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Summer Graduate School

Stochastic Quantization (SLMath) July 01, 2024 - July 12, 2024
Parent Program: --
Location: SLMath: Eisenbud Auditorium
Organizers Massimiliano Gubinelli (University of Oxford), Martina Hofmanova (Universität Bielefeld), LEAD Hao Shen (University of Wisconsin-Madison), Lorenzo Zambotti (Sorbonne Université)
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Teaching Assistants(s)

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Description
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This summer school will familiarize students with the basic problems of the mathematical theory of Euclidean quantum fields. The lectures will introduce some of its prominent models and analyze them via the so called “stochastic quantization” methods, involving recently developed stochastic and PDE techniques. This is an area which is highly interdisciplinary combining ideas ranging from the theory of partial differential equations, to stochastic analysis, to mathematical physics. Our goal is to bring together students who are perhaps familiar with some but not all of these subjects and teach them how to integrate these different tools to solve cutting-edge problems of Euclidean quantum field theory.

School Structure

The organizers plan to introduce gradually the basic tools and ideas needed in this area of research, which will cover at least the first 3/4 of the school.  The remaining time will be left to discuss other models and more recent developments. There will be two or three lectures per day, 1.5 hours each, plus one or two 1.5 hours long TA/problem session.

Bibliography

Prerequisites

A basic knowledge of finite-dimensional probability and of Brownian motion (without stochastic calculus), of basic functional analysis and basic PDE theory: linear equations, Fourier transform, Sobolev spaces on R^d. This the ideal background but it is not strictly necessary: we are well aware that the participants will come from a wide range of knowledge and abilities, so we will work to bring them to a common ground. A few weeks before the school we plan to assign recommended readings that would help the participants get better prepared, for instance, reading material along the line of Section 2.2.1 and 2.3.1 of “Partial Differential Equations” by Evans, and Chapter 2 of “Stochastic Differential Equations: An Introduction with Applications” by Oksendal.

Recommended Readings

For eligibility and how to apply, see the Summer Graduate Schools homepage.

Keywords and Mathematics Subject Classification (MSC)
Tags/Keywords
  • Euclidean quantum field theory

