18.177 | Fall 2015 | Graduate

Universal Random Structures in 2D


Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session


Basic probability at the level of an introductory graduate course, 18.175 Theory of Probability or equivalent. Useful background material can also be found in the online Probability: Theory and Examples (PDF - 1.9MB) notes by Rick Durrett at Duke University.


This graduate-level course introduces students to some fundamental 2D random objects, explains how they are related to each other, and explores some open problems in the field.

Topics include:

  • A brief review of universal random structures in 1D, including Brownian motion, Bessel processes, stable Levy processes, ranges of stable subordinators, and Ito’s formula.
  • An introduction to universal random structures that are (at least in some sense) two dimensional or planar, including planar trees, generalized functions on planar domains, Riemannian surfaces, planar growth models, planar loop ensembles, and planar connections.
  • Discussion of motivating problems from statistical physics, quantum field theory, conformal field theory, string theory, and early universe cosmology.

Textbook and Notes

There is no required text; references are assigned in the readings section. A set of lecture notes that is still in development is also available.

Problem Sets & Final Project

There will be two problem sets and one final project. The final project may be either expository or original-research based. Several suggested research problems will be presented. Collaborative efforts will be allowed.


Your grade will be determined by your performance on the problem sets and the final project.

Course Info

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