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Summer Graduate School

Metric Geometry and Geometric Analysis (Oxford, United Kingdom) July 11, 2022 - July 22, 2022
Parent Program: --
Location: University of Oxford, United Kingdom
Organizers LEAD Cornelia Drutu (University of Oxford), Panos Papazoglou (University of Oxford)
Lecturer(s)

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Speaker(s)

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Description
Cornelia picture 2
Several geometric ideas in the context of a surface: hyperbolic metric, CAT(0) inequality, Gromov hyperbolicity/coarse median geometry, nonpositively-curved square tiling, Besikovitch inequality. (Picture by M. Hagen and A. Sisto.)

The purpose of the summer school is to introduce graduate students to key mainstream directions in the recent development of geometry, which sprang from Riemannian Geometry in an attempt to use its methods in various contexts of non-smooth geometry. This concerns recent developments in metric generalizations of the theory of nonpositively curved spaces and discretizations of methods in geometry, geometric measure theory and global analysis. The metric geometry perspective gave rise to new results and problems in Riemannian Geometry as well.

 

All these themes are intertwined and have developed either together or greatly influencing one another. The summer school will introduce some of the latest developments and the remaining open problems in these very modern areas, and will emphasize their synergy.

Suggested prerequisites

We anticipate that the participating students will come from diverse backgrounds in Geometric Group Theory, Riemannian Geometry and Geometric Analysis.

The following textbooks would give adequate preparation but we don’t expect students to have necessarily studied all of them:

  • Chapters 3, 8, 10, 11 of “Geometric Group Theory” by C. Drutu and M. Kapovich
  • First 4 chapters of "Riemannian Geometry" by M. P. Do Carmo and the first 2 chapters of "Morse Theory" by J. Milnor
  • Parts I.2, I.3, I.8, and II.1 of “Metric Spaces of Non-Positive Curvature” by Bridson-Haefliger

The students who have been selected to take part in the school will be sent a compilation of introductory texts extracted from the books above and other existing lecture notes, for study beforehand, by email and, if they so wish, a paper version by ordinary mail as well. A more definite and specific list of prerequisites will be given later on after consulting with the lecturers so that background material matches the specific topics they plan to cover in their lectures.

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

Due to the small number of students supported by MSRI, only one student per institution will be funded by MSRI. You may must also apply directly with The Mathematical Insitute, University of Oxford, take a look at this page for more information.