  • stochastic (partial) differential equations

  • paracontrolled distributions

  • a priori estimates

  • renormalization theory

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
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Schedule, Notes/Handouts & Videos
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Jul 01, 2024
Monday
09:15 AM - 09:30 AM
  Introduction to SLMath
09:30 AM - 10:45 AM
  From quantum mechanics to measures
Hao Shen (University of Wisconsin-Madison), Lorenzo Zambotti (Sorbonne Université)
10:45 AM - 11:15 AM
  Break
11:15 AM - 12:30 PM
  Euclidean quantum field theory and Gaussian free field
Hao Shen (University of Wisconsin-Madison), Lorenzo Zambotti (Sorbonne Université)
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  TA Session: "Fourier analysis"
Sky Cao (Massachusetts Institute of Technology)
03:00 PM - 03:10 PM
  Group Photo
03:10 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  TA Session: "Tightness of probability measures"
Evan Sorensen (Columbia University)
Jul 02, 2024
Tuesday
09:30 AM - 10:45 AM
  Langevin dynamic on finite lattice
Hao Shen (University of Wisconsin-Madison), Lorenzo Zambotti (Sorbonne Université)
10:45 AM - 11:15 AM
  Break
11:15 AM - 12:30 PM
  Stationary coupling, existence of infinite volume limit
Hao Shen (University of Wisconsin-Madison), Lorenzo Zambotti (Sorbonne Université)
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  TA Session: "Littlewood-Paley theory, Part I"
Sarah-Jean Meyer (University of Oxford)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  TA Session: "Hermite polynomials"
Sky Cao (Massachusetts Institute of Technology)
Jul 03, 2024
Wednesday
09:30 AM - 10:45 AM
  Convergence to equilibrium / uniqueness of infinite volume limit
Hao Shen (University of Wisconsin-Madison), Lorenzo Zambotti (Sorbonne Université)
10:45 AM - 11:15 AM
  Break
11:30 AM - 02:00 PM
  Lunch
02:00 PM - 03:15 PM
  Integration by parts, Wick theorem, Introduction to small scale limit
Hao Shen (University of Wisconsin-Madison), Lorenzo Zambotti (Sorbonne Université)
03:15 PM - 03:45 PM
  Afternoon Tea
03:45 PM - 04:45 PM
  TA Session: "Wiener chaos & practice with Wick’s theorem"
Evan Sorensen (Columbia University)
Jul 04, 2024
Thursday
All Day
  SLMath Closed
Jul 05, 2024
Friday
09:30 AM - 10:45 AM
  Small scale limit - probabilistic estimates
Hao Shen (University of Wisconsin-Madison), Lorenzo Zambotti (Sorbonne Université)
10:45 AM - 11:15 AM
  Break
11:15 AM - 12:30 PM
  Small scale limit - PDE estimates
Hao Shen (University of Wisconsin-Madison), Lorenzo Zambotti (Sorbonne Université)
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  TA Session: "Littlewood-Paley theory, Part II"
Sarah-Jean Meyer (University of Oxford)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  TA Session: "Estimates on some stochastic terms"
Evan Sorensen (Columbia University)
Jul 08, 2024
Monday
09:30 AM - 10:45 AM
  $\Phi^4_2$ without cutoffs, computations with stochastic terms (like the cube)
Massimiliano Gubinelli (University of Oxford)
10:45 AM - 11:15 AM
  Break
11:15 AM - 12:30 PM
  Perturbation theory for $\Phi^4_2$, Part I
Hao Shen (University of Wisconsin-Madison)
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  TA Session: "Paraproducts "
Sarah-Jean Meyer (University of Oxford)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  TA Session: "Weighted $L^p$ estimates for $\Phi^4_2$"
Sky Cao (Massachusetts Institute of Technology)
Jul 09, 2024
Tuesday
09:30 AM - 10:45 AM
  Perturbation theory for $\Phi^4_2$, Part II
Hao Shen (University of Wisconsin-Madison)
10:45 AM - 11:15 AM
  Break
11:15 AM - 12:30 PM
  $\Phi^4_3$, paracontrolled energy estimates
Massimiliano Gubinelli (University of Oxford)
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  TA Session: "Lattice Schauder estimates"
Sky Cao (Massachusetts Institute of Technology)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  TA Session: "Weighted Besov spaces"
Sarah-Jean Meyer (University of Oxford)
Jul 10, 2024
Wednesday
09:30 AM - 10:45 AM
  $\Phi^4_3$, the second renormalization and commutator lemmas
Massimiliano Gubinelli (University of Oxford)
10:45 AM - 11:15 AM
  Break
11:15 AM - 12:30 PM
  Subgaussian tails for $\Phi^4_2$
Hao Shen (University of Wisconsin-Madison)
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  TA Session: "Wavelet representation and the localization operators"
Sarah-Jean Meyer (University of Oxford)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  TA Session: "$L^\infty$ estimates and coming down from infinity for $\Phi^4_2$"
Sky Cao (Massachusetts Institute of Technology)
Jul 11, 2024
Thursday
09:30 AM - 10:45 AM
  The Abelian Higgs model in two dimensions, local theory
Hao Shen (University of Wisconsin-Madison)
10:45 AM - 11:15 AM
  Break
11:15 AM - 12:30 PM
  $\Phi^4_3$, tightness, infinite volume limit and some properties of the measure#
Massimiliano Gubinelli (University of Oxford)
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  TA Session: "The Abelian Higgs model in two dimensions, global theory"
Sky Cao (Massachusetts Institute of Technology)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  TA Session: "Approximate duality and commutator lemma"
Sarah-Jean Meyer (University of Oxford)
Jul 12, 2024
Friday
09:30 AM - 10:45 AM
  An invitation to nonabelian gauge theories
Hao Shen (University of Wisconsin-Madison)
10:45 AM - 11:15 AM
  Break
11:15 AM - 12:30 PM
  Stochastic estimates in $\Phi^4_3$
Massimiliano Gubinelli (University of Oxford)
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  Participant Survey
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:00 PM
  TA Session or Free Discussion