FALL 2023
Daniel Ketover
Subtitle:
Topics in geometric analysis
Course Description:
TBD
Text:
TBD
Prerequisites:
TBD
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SPRING 2023
Jian Song
Subtitle:
Ricci flow and heat equations
Course Description:
We will discuss some of Perelman's work and recent development on the Ricci flow with an analytic approach.
Text:
My own notes
Prerequisites:
Differential Geometry
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FALL 2022
Li Chi
Subtitle:
Topics in Complex Geometry
Course Description:
This is an introductory course to complex geometry. We plan to cover the following topics, which may vary depending on the progress and the students interests.
1. Complex manifolds and sheaf theory
2. Kahler geometry and Hodge theory
3. Kodaira projective embedding theorem
4. L2 methods and extension theorems
Text:
R.O. Wells: Differential Analysis on Complex Manifolds, P. Griffiths and J. Harris: Principles of algebraic geometry, J.P. Demailly: Analytic methods in algebraic geometry
Prerequisites:
complex analysis, differential geometry
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FALL 2020
Feng Luo
Subtitle:
Collapsed Riemannian and non-Riemannian spaces
Course Description:
We will present a topic course that covers on the collapsing theory in Metric Riemannian geometry. We will start with the Gromov’s almost flat manifolds, the nilpotent structure theory of Cheeger-Fukaya-Gromov. We will discuss similar nilpotent structures found on collapsed Riemannian manifolds with local Ricci bounded covering geometry, and on Alexandrov spaces (non-Riemannian spaces) with local bounded covering geometry. CONTENTS: I. Collapsed Manifolds with Bounded Sectional Curvature - 1.1. Examples 1.2. The almost flat manifolds 1.3. The nilpotent fiber bundle theorem 1.4. The singular nilpotent fibration theorem 1.5. Applications - II. Collapsed Manifolds with Local Bounded Ricci Covering Geometry - 2.1. Examples 2.2. The Cheeger-Colding-Naber theory on Ricci limit spaces 2.3. The Margulis lemma 2.4. Maximally collapsed manifolds with local bounded Ricci covering geometry 2.5. The nilpotent fibration theorem 2.6. Applications - III. Collapsed Alexandrov Spaces - 3.1. Basic Alexandrov spaces 3.2. The fibration theorem 3.3. Maximally collapsed Alexandrov space with local bounded covering geometry 3.4. The nilpotent fibration theorem 3.5. Singular fibration structures 3.6. Applications
Text:
Reference: Notes on convergence and collapsing theorems in Riemannian geometry, Handbook of Geometric Analysis (Vol II) ALM 13, pp. 193-298.
Prerequisites:
Riemannian geometry
Schedule of Sections
Previous Semesters
- Fall 2020 Prof. F. Luo
- Fall 2017 Prof. J. Song
