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Summer Graduate School

Geometric Flows (Crete, Greece) June 19, 2022 - July 01, 2022
Parent Program: --
Location: Crete, Greece
Organizers Nicholas Alikakos (National and Kapodistrian University of Athens (University of Athens)), Panagiota Daskalopoulos (Columbia University)
Speaker(s)

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Description
Image
photo courtesy of Panagiota Daskalopoulos

[The image on this vase from Minoan Crete, dated on 1500-2000 BC, resembles an ancient solution to the Curve shortening flow - one of the most basic geometric flows. The vase is at Heraklion Archaeological Museum]

This summer graduate school is a collaboration between MSRI and the FORTH-IACM Institute in Crete. The purpose of the school is to introduce graduate students to some of the most important geometric evolution equations. Information about the location of the summer school can be found here.

This is an area of geometric analysis that lies at the interface of differential geometry and partial differential equations. The lectures will begin with an introduction to nonlinear diffusion equations and continue with classical results on the Ricci Flow, the Mean curvature flow and other fully non-linear extrinsic flows such as the Gauss curvature flow. The lectures will also include geometric applications such as isoperimetric inequalities, topological applications such as the Poincaré onjecture, as well as recent important developments related to the study of singularities and ancient solutions.

For more information, please see this LINK.

The schedule is now available.

Suggested Prerequisites

Students are expected to have a basic (advanced undergraduate or beginning graduate) background on elliptic and parabolic partial equations and Classical Riemannian Geometry. More precisely the students are expected to have the following background:

1. A basic course on Partial Differential Equations such as: L.C. Evans, Partial Differential Equations, Chapters 2 and 5-7

2. A basic course on Riemannian Geometry such as: Manfredo do Carmo, Riemannian Geometry, Chapters 1-8 or John Lee, Riemannian Manifolds.

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.

Support for this school is provided by the Stavros Niarchos Foundation and MSRI.

