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.
This course covers most of the materials in the first seven chapters of Probability and Random Processes by Grimmett and Stirzaker. In particular, it covers topics such as generating and characteristic functions and their applications in random walk and branching process, different modes of convergence and an introduction of martingales.
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.
This course develops the theory of high-dimensional probability: random vectors and matrices and the mathematics of how these transform when one applies transformations (especially convex transformations) to them, such as norms and seminorms, eigenvalue maps, and others. This reveals fundamental geometric properties of normed spaces and convex sets in high dimensions, and it is also deeply connected to modern application in statistics, computer science, and data science. The material has substantial applications to topics in statistics and machine learning. Topics covered will be: concentration of measure, net arguments and norm bounds, Gaussian processes and Gaussian concentration, chaining, and many applications of the above.
Espace de probabilité, variables aléatoires, indépendance, espérance mathématique, modes de convergence, lois des grands nombres, théorème central limite, espérance conditionnelle et martingales. Introduction au mouvement brownien.
Tribus et variables aléatoires. Théorie de l'intégration: théorème de Lebesgue, espace Lp, théorème de Fubini. Construction de mesures, mesure de Radon. Indépendance. Conditionnement.
The first part of this course covers materials such as Markov chain, branching processes and optimal stopping for Markov chains. The second part covers Brownian motion and its properties, continuous time martingales and stochastic integral. Girsanov transform, Feynman-Kac formula and stochastic differential equations will also be introduced.
This course develops the main topics in (discrete time) stochastic process theory: Markov chains, random walks, branching processes, and martingales.
Characteristic functions: elementary properties, inversion formula, uniqueness, convolution and continuity theorems. Weak convergence. Central limit theorem. Additional topic(s) chosen (at discretion of instructor) from: Martingale Theory; Brownian motion, stochastic calculus.
Mouvement brownien, intégrale stochastique, formule d’Itô, équations différentielles stochastiques, théorèmes de représentation, théorème de Girsanov. Formule de Black et Scholes.