Point Processes

Graduate Topics in Math/Stat I, MATH8048, Fall semester, 2022

Instructor: Yizao Wang
Email: yizao.wang@uc.edu
Office: 4302 French Hall
Office Hours: by appointment.
Class meeting: MWF 2:30pm - 3:25pm, Room 140 WCharlton.

Textbook

• Lecture notes available on Canvas.

Course description

The first part of the course will be on Poisson point processes, starting with a quick review on Poisson processes. Poisson point processes are the fundamental tools to investigate rare/extremal events, and they show up in various areas within and related to probability and statistics, including spatial statistics, stochastic geometry, stochastic combinatorial structures, and number theory.

The second part will be on determinantal point processes, which are extensively used in random matrix theory and integrable probability.

Various examples/applications will be provided from the aforementioned fields.

Pre-req: Probabilith Theory (MATH/STAT7032).

Assignments and Grades

• 2-4 HW problem sets, 50%.
• invididual study project, 50%.
  I will provide a list of topics and references for each student to pick, with a few guideline questions, at the beginning of the semester. Each topic is provided with one or two papers, and the task is to either read some proofs, or to implement certain algorithm to simulate the point processes. The student is required to study the materials related to the topic picked and give a presentation at the end.
• No exams.
• Letter grades: A (90), A- (85), B+ (80), B (75), B- (70), etc.

Tentative Schedule and Homework

HWs will be posted, downloaded, submitted and graded via Canvas. See info/deadlines there.

• Week 1 (Aug 22): Review of Poisson processes.
• Week 2 (Aug 29): Poisson point processes.
• Week 3 (Sep 5): No class on Labor Day, Sep 5.
• Week 4 (Sep 12): Stochastic processes based on Poisson point processes.
• Week 5 (Sep 19): Basics of extreme value theory.
• Week 6 (Sep 26): Point processes in extreme value theory.
• Week 7 (Oct 3): More on convergence of point processes.
• Week 8 (Oct 10): No class on Reading Day, Oct 10.
• Week 9 (Oct 17): Determinantal point processes.
• Week 10 (Oct 24):
• Week 11 (Oct 31):
• Week 12 (Nov 7):
• Week 13 (Nov 14):
• Week 14 (Nov 21): Thanks giving, no class on Nov 25.
• Week 15 (Nov 28):
• Week 16 (Dec 5): No final exam.

---