**Lectures:** Tuesday and Thursday 1:00-2:30, Room 2-142.
First lecture Thursday, September 7.

**Office hours:** Tuesday and Thursday, 2:30-3:30, Room
2-249.

**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: ** various readings, including some topics in
lecture notes with Jason Miller in progress

**Course topics:** first an introduction to Yang-Mills lattice
gauge theory and some of its interesting variants, along with
their relationships to embedded planar maps and discretized string
trajectories. Second, an overview of universal random
structures in
1D and 2D, including Brownian motion, Bessel processes, stable Levy
processes,
ranges of stable subordinators, continuum random trees, Gaussian random
distributions
fields, and random curves and loop ensembles. Some discussion of
motivating problems from statistical physics, quantum
field theory, conformal field
theory, string theory, and early universe cosmology.

1. ** Lattice and continuum connections: ** defining lattice
Yang-Mills, compact and non-compact gauge groups, gauge
invariance and gauge fixing, Gaussian ensembles

2. ** Planar maps and random matrix integrals: **
planar map enumeration calculations involving Wick's theorem, variants
that encode statistical physics models, t'Hooft limits

3. ** String trajectories and embedded planar maps: ** Chatterjee's
discretized string trajectories along with related earlier work,
1/N expansion

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 probability background **

** Yang Mills **

**Continuum random tree **

**Random planar maps and the Brownian map**

** Gaussian free field**

** Liouville quantum gravity **

** Schramm-Loewner evolution and discrete analogs **

** Growth models **

** GFF + SLE **

** LQG + LQG = LQG + SLE **

** CRT + CRT = LQG + SLE **

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

** LQG = TBM **

Open problem document in progress