Spring 2026
Xiaochun Rong
Subtitle:
Introduction to collapsed Riemannian Manifolds with Curvature Bound
Course Description:
We give a quick introduction to the study of geometric and topological structures on an ϵ-collapsed Riemannian n-manifold, M , i.e., every unit ball on M has the
volume < ϵ (a small constant depending on n), while a suitable bound on curvature is imposed preventing a scaling of the metric.
This course consists of two parts: in the first part, we will present new proofs for the Gromov’s theorem on almost flat manifolds, and the nilpotent fiberation
theorem of Cheeger-Fukaya-Gromov on collapsed n-manifolds with sectional curvature bounded in absolute value, and applications. In the second part of this course,
we will review basic Cheeger-Colding-Naber theory on a Ricci limit space i.e ., the Gromov-Hausdorff limit of a sequence of Riemannian manifolds with Ricci curvature bounded below. We will focus on the class of collapsing with local bounded covering geometry where we extend the collapsing theory of Cheeger-Fukaya-Gromova
collapsed manifold. We also discuss recent applications.
CONTENTS
Part I. Collapsing with bounded sectional curvature
1. Overview on collapsed Riemannian manifolds with local bounded covering geometry
2. Introduction to the Gromov precompactness in metric geometry
3. Fiber bundle structures on manifolds collapse to a manifold.
4. A new proof of Gromov’s almost flat manifolds
5. Applications
Part II. Collapsing with local Ricci bounded covering geometry
6. Ricci-limit spaces, and the subclass of local bounded covering geometry
7. Strong (Perelman) pseudo-locality on Ricci flows on collapsed manifolds of local Ricci bounded covering geometry
8. Quantitative maximal rigidity of Ricci curvature bounded below
9. Compact Ricci limit spaces of solvable manifolds
Text:
None
Prereq:
Basic knowledge in Differential Geometry (Manifolds, tangent/cotangent bundles, Riemannian structures, etc), basic knowledge in Topology (covering spaces, homotopy groups, fiber bundles, etc)
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Fall 2025
Paul Feehan
Subtitle:
Gauge theory in analysis and geometry.
Course Description:
First, we will first introduce the fundamentals of gauge theory over closed manifolds, including a review of the required background in complex geometry, differential geometry, functional analysis, partial differential equations, and topology. Second, we will survey the gauge theory equations on manifolds with dimensions in the range 3 to 8 that are actively studied today, including the Hermitian Yang-Mills or Hermitian-Einstein equations, Seiberg-Witten equations, Kapustin-Witten equations, Vafa-Witten equations, G_2 instanton equations, and others. Third, we shall introduce the work of Taubes on the relationship between the Seiberg-Witten and Gromov-Witten invariants of 4-dimensional symplectic manifolds. The course will be tailored to the interests and background of students and the preceding topic list adjusted accordingly.
Text:
TBA
Prerequisites:
Permission of instructor
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Spring 2025
Xiaochun Rong
Subtitle:
Collapsed Manifolds of Bounded Curvature
Course Description:
A Riemannian n-manifold M is called ϵ-collapsed, if every unit ball on M has a volume < ϵ(n) (a small constant depending on n), while a suitable bound on curvature is imposed in preventing a possible scaling.
This graduate topic course consists of two parts: In the first part, we will start with the Gromov’s theorem on almost flat manifolds, then cover basics of the nilpotent structure theory of Cheeger-Fukaya-Gromov on collapsed n-manifolds with sectional curvature bounded in absolute value, as well as applications. In the second part of this course, we will present partial nilpotent structures on collapsed n-manifolds with Ricci curvature bounded in absolute value, or bounded below, under additional restrictions.
CONTENTS
Part I. Collapsed Manifolds with Sectional Curvature Bounded in Absolute Value
0. Preliminaries
1. The Gromov-Hausdorff topology and equivariant GH-convergence
2. Fiber bundle theorems
3. The stability of compact Lie group actions
4. Gromov’s almost Flat Manifolds 5. The Singular nilpotent fibration
Part II. Collapsed Manifolds with Ricci Curvature Bounded Below and Local Bounded Covering Geometry
6. Basics in the theory of Ricci limit spaces by Cheeger-Colding-Naber
7. Nilpotent structures on collapsed manifolds with local Ricci bounded covering geometry
8. Application: quantitative maximal rigidity of Ricci curvature bounded below
Text:
The instructor prepares to write lecture notes for this course.
Prerequisites:
A basic knowledge in Riemannian geometry is required for this course, we will try to make the topics self-contained and accessible to some early graduate students.
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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:
16:640:537 Schedule of Classes
Previous Semesters
- Fall 2020 Prof. F. Luo
- Fall 2017 Prof. J. Song