Probability theory is the branch of mathematics concerned with the analysis of random phenomena. The members of the ISM Probability Group are involved in research in a broad range of areas spanning theoretical and applied, continuous and discrete probability. A particular focus is on the development and analysis of probabilistic models for real-world phenomena from physics, biology, statistics and computer science. Some specific topics of interest are: statistical physics in a random environment, branching systems in biology, distances and random energy landscapes, data structure analysis using random trees, genetics and population biology.
Many members of the group are also members of the CRM Probability Lab.
Students interested in graduate study in any of the areas cited above are invited to apply for admission to the program. There are no formal prerequisites other than those required by the departments. The following guidelines should be followed, however, and courses selected in consultation with an advisor from the group.
Students in the program are expected to have mastered the subject matter of the undergraduate curriculum in probability theory. All students are required to take the basic courses: Real Analysis Measure Theory and Probability Theory. Students are then expected to take a number of more specialized courses.
Probability spaces. Random variables and their expectations. Convergence of random variables in Lp. Independence and conditional expectation. Introduction to Martingales. Limit theorems including Kolmogorov's Strong Law of Large Numbers.
Characteristic functions: Elementary properties, Inversion formula, Uniqueness, convolution and continuity theorems.
Weak convergence: Portmanteau theorem, Sequential compactness, tightness, Prohorov's theorem, Polish spaces, Central limit theorem, Skorokhod's representation theorem.
Stochastic processes: General theory. Kolmogorov Extension theorem, Kolmogorov continuity theorem. Regular probability spaces and conditional distributions, probability kernels. Construction of Brownian motion, Donsker's theorem
Exchangeability: De Finetti's theorem, The Aldous-Hoover theorem.
Other topics if time permits.