• Instructor: Lionel Levine
  • Office hours: Thursday 2-3 and Friday 2:30-3:30 in Malott 438


• TA: Swee Hong Chan
  • Office hour: Wednesday 3:30-5:30 in Malott 218

Syllabus

This is the second semester of probability at the graduate level. Topics include Markov chains, probability on trees and networks, Brownian motion, martingales in continuous time, stationary sequences and ergodic theory. The syllabus has a more detailed list of topics (subject to change!)

Textbooks

• Probability on Trees and Networks, by Russell Lyons and Yuval Peres

• Markov Chains and Mixing Times (2nd edition), by David Levin and Yuval Peres with Elizabeth Wilmer

• The Probabilistic Method (3rd edition), by Noga Alon and Joel Spencer

• Brownian Motion, by Peter Moters and Yuval Peres

• Probability: Theory and Examples (4th edition), by Rick Durrett

Problem Sets

• Problem set 1, due Friday 2/16

• Problem set 2, due Friday 3/23

• Problem set 3, due Weds 5/9

Aim for the writing style of a research paper. Many scientfic papers end with "acknowledgements'' where you can thank X and Y for inspiring conversations, Z for pointing out a gap in your proof, W for feeding and encouraging you when the going got tough, and your favorite coffee shop for providing an essential raw material.

The vast majority of math papers are typeset using LaTeX.

Group work policy

Working in groups is strongly encouraged! Discuss all you want, then write the solution in your own words. (Hint: It will be hard for your classmate to write the solution in their own words if you give them your written solution to look at.) You are free to use any online or offline resource for the problem sets, provided you clearly state when you've done so. If someone gave you a good idea, it would be polite to thank them. And yes, "them" can be a singular pronoun.

Presentations

Toward the end of the semester, you'll present in a group of 2-5 on a probability research topic of your choice. You can choose between two fora:

• Present in class during the last two weeks of the semester; or
• Present in the arXiv seminar, which meets Tuesdays 11:40-12:55.

The length of the presentation is 50 minutes if you present in class, or up to 75 minutes if you present in the arXiv seminar. Your group can choose how to split up the time among the group members. In that time the group should aim to state one theorem, place it in context (why is it interesting? what gap in human knowledge does it fill? what related things are true/false/easy/hard/known/unknown?), and convey the main idea of the proof.

Expect a lot of questions from me and your classmates that will slow you down! If a practice presentation takes 30 minutes with no questions, then you're in good shape.

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