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Fall 2024 Graduate Courses

Math 6510 - Representation Theory Charles Conley

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Math 6810 - Probability Pieter Allaart

Overview: The purpose of this course is to explore the interface between probability theory and fractal geometry. The first four weeks or so will be spent on a rigorous development of probability based on measure theory. Topics will include the law of large numbers, conditional expectation and martingales. In the middle part of the course, we will introduce Hausdorff dimension, iterated function systems, self-similar sets (including the ternary Cantor set) and their generalizations. We will then learn some important probabilistic techniques for computing the Hausdorff dimension of these sets. In the third part of the course, probability and fractal geometry come together in the study of random fractals, with randomized versions of the classical Weierstrass function as one of the main examples. The course ends with a detailed study of the various fractal properties of Brownian motion.

Text: There is no required text for this course, but recommended reading includes:

  • P. Billingsley, Probability and Measure (3rd edition), Wiley
  • K. Falconer, Fractal geometry. Mathematical foundations and applications (2nd or 3rd edition), Wile

Prerequisites: Math 5320 (measure theory). Some knowledge of undergraduate-level probability is helpful, but not required.

Grading: Your grade will be based on attendance, active class participation and one or two 30-50 minute presentations, depending on the number of students in the class.

Breadth requirement: This course satisfies the Analysis breadth requirement.