Problem Set 1
- Let Bt be standard Brownian motion and f(x) = x3+ex. Compute df(Bt) using the formalism of Ito's formula.
- Work out an explicit example (as small as you like) of the Cori-Vaquelin-Schaeer, the Mullin bijection, the hamburger cheeseburger bijection, and the bipolar planar map bijection.
- Generate a uniform random spanning tree on a small graph (by hand) using coin tosses and Wilson's algorithm.
- If you have the range of a stable subordinator of parameter α and the range of another stable subordinator of parameter β, what conditions on β and α ensure the intersection of these two sets is almost surely empty? Use the Bessel process relationship to explain your answer.
- Read and take notes on Berestycki's
Introduction to the Gaussian Free Field and Liouville Quantum Gravity (PDF - 2.8MB). (You may also consult Gaussian free fields for mathematicians and the introduction to Liouville quantum gravity and KPZ.) Try working through some of the exercises. - Come talk to me at some point about open problems and / or your final project.
Problem Set 2
- Read and take notes on Schramm-Loewner Evolution (PDF) by Berestycki and Norris. (You may also consult the notes by Lawler, by Kager / Neinhuis, and by Werner in the readings section.) Try working out a few of the things left as exercises.
- Read and take notes on one of the references (your choice) from the Selected References on Universal Object Relationships portion in the readings section.
- Read and take notes on one of the papers (your choice) about growth models in the readings section.
- Come talk to me at some point about open problems and / or your final project.
Final Project
The final project may be either expository or original-research based. Several research problems are suggested in the Open Problems (PDF) document. Collaborative efforts will be allowed. The final project is due on last day of class.