Schedule, Notes/Handouts & Videos
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Jun 19, 2022
Sunday
11:45 PM - 12:15 AM
  Registration
Jun 20, 2022
Monday
12:15 AM - 12:30 AM
  Welcome
Hélène Barcelo (MSRI - Mathematical Sciences Research Institute), Charalambos Makridakis (Institute of Applied & Computational Mathematics)
12:30 AM - 01:30 AM
  Pre-Lecture: Review of Curvature, Commutator Identities for Covariant Derivatives
01:30 AM - 02:00 AM
  Coffee Break
02:00 AM - 03:15 AM
  Ricci Flow: Definition of Ricci Flow, Some Basic Examples, Self-Similar Solutions, Evolution of Curvature, Curvature Algebra in Three Dimensions
Simon Brendle (Stanford University)
03:15 AM - 03:45 AM
  Informal Questions
03:45 AM - 05:30 AM
  Lunch
05:30 AM - 06:30 AM
  Post-Lecture & Problem Session
11:45 PM - 12:45 AM
  Pre-Lecture: Heat Equation on R^n and on Manifolds
Jun 21, 2022
Tuesday
12:45 AM - 01:15 AM
  Coffee Break
01:15 AM - 02:30 AM
  Ancient Solutions to Parabolic Equations: Ancient Solutions to the Heat Equation and to the Semi-Linear Heat Equation
Panagiota Daskalopoulos (Columbia University)
02:30 AM - 03:00 AM
  Informal Questions
03:00 AM - 05:00 AM
  Lunch
05:00 AM - 06:00 AM
  Post-Lecture & Problem Session
06:00 AM - 07:00 AM
  Informal Discussions between Students
11:45 PM - 12:45 AM
  Pre-Lecture: Maximum Principle for Scalar Heat Equations, Interior Estimates
Jun 22, 2022
Wednesday
12:45 AM - 01:15 AM
  Coffee Break
01:15 AM - 02:30 AM
  Ricci Flow: Hamilton’s Maximum Principle for Systems, Preservation of Nonnegative Ricci in Three Dimensions, Pinching and Convergence in Three Dimensions, Curvature Conditions in Higher Dimensions
Simon Brendle (Stanford University)
02:30 AM - 03:00 AM
  Informal Questions
03:00 AM - 05:00 AM
  Lunch
05:00 AM - 06:00 AM
  Post-Lecture & Problem Session
06:00 AM - 07:00 AM
  Informal Discussion between Students
11:45 PM - 12:45 AM
  Pre-Lecture: Log-Sobolev Entropy and Nash Entropy
Jun 23, 2022
Thursday
12:45 AM - 01:15 AM
  Coffee Break
01:15 AM - 02:30 AM
  Ricci Flow: Perelman’s Entropy, and Noncollapsing
Simon Brendle (Stanford University)
02:30 AM - 03:00 AM
  Informal Questions
03:00 AM - 05:00 AM
  Lunch
05:00 AM - 06:00 AM
  Post-Lecture & Problem Session
06:00 AM - 07:00 AM
  Informal Discussions between Students
11:45 PM - 12:45 AM
  Pre-Lecture: Extrinsic Curvature of Hypersurfaces, and Evolution Equations for Extrinsic Geometric Flows
Jun 24, 2022
Friday
12:45 AM - 01:15 AM
  Coffee Break
01:15 AM - 02:30 AM
  Inverse Mean Curvature Flow: Properties of Smooth Solutions to IMCF
Gerhard Huisken (Math. Forschungsinstitut Oberwolfach)
02:30 AM - 03:00 AM
  Informal Questions
03:00 AM - 05:00 AM
  Lunch
05:00 AM - 06:00 AM
  Post-Lecture & Problem Session
06:00 AM - 07:00 AM
  Informal Discussions between Students
Jun 26, 2022
Sunday
11:45 PM - 12:45 AM
  Pre-Lecture: Level Set Approach to Extrinsic Geometric Flows (and the Relation to the Parametrized Version of the Flow); Weak Mean Curvature
Jun 27, 2022
Monday
01:00 AM - 02:15 AM
  Inverse Mean Curvature Flow: Weak Solutions for Inverse Mean Curvature Flow
Gerhard Huisken (Math. Forschungsinstitut Oberwolfach)
02:30 AM - 03:00 AM
  Informal Questions
03:00 AM - 05:00 AM
  Lunch
05:00 AM - 06:00 AM
  Post-Lecture & Problem Session
06:00 AM - 07:00 AM
  Informal Discussions between Students
11:45 PM - 12:45 AM
  The 2-Dimensional Ricci Flow
Jun 28, 2022
Tuesday
12:45 AM - 01:15 AM
  Coffee Break
01:15 AM - 02:30 AM
  Ancient Solutions to Geometric Flows: Ancient Solutions to the 2-Dim Ricci Flow
Panagiota Daskalopoulos (Columbia University)
02:30 AM - 03:00 AM
  Informal Questions
03:00 AM - 05:00 AM
  Lunch
05:00 AM - 06:00 AM
  Post-Lecture & Problem Session
06:00 AM - 07:00 AM
  Informal Discussions between Students
11:45 PM - 12:45 AM
  Pre-Lecture: Asymptotically Flat 3-Manifolds, their Role in General Relativity, the ADM-Mass and the Dominant Energy Condition
Jun 29, 2022
Wednesday
12:45 AM - 01:15 AM
  Coffee Break
01:15 AM - 02:30 AM
  Inverse Mean Curvature Flow: Applications to General Relativity
Gerhard Huisken (Math. Forschungsinstitut Oberwolfach)
02:30 AM - 03:00 AM
  Informal Questions
03:00 AM - 05:00 AM
  Lunch
05:00 AM - 06:00 AM
  Post-Lecture & Problem Session
06:00 AM - 07:00 AM
  Informal Discussions between Students
11:45 PM - 12:45 AM
  Pre-Lecture: Curve Shortening Flow on the Plane
Jun 30, 2022
Thursday
12:45 AM - 01:15 AM
  Coffee Break
01:15 AM - 02:30 AM
  Ancient Solutions to Geometric Flows: Ancient Solutions to the Curve Shortening Flow
Theodora Bourni (University of Tennessee)
02:30 AM - 03:00 AM
  Informal Questions
03:00 AM - 05:00 AM
  Lunch
06:00 AM - 07:00 AM
  Informal Discussions between Students