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Summer Graduate School

Random Conformal Geometry (Virtual School) July 19, 2021 - July 30, 2021
Parent Program:
Location: MSRI: Simons Auditorium, Atrium
Organizers Mario Bonk (University of California, Los Angeles), Steffen Rohde (University of Washington), LEAD Fredrik Viklund (Royal Institute of Technology)
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Speaker(s)

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Description
Graphisc
a random quasiconformal map obtained from Beltrami equation by randomly assigning the values of +-1/2 for the Beltrami coefficient on small squares subdividing the unit square

This Summer Graduate School will cover basic tools that are instrumental in Random Conformal Geometry (the investigation of analytic and geometric objects that arise from natural probabilistic constructions, often motivated by models in mathematical physics) and are at the foundation of the subsequent semester-long program  "The Analysis and Geometry of Random Spaces".  Specific topics are Conformal Field Theory, Brownian Loops and related processes, Quasiconformal Maps, as well as Loewner Energy and Teichmüller Theory.

School Structure

There will be two lectures per day, each followed by participants presentations and/or problem sessions. The school will be in the form of a learning seminar with active involvement of the participants.  In particular, the talks by the main lecturers will be supplemented by some of the participants on additional material related to the main lectures.  This will also include discussion problems and their solutions related to the topics.  The organizers will coordinate these activities with lecturers and the participants, and will assign specific topics to some of the participants in advance for presentation at the sumer school.  Participants who will not give longer talks will contribute to the Problem Sessions with the presentation of solutions to select problems.

Suggested Prerequisites

We expect that the students have some solid knowledge of real and complex analysis corresponding to basic first-year graduate courses (as presented in the first 16 Chapters of Rudin’s “Real and Complex Analysis”, for example). The students should also have some background in basic probability up to the central limit theorem (corresponding to Chapters 1-3 in Durrett’s “Probability”), but we do not expect knowledge of more advanced topics such as Brownian motion. 

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

Keywords and Mathematics Subject Classification (MSC)
Tags/Keywords
  • Conformal invariance

