18.177 | Fall 2015 | Graduate

Universal Random Structures in 2D


Problem Set 1

  1. Let Bt be standard Brownian motion and f(x) = x3+ex. Compute df(Bt) using the formalism of Ito’s formula.
  2. 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.
  3. Generate a uniform random spanning tree on a small graph (by hand) using coin tosses and Wilson’s algorithm.
  4. 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.
  5. 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.
  6. Come talk to me at some point about open problems and / or your final project.

Problem Set 2

  1. 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.
  2. Read and take notes on one of the references (your choice) from the Selected References on Universal Object Relationships portion in the readings section.
  3. Read and take notes on one of the papers (your choice) about growth models in the readings section.
  4. 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.

Course Info

As Taught In
Fall 2015
Learning Resource Types
Problem Sets
Lecture Notes
Written Assignments