MATH 758 -- Combinatorial Stochastic Processes Winter 2010 -- Course syllabus
Professor	Louigi Addario-Berry | louigi@math.mcgill.ca | Tel: (514) 398-3831 (office) | 1219 Burnside Hall | Office Hours: By appointment.
Time and Location	TTh 13:05-14:25 | Burnside Hall 1205 | 5 January -- 13 April, 2010
Description	Mostly based on Pitman, Chapters 1,2,3,6,7. Stirling numbers, Bell polynomials, composite structures, Gibbs partitions. Connection with moments and cumulants; infinitely divisible distributions, the Lévy-Khintchine Theorem. Random sampling, de Finetti's theorem. Pólya's urn scheme. Exchangeable random partitions, infinite partitions, Kingman's paintbox. The Chinese restaurant process, the two-parameter model. Random trees and forests; connection with random walks and excursions. Brownian forest, the continuum random tree, definition and constructions. Gromov-Hausdorff converence. Sampling random leaves. Cutting down random trees.
Evaluation	Two percent for each full solution to an exercise from Pitman, to a maximum of 100% of your grade. Solutions must be submitted in LaTeX and multiple attempts at the same question are allowed. You may work in groups on the questions from Pitman; if you do then the group should hand in a single solution to the question, with the name of all group members. The remainder of the mark will be bsaed on an in-class presentation of a publication or of a section of the book.
Course book	Pitman's Combinatorial stochastic processes, available for download here. Note: some course material may not be covered in the textbook. My notes on Gibbs fragmentations and on infinite exxchangeable partitions
Prerequisites	Tao's review of probability theory and some of the links therein (notably to the Fubini-Tonelli theorem and the Radon-Nikodym derivative). You may want to read Tao's review of measure and integration theory first. If you do not already know this material at the start of the course, be prepared to catch up quickly. The section on conditional probability and conditional expectation is particularly important. We will cover further aspects of this area in the course. The following notes on stochastic processes, while not a prerequisite, may be of use during the course.
Assignments	Any full solutions to exercises from Pitman; see the grading scheme.
Course Outline	PDF version.
Other resources	Words of wisdom from Fields medalist Terence Tao, for graduate mathematics students: It is important to work hard, and work professionally. But it is also important to enjoy your work. Think ahead to understand the way forward; ask yourself dumb questions to understand the way before. Attend talks and conferences, even those not directly related to your own work. Talk to your advisor, but also take the initiative. Don’t prematurely obsess on a single “big problem” or “big theory”. Write down what you’ve done, and make your work available. In this regard, I have some advice on how to write and submit papers. George Polya's How to solve it and the Wikipedia version Slightly off-base: Tim Gowers' mathematical discussions.
Additional Information	In accord with McGill University's Charter of Students' Rights, students in this course have the right to submit in English or in French any written work that is to be graded. McGill University values academic integrity. Therefore all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/students/srr/honest/ ) for more information).