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
Exercises
Schedule and topics: (tentative)
Part 1 (September): Random Planar Maps
Discrete and continuum random trees
Random quadrangulations
The Brownian map
FK-weighted maps / hamburger-cheeseburger bijection
Part 2 (October): Liouville Quantum Gravity
Discrete and continuum GFF
Liouville quantum gravity measure
KPZ formula
Liouville Brownian motion
Quantum wedges, cones, disks, and spheres
Part 3 (November, December): SLE and its connections to the GFF/LQG
Derivation and construction of SLE
Continuity of the SLE trace
Forward / reverse couplings of SLE with the GFF
Conformal welding of quantum surfaces
Imaginary geometry
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)
