Analysis Student Seminar Spring 2020

During this seminar we will investigate Liouville Quantum Gravity in some depth.

This is a graduate student seminar, with talks given by graduate students participants. We all attempt together to understand the topic being discussed. Of course, people with any level of background are welcome, and everyone is especially encouraged to ask questions.

Main Sources

These are the main articles we’ll make use of this semester. Other potentially useful sources are listed below the schedule.

• Lawler, Notes on Probability Prob1
• Berestycki, Introduction to the Gaussian Free Field and Liouville Quantum Gravity LQG2
• Le Gall, Brownian Geometry TBM2

Schedule

All meetings are held on Wednesdays at 4pm in the Math Tower, Room 5-127.

Reading sections in parentheses are optional, and we’ll likely exclude them unless we have spare time.

Potentially Useful Materials

A few of these resources we’ll work through in detail. The rest should be thought of as context for those interested or as reference documents containing proofs of certain results we’ll use along the way.

This course website from a 2017 MIT class taught by Scott Sheffield contains several really useful resources.

Probability

• Lawler, Notes on Probability Prob1
  • A crash course to terminology common in probability. Particularly useful as a reference.
  • Some extra terminology to be aware of which is not used in Lawler:
    1. The law of a random variable is termed the distribution of the random variable in Lawler - both mean the same thing. If $f: \Omega \to X$ is a random variable with $P$ the probability measure on $\Omega$, then the law of $f$ is the resulting pushforward measure on $X$.
    2. If $(\Omega_1, P_1)$ and $(\Omega_2, P_2)$ are probability spaces with $f: \Omega_1 \times \Omega_2 \to X$ (for $X$ a vector space), then the marginal or marginal distribution or marginal law of $f$ on $\Omega_1$ is the random variable $\int_{\Omega_2} f(x,y) dP_2(y)$.
• Lawler, Conformally Invariant Processes in the Plane Prob2
  • Chapter 2 is a useful reference for some of the basic properties of Brownian Motion, particularly conformal invariance.
• Uncertain Author, Stochastic Analyis, An Introduction Prob3
  • The proof of the martingale convergence theorem and some useful other results are contained within.
• Bell, The Kolmogorov Extension Theorem Prob4
  • This contains some nice review about $\sigma$-algebras and leads to the proof of the Kolmogorov Extension Theorem.
• Berestycki, Stochastic Calculus and Applications Prob5
  • A thorough introduction to stochastic calculus. Largely unneeded by us, it’s still useful as a reference.
• Taylor, Random Fields Prob6
  • This is included as a reference mainly for sections 2.2.1 and 2.2.2, which give a good self-contained definition of Gaussian Random Variables.
    • Extra terminology: A centered Gaussian random variable is a Gaussian random variable with mean 0.
• Morters, Peres, Brownian Motion Prob7
  • A very thorough discussion of the myriad properties of Brownian Motion. A good reference, even if we’ll only skim it during the seminar.

Gaussian Free Fields

• Sheffield, Gaussian Free Fields for Mathematicians GFF1
  • A formal, functional-analytic description of Gaussian Free Fields.
• Werner, Topics on the two-dimensional Gaussian Free Field GFF2
  • A mostly probabilistic description of the 2D Gaussian Free Field with some intuition from Brownian Motion.

Liouville Quantum Gravity

• Gwynne, Random Surfaces and Liouville Quantum Gravity LQG1
  • This is a survey article written to be readable to graduate students. It’s useful to skim as an explanation of a lot of the modern work on LQG.
• Berestycki, Introduction to the Gaussian Free Field and Liouville Quantum Gravity LQG2
  • This is written to be a softer introduction to LQG, assuming only a background in probability.
• Chern, An Elementary Proof of the Existence of Isothermal Parameters on a Surface LQG3
  • For any so curious, this paper proves the existence of isothermal coordinates using only the assumption that the Riemannian metric is Holder continuous.
• Duplantier, Sheffield, Liouville Quantum Gravity and KPZ LQG4
  • A very detailed, but somewhat more difficult to parse explanation of LQG.
• Rhodes, Vargas, Gaussian Multiplicative Chaos and Applications: A Review LQG5
  • While on its face this topic does not seem clearly related to LQG, multiplicative chaos is in fact the general case, with LQG being an example of multiplicative chaos applied to the GFF. We won’t use this in the seminar, but it may be of interest.

The Brownian Map

• Le Gall, Miermont, Scaling Limits of Random Trees and Planar Maps TBM1
  • This article goes into exhaustive detail into each of the constituent types of discrete and continuous objects which are used in a thorough construction of the Brownian Map. It also features some very descriptive pictures.
• Le Gall, Brownian Geometry TBM2
  • A more concise description of the material leading to the Brownian Map, with a bit less detail in the proofs but more focus on intuition.