Schedule, Notes/Handouts & Videos
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Jul 11, 2022
Monday
09:30 AM - 10:30 AM
  Lecture & Mini-Course 1: "Geometric Inequalities: Homotopies, Fillings and Geodesics"
Regina Rotman (University of Toronto)
10:30 AM - 11:00 AM
  Coffee Break
11:00 AM - 12:00 PM
  Lecture & Mini Course 2: Isoperimetric Filling Inequalities in CAT(0) Spaces
Urs Lang (ETH Zurich)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Research Talk: "Smooth and Non-Smooth Aspects of Ricci Curvature Lower Bounds: an Optimal Transport Point of View"
Andrea Mondino (University of Warwick)
02:00 PM - 02:30 PM
  Coffee Break
02:30 PM - 03:30 PM
  Research Talk: "Random Walks on Gromov-Hyperbolic Spaces: a Survey and Results in Large Deviations"
Cagri Sert
04:00 PM - 05:00 PM
  Meet Your Mentor Session
05:00 PM - 05:30 PM
  Further Explanation of Course Material by a Mentor or a Lecturer
Jul 12, 2022
Tuesday
09:30 AM - 10:30 AM
  Lecture & Mini-Course 1: "Geometric Inequalities: Homotopies, Fillings and Geodesics"
Regina Rotman (University of Toronto)
10:30 AM - 11:00 AM
  Coffee Break
11:00 AM - 12:00 PM
  Lecture & Mini Course 2: Isoperimetric Filling Inequalities in CAT(0) Spaces
Urs Lang (ETH Zurich)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Research Talk: Hyperbolic Actions and Relative Free Factor Complexes
Richard Wade (University of British Columbia)
02:00 PM - 02:30 PM
  Coffee Break
02:30 PM - 04:30 PM
  TA Session
05:00 PM - 05:30 PM
  Further Explanations of Course Material by a Mentor or a Lecturer
Jul 13, 2022
Wednesday
09:30 AM - 10:30 AM
  Lecture & Mini-Course 1: "Geometric Inequalities: Homotopies, Fillings and Geodesics"
Regina Rotman (University of Toronto)
10:30 AM - 11:00 AM
  Coffee Break
11:00 AM - 12:00 PM
  Lecture & Mini Course 2: Isoperimetric Filling Inequalities in CAT(0) Spaces
Urs Lang (ETH Zurich)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Research Talk: Rigidity for Automorphism Groups of Free Groups
Martin Bridson (University of Oxford)
02:00 PM - 02:30 PM
  Coffee Break
02:30 PM - 04:30 PM
  TA Session
05:00 PM - 05:30 PM
  Further Explanations of Course Material by a Mentor or a Lecturer
Jul 14, 2022
Thursday
09:30 AM - 10:30 AM
  Lecture & Mini-Course 1: "Geometric Inequalities: Homotopies, Fillings and Geodesics"
Regina Rotman (University of Toronto)
10:30 AM - 11:00 AM
  Coffee Break
11:00 AM - 12:00 PM
  Lecture & Mini Course 2: Isoperimetric Filling Inequalities in CAT(0) Spaces
Urs Lang (ETH Zurich)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Research Talk: Old and New Trends in Systolic Geometry
Alexander Nabutovsky
02:00 PM - 02:30 PM
  Coffee Break
02:30 PM - 04:30 PM
  TA Session
05:00 PM - 05:30 PM
  Further Explanations of Course Material by a Mentor or a Lecturer
Jul 15, 2022
Friday
09:30 AM - 10:30 AM
  Lecture & Mini-Course 1: "Geometric Inequalities: Homotopies, Fillings and Geodesics"
Regina Rotman (University of Toronto)
10:30 AM - 11:00 AM
  Coffee Break
11:00 AM - 12:00 PM
  Lecture & Mini Course 2: Isoperimetric Filling Inequalities in CAT(0) Spaces
Urs Lang (ETH Zurich)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Research Talk: Balls in Essential Manifolds and Actions on Cantor Spaces
Roman Sauer (Karlsruhe Institute of Technology)
02:00 PM - 02:30 PM
  Coffee Break
02:30 PM - 04:30 PM
  TA Session
05:00 PM - 05:30 PM
  Further Explanations of Course Material by a Mentor or a Lecturer
Jul 18, 2022
Monday
09:30 AM - 10:30 AM
  Lecture & Mini Course 1: Metric Geometry and Analysis on Boundaries of Gromov Hyperbolic Spaces, and Applications
Bruce Kleiner (New York University, Courant Institute)
10:30 AM - 11:00 AM
  Coffee Break
11:00 AM - 12:00 PM
  Lecture & Mini Course 2: Projection Complexes and Applications to Mapping Class Groups
Mladen Bestvina (University of Utah)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Research Talk: An Introduction to Hierarchical Hyperbolicity
Alessandro Sisto (ETH Zürich)
02:00 PM - 02:30 PM
  Coffee Break
02:30 PM - 03:30 PM
  Research Talk: Knot Theory and Machine Learning
Marc Lackenby
04:00 PM - 05:00 PM
  Meet Your Mentor Session
05:00 PM - 05:30 PM
  Further Explanations of Course Material by a Mentor or a Lecturer
Jul 19, 2022
Tuesday
09:30 AM - 10:30 AM
  Lecture & Mini Course 1: Metric Geometry and Analysis on Boundaries of Gromov Hyperbolic Spaces, and Applications
Bruce Kleiner (New York University, Courant Institute)
10:30 AM - 11:00 AM
  Coffee Break
11:00 AM - 12:00 PM
  Lecture & Mini Course 2: Projection Complexes and Applications to Mapping Class Groups
Mladen Bestvina (University of Utah)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Research Talk: Product Set Growth in Mapping Class Groups
Alice Kerr (University of Oxford)
02:00 PM - 02:30 PM
  Coffee Break
02:30 PM - 04:30 PM
  TA Session
05:00 PM - 05:30 PM
  Further Explanations of Course Material by a Mentor or a Lecturer
Jul 20, 2022
Wednesday
09:30 AM - 10:30 AM
  Lecture & Mini Course 1: Metric Geometry and Analysis on Boundaries of Gromov Hyperbolic Spaces, and Applications
Bruce Kleiner (New York University, Courant Institute)
10:30 AM - 11:00 AM
  Coffee Break
11:00 AM - 12:00 PM
  Lecture & Mini Course 2: Projection Complexes and Applications to Mapping Class Groups
Mladen Bestvina (University of Utah)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Research Talk: Every Countable Group is an Outer Automorphism Group of an Acylindrically Hyperbolic Group with Kazhdan's Property (T)
Bin Sun (University of Oxford)
02:00 PM - 02:30 PM
  Coffee Break
02:30 PM - 04:30 PM
  TA Session
05:00 PM - 05:30 PM
  Further Explanations of Course Material by a Mentor or a Lecturer
Jul 21, 2022
Thursday
09:30 AM - 10:30 AM
  Lecture & Mini Course 1: Metric Geometry and Analysis on Boundaries of Gromov Hyperbolic Spaces, and Applications
Bruce Kleiner (New York University, Courant Institute)
10:30 AM - 11:00 AM
  Coffee Break
11:00 AM - 12:00 PM
  Lecture & Mini Course 2: Projection Complexes and Applications to Mapping Class Groups
Mladen Bestvina (University of Utah)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Research Talk: Large Scale Geometry of Hecke Pairs
Clément Dell'Aiera (Université de Metz)
02:00 PM - 02:30 PM
  Coffee Break
02:30 PM - 04:30 PM
  TA Session
05:00 PM - 05:30 PM
  Further Explanations of Course Material by a Mentor or a Lecturer
Jul 22, 2022
Friday
09:30 AM - 10:30 AM
  Lecture & Mini Course 1: Metric Geometry and Analysis on Boundaries of Gromov Hyperbolic Spaces, and Applications
Bruce Kleiner (New York University, Courant Institute)
10:30 AM - 11:00 AM
  Coffee Break
11:00 AM - 12:00 PM
  Lecture & Mini Course 2: Projection Complexes and Applications to Mapping Class Groups
Mladen Bestvina (University of Utah)
12:00 PM - 01:00 PM
  Lunch
01:00 PM - 02:00 PM
  Research Talk: Boundaries of Random Groups
John Mackay (University of Bristol)
02:00 PM - 02:30 PM
  Coffee Break
02:30 PM - 04:30 PM
  TA Session
05:00 PM - 05:30 PM
  Further Explanations of Course Material by a Mentor or a Lecturer