis background.
– C∗ - and von Neumann algebras. Spectral theorem.
– Gelfand-Naimark-Segal construction.
– Distributions and moments. Group algebras. Free groups. Tensor products. Unbounded operators. Moment problem.
• Free independence.
– Independence. Free independence.
– Free products: groups, algebras, Hilbert spaces. Positivity.
• Set partition lattices.
– All, non-crossing, interval partitions. Enumeration.
– Incidence algebra. Möbius inversion. Kreweras complement. Multiplicative functions.
• Free cumulant machinery.
– Free cumulants. Generating functions. Möbius function.
– Free cumulants of products.
– Free independence as vanishing mixed free cumulants.
– R-transform. Free convolution. Semicircular, free Poisson, and free compound Poisson
distributions. Limit theorems.
• Lattice paths and Fock space models.
– I. Dyck paths. Full Fock space. Creation and annihilation operators. Free Wick formula. Multivariate free central limit theorem. Chebyshev polynomials.
– II. Lukasiewicz paths. Voiculescu’s operator model.
– III. Motzkin paths. Schürmann’s operator model. Freely infinitely divisible distributions. Free convolution semigroups. Conditionally positive functionals. Schönberg
correspondence. Bercovici-Pata bijection.
– IV. Motzkin paths and orthogonal polynomials. A general Fock space construction.
Free Meixner distributions.
– V. q-deformed free probability. Dyck paths with q-weights. q-Gaussian distribution.
q-Fock space. q-moment-cumulant formula.
• Free Lévy processes.
– Free Appell polynomials. Markov property.
• Multiplicative free convolution and the R-transform.
– Compound Poisson realization. Free compression.
• Belinschi-Nica evolution.
– Relation between free and Boolean cumulants. Bt transformation. Φ transformation.
Evolution equation.
• Combinatorics of random matrices.
– Random matrices. Gaussian random variables. Wick formula. GUE.
– Map enumeration. Matrix models.
– Partitions and permutations.
– Asymptotic free independence for Gaussian random matrices.
– Asymptotic free independence for unitary random matrices.
• Operator-valued free probability.
– B-valued probability spaces. Complete positivity. Free products with amalgamation.
Operator-valued free cumulants. Operator-valued semicircular distributions. Relation
between different operator-valued cumulants.
• Survey of analytic results.
– Complex analytic methods. Limit theorems. Stable distributions.
– Free entropy. Microstates and non-microstates approaches.
– Operator algebras. Interpolated free group factors.
All printed handouts and web-materials are protected by US Copyright Laws. No multiple copies can be
made without written permission by the instructor.