18.177 Topics in Stochastic Processes: Random Planar Maps, Liouville Quantum Gravity, and SLE (Fall 2014)

(The first image is a random quadrangulation with 25,000 faces embedded (non-isometrically) into 3-dimensional space and was taken from here. The second image is a circle packing, generated with CirclePack, of a random quadrangulation. The final image is a space-filling SLE(6). Additional images can be found here and here.)

Instructor: Jason Miller (E18-470)

Lectures: Tuesday and Thursday 1-2:30 (E17-136)

Office hours: After class on Thursdays

Contact: jpmiller@mit.edu

Course overview

Prerequisites: graduate level probability, Ito calculus, and complex analysis


Schedule and topics: (tentative)

  • Part 1 (September): Random Planar Maps
  • Part 2 (October): Liouville Quantum Gravity
  • Part 3 (November, December): SLE and its connections to the GFF/LQG
  • Wikipedia links:
  • Planar map
  • Gaussian free field
  • SLE
  • CLE
  • References:
  • Scaling limits of random trees and planar maps (Le Gall and Miermont)
  • Random geometry on the sphere (Le Gall)
  • Quantum gravity and inventory accumulation (Sheffield)
  • Gaussian free fields for mathematicians (Sheffield)
  • Liouville Quantum Gravity and KPZ (Duplantier and Sheffield)
  • Quantum gravity and the KPZ formula (Garban)
  • Random planar curves and Schramm-Loewner evolutions (Werner)
  • Conformally Invariant Processes in the Plane (Lawler)
  • A Guide to Stochastic Loewner Evolution and its Applications (Kager and Nienhuis)
  • Conformal weldings of random surfaces: SLE and the quantum gravity zipper (Sheffield)
  • A contour line of the continuum Gaussian free field (Schramm and Sheffield)
  • Imaginary Geometry I: Interacting SLEs (Miller and Sheffield)
  • Quantum Loewner Evolution (Miller and Sheffield)