  • quasiconformal map

  • Brownian motion

  • conformal field theory

  • quantum field theory

  • stochastic processes

  • Teichmüller theory

  • Loewner equation

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification
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Schedule, Notes/Handouts & Videos
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Jul 19, 2021
Monday
07:15 AM - 07:30 AM
  Welcome and Introductions
07:30 AM - 08:30 AM
  Geometric function theory and quasiconformal maps: Lecture 1
Mario Bonk (University of California, Los Angeles), Steffen Rohde (University of Washington), Fredrik Viklund (Royal Institute of Technology)
08:30 AM - 09:00 AM
  Break
09:00 AM - 10:00 AM
  Geometric function theory and quasiconformal maps: Lecture 2
Mario Bonk (University of California, Los Angeles), Steffen Rohde (University of Washington), Fredrik Viklund (Royal Institute of Technology)
10:00 AM - 11:00 AM
  Break
11:00 AM - 12:00 PM
  Exercise session 1
12:00 PM - 01:00 PM
  Exercise session 2
Jul 20, 2021
Tuesday
07:30 AM - 08:30 AM
  Planar lattice models and Conformal Field Theory: Lecture 1
Clément Hongler (École Polytechnique Fédérale de Lausanne (EPFL))
08:30 AM - 09:00 AM
  Break
09:00 AM - 10:00 AM
  Geometric function theory and quasiconformal maps: Lecture 3
Mario Bonk (University of California, Los Angeles), Steffen Rohde (University of Washington), Fredrik Viklund (Royal Institute of Technology)
10:00 AM - 11:00 AM
  Break
11:00 AM - 12:00 PM
  Exercise session 1
12:00 PM - 01:00 PM
  Exercise session 2
Jul 21, 2021
Wednesday
07:30 AM - 08:30 AM
  Planar lattice models and Conformal Field Theory: Lecture 2
Clément Hongler (École Polytechnique Fédérale de Lausanne (EPFL))
08:30 AM - 09:00 AM
  Break
09:00 AM - 10:00 AM
  Lectures on the Brownian loop-soup: Lecture 1
Wei Qian (Université Paris-Saclay)
10:00 AM - 11:00 AM
  Break
11:00 AM - 12:00 PM
  Exercise session 1
12:00 PM - 01:00 PM
  Exercise session 2
Jul 22, 2021
Thursday
07:30 AM - 08:30 AM
  Geometric function theory and quasiconformal maps: Lecture 4
Mario Bonk (University of California, Los Angeles), Steffen Rohde (University of Washington), Fredrik Viklund (Royal Institute of Technology)
08:30 AM - 09:00 AM
  Break
09:00 AM - 10:00 AM
  Lectures on the Brownian loop-soup: Lecture 2
Wei Qian (Université Paris-Saclay)
10:00 AM - 11:00 AM
  Break
11:00 AM - 12:00 PM
  Exercise session 1
12:00 PM - 01:00 PM
  Exercise session 2
Jul 23, 2021
Friday
07:30 AM - 08:30 AM
  Lectures on the Brownian loop-soup: Lecture 3
Wei Qian (Université Paris-Saclay)
08:30 AM - 09:00 AM
  Break
09:00 AM - 10:00 AM
  Geometric function theory and quasiconformal maps: Lecture 5
Mario Bonk (University of California, Los Angeles), Steffen Rohde (University of Washington), Fredrik Viklund (Royal Institute of Technology)
10:00 AM - 11:00 AM
  Break
11:00 AM - 12:00 PM
  Exercise session 1
12:00 PM - 01:00 PM
  Exercise session 2
Jul 26, 2021
Monday
07:30 AM - 08:30 AM
  Geometric function theory and quasiconformal maps: Lecture 6
Mario Bonk (University of California, Los Angeles), Steffen Rohde (University of Washington), Fredrik Viklund (Royal Institute of Technology)
08:30 AM - 09:00 AM
  Break
09:00 AM - 10:00 AM
  The TLAs of the Conformally Invariant World: Lecture 1
Gregory Lawler (University of Chicago)
10:00 AM - 11:00 AM
  Break
11:00 AM - 12:00 PM
  Exercise session 1
12:00 PM - 01:00 PM
  Exercise session 2
Jul 27, 2021
Tuesday
07:30 AM - 08:30 AM
  The TLAs of the Conformally Invariant World: Lecture 2
Gregory Lawler (University of Chicago)
08:30 AM - 09:00 AM
  Break
09:00 AM - 10:00 AM
  The TLAs of the Conformally Invariant World: Lecture 3
Gregory Lawler (University of Chicago)
10:00 AM - 11:00 AM
  Break
11:00 AM - 12:00 PM
  Exercise session 1
12:00 PM - 01:00 PM
  Exercise session 2
Jul 28, 2021
Wednesday
07:30 AM - 08:30 AM
  Around the Loewner energy: Lecture 1
Yilin Wang (Massachusetts Institute of Technology)
08:30 AM - 09:00 AM
  Break
09:00 AM - 10:00 AM
  The TLAs of the Conformally Invariant World: Lecture 4
Gregory Lawler (University of Chicago)
10:00 AM - 11:00 AM
  Break
11:00 AM - 12:00 PM
  Exercise session 1
12:00 PM - 01:00 PM
  Exercise session 2
Jul 29, 2021
Thursday
07:30 AM - 08:30 AM
  Around the Loewner energy: Lecture 2
Yilin Wang (Massachusetts Institute of Technology)
08:30 AM - 09:00 AM
  Break
09:00 AM - 10:00 AM
  The TLAs of the Conformally Invariant World: Lecture 5
Gregory Lawler (University of Chicago)
10:00 AM - 11:00 AM
  Break
11:00 AM - 12:00 PM
  Exercise session 1
12:00 PM - 01:00 PM
  Exercise session 2
Jul 30, 2021
Friday
07:30 AM - 08:30 AM
  Around the Loewner energy: Lecture 3
Yilin Wang (Massachusetts Institute of Technology)
08:30 AM - 09:00 AM
  Break
09:00 AM - 10:00 AM
  Lecture 2
10:00 AM - 11:00 AM
  Break
11:00 AM - 12:00 PM
  Exercise session 1
12:00 PM - 01:00 PM
  Exercise session 2