**Lectures:** Tuesday and Thursday 1:00-2:30, TR 1-2:30.
First lecture Thursday, September 10.

**Office hours:** Tuesday 2:30-3:30 and Thursday 2:30-3:30.

**Assignments:** three problem sets and one final project. Final
project may be either expository or original-research based.
Several suggested research problems will be presented. Collaborative
efforts will be
allowed.

**Texts:** course notes and references to be assigned.

** Prerequisites:** basic probability at the level of an
introductory
graduate course (18.175 or equivalent).

** Textbook: **
Lecture notes with Jason Miller in progress

**Course topics:** first 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. Then 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.

1. ** Random planar trees: ** Aldous's continuum random tree,
Levy trees, loop trees,
Brownian snakes, Galton Watson trees and uniform
random trees.

2. ** Random generalized functions: ** Gaussian free fields (free
boundary, fixed
boundary, massive), fractional Gaussian fields, log correlated
free fields, discrete Gaussian
free field, dimer model height functions, uniform spanning tree height
functions, non-intersecting lattice paths and determinants, Laplacian
determinants.

3. ** Random curves: ** Schramm-Loewner evolution (SLE), conformal
loop
ensembles (CLE), 2D Brownian motion, percolation,
Ising and Potts models, FK cluster models, GFF level lines/harmonic
explorer, uniform spanning tree boundary, loop-erased random walk,
Wilson's algorithm.

4. ** Random surfaces: ** Brownian map and Liouville quantum
gravity, multiplicative chaos, random planar maps, random
quadrangulations,
random triangulations, the Schaeffer bijection, the Mullin bijection, the
FK bijection.

5. ** Random growth models: ** KPZ growth,
Brownian web, Hastings-Levitov,
DLA, Eden model, internal DLA.

6. ** Random connections: ** Yang Mills, quantum electromagnetism,
lattice Yang Mills.

1. **Imaginary geometry:** generalized functions and
curves.

2. **Conformal welding:** surfaces, generalized
functions and curves.

3. ** Mating trees and the peanosphere: ** trees,
surfaces, generalized functions and curves.

4. ** Quantum Loewner evolution and the Brownian map:** growth
models, trees, surfaces, generalized functions and curves.

** Overview **

** Graduate probabiliy background **

**Continuum random tree **

**Random planar maps and the Brownian map**

** Gaussian free field**

** Liouville quantum gravity **

** Schramm-Loewner evolution and discrete analogs **

** Growth models **

** Yang Mills **

** GFF + SLE **

** LQG + LQG = LQG + SLE **

** CRT + CRT = LQG + SLE **

** LQG + reshuffled SLE = LQG + DBM **

** LQG = TBM **

Open problem document in progress