18.177 Gauge theory and random surfaces: Fall, 2017

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.

Lattice gauge theory: Yang-Mills and its variants

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

Summary of universal objects and discrete analogs:

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.