MATH 523 - Advanced Probability

Math 523 - Advanced Probability

Spring 2009

Instructors: Yuval Peres and Eyal Lubetzky
Email: eyal [at] math [dot] washington [dot] edu
Office: Padelford C-529
Office hours: by appointment

Meeting Times and Locations
Mon,Fri 11:00-12:20 at LOW 106.

Course description
The applications of probability to many areas in Mathematics and in other fields have multiplied dramatically in recent years. Rich interactions with classical analysis have been found in the study of random fractals; researchers in Combinatorics, Theoretical CS, machine learning, high-dimensional geometry (and, of course, Statistics) increasingly need sophisticated probabilistic tools. The aim of this course is to provide such tools, while also preparing the students for more advanced courses in Probability.

The main focus will be on martingales and their applications in Discrete Mathematics and Computer Science. These include: concentration inequalities, optional stopping, maximal inequalities, L2 martingales, and much more.

Homework (35%), Take-home exam (35%) [done individually, but with open books], Presentations (30%).
Students can opt to be graded solely on the HW and take-home exam, 50% each.
Group discussions on the material and HW are encouraged, yet writing up the HW must be done individually.



Topics for presentations
  • A constructive proof of the Lovasz Local Lemma (arXiv article)
  • Hoeffding's inequality for hypergeometric variables (JSTOR article, Section 6 [pp. 28-29])
  • Lower bound on the size of the critical giant component (arXiv article, Theorem 2, proved in Section 5)
  • Coalescence in continuous time: bound for n particles (book chapter, Section 2, Proposition 5)
  • Evolving sets (book chapter, Chapter 17.4 [pp. 235-239])