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.

Relationships among universal objects:

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.

Selected references on universal objects


  • Introductory slides

    Graduate probability background

  • Probability: theory and examples (Durrett)
  • Slides and other references from 18.175

    Yang Mills

  • Quantum Yang-Mills Theory (Jaffe and Witten)
  • SO(N) lattice gauge theory (Chatterjee)
  • SU(N) lattice gauge theory (Jafarov)
  • Large N expansion (Chatterjee and Jafarov)
  • Large N master field in 2D (Levy)
  • Formal matrix integrals and combinatorics of maps (Eynard)
  • Random matrices and enumeration of maps (Guionnet)
  • Wishart matrices (Lalley)
  • Continuum random tree

  • Random trees, Levy processes, and spatial branching processes (Duquesne and Le Gall)
  • Levy processes, stable processes, and subordinators (Lalley)
  • Bessel processes (Lawler)
  • Continuous martingales and Brownian motion (Revuz and Yor)
  • Levy processes (Bertoin)
  • Poisson point proccesses (Johnson)
  • Stable loop trees (Curien and Kortchemski)
  • Continuum random tree I, Continuum random tree II, Continuum random tree III (Aldous)
  • Brownian motion (Morters and Peres)
  • Moore's theorem (Timorin)
  • Random planar maps and the Brownian map

  • Scaling limits of random trees and planar maps (Le Gall and Miermont)
  • Random geometry on the sphere (Le Gall)
  • Slides on Cori-Vauquelin-Schaeffer bijection and Brownian map convergence (Bernardi)
  • Quantum gravity and inventory accumulation (Sheffield)
  • Bipolar orientations (Kenyon, Miller, Sheffield, Wilson)
  • Schnyder woods (Li, Sun, Watson)
  • An axiomatic characterization of the Brownian map (Miller, Sheffield)
  • Gaussian free field

  • Gaussian free fields for mathematicians (Sheffield)
  • Topics on the two-dimensional Gaussian free field (Werner)
  • Log-correlated free field in general dimension (Duplantier, Rhodes, Sheffield, Vargas)
  • Fractional Gaussian fields: a survey (Lodhia, Sheffield, Sun, Watson)
  • Liouville quantum gravity

  • Liouville Quantum Gravity and KPZ (Duplantier and Sheffield)
  • Quantum gravity and the KPZ formula (Garban)
  • Introduction to the Gaussian Free Field and Liouville Quantum Gravity (Berestycki)
  • Polyakov's formulation of 2d Bosonic string theory (Guillarmou, Rhodes, Vargas)
  • Gaussian multiplicative chaos and applications: a review
  • Determinants of Laplacians; Heights and Finiteness (Sarnak)

    Schramm-Loewner evolution and discrete analogs

  • Random planar curves and Schramm-Loewner evolutions (Werner)
  • Conformally Invariant Processes in the Plane: Summer School Lecture Notes (Lawler)
  • Conformally Invariant Processes in the Plane: Book (Lawler --- save and use online ps2pdf if your machine doesn't have postscript).
  • A Guide to Stochastic Loewner Evolution and its Applications (Kager and Nienhuis)
  • Lectures on Schramm-Loewner evolution (Berestycki and Norris)
  • Growth models

  • Diffusion limited aggregation (Witten and Sander)
  • DLA bounds (Kesten)
  • Dielectric breakdown model (Niemeyer, Pietronero, Weismann)
  • Introduction to KPZ (Quastel)
  • Renormalization fixed point of the KPZ universality class (Corwin, Quastel, Remenik)
  • KPZ equation and universality class (Corwin)
  • Directed polymers (Borowin, Corwin, Ferrari)
  • Selected references on universal object relationships

    GFF + SLE

  • A contour line of the continuum Gaussian free field (Schramm and Sheffield)
  • Level lines of Gaussian Free Field I: Zero-boundary GFF (Wang and Wu)
  • Imaginary geometry I: Interacting SLEs (Miller and Sheffield)
  • Imaginary geometry II: reversibility results for kappa in (0,4) (Miller and Sheffield)
  • Imaginary geometry III: reversibility results for kappa in (4,8) (Miller and Sheffield)
  • Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees (Miller and Sheffield)
  • LQG + LQG = LQG + SLE

  • Conformal weldings of random surfaces: SLE and the quantum gravity zipper (Sheffield)
  • Notes on Sheffield's quantum zipper (Benoist)
  • Introduction to the Gaussian Free Field and Liouville Quantum Gravity (Berestycki)
  • CRT + CRT = LQG + SLE

  • Liouville quantum gravity as a mating of trees (Miller and Sheffield)
  • Liouville quantum gravity spheres as matings of finite-diameter trees (Miller and Sheffield)
  • LQG + reshuffled SLE = LQG + DBM

  • Quantum Loewner Evolution (Miller and Sheffield)
  • LQG = TBM

  • Liouville quantum gravity and the Brownian map I (Miller and Sheffield)
  • Liouville quantum gravity and the Brownian map II (Miller and Sheffield)
  • Liouville quantum gravity and the Brownian map III (Miller and Sheffield)

    Open problems:

    Open problem document in progress

    Problem sets and final project:

    Problem set document