Course Descriptions

16:640:534 - Select Topics Geometry I

Spring 2021 - Chris Woodward


Introduction to Symplectic Topology

Course Description:

Symplectic manifolds, Lagrangian submanifolds, examples from Hamiltonian dynamics and algebraic geometry, pseudoholomorphic maps. Further topics may include: symplectic capacities; Floer cohomology, relations with mirror symmetry and tropical geometry.  Please contact the instructor to discuss topic choice.


Basics of manifolds and algebraic topology




Schedule of Sections:

Previous Semesters:

  • Spring 2021 Prof. Woodward
  • Spring 2019 Prof. X.C. Rong


Xiaochun Rong


Structures on Manifolds of Ricci Curvature Bounded Below


The prerequisites of this course is a basic knowledge of Riemannian geometry (surfaces in R3). The main purpose of this course is to give an introduction to structures on manifolds with Ricci curvature bounded below, and their limit spaces. We start with classical results on Laplacian comparison, volume and splitting rigidity, and the Gromov-Haudorff topology on metric spaces. We will cover basic Cheeger-Colding theory on structures of Ricci limit space, and basic nilpotent structures on collapsed manifolds with Ricci curvature bounded below.


 No textbook. Will distribute notes.

Course Description:


I. Ricci curvature comparison, Rigidities and Quantitative Rigidity

1.1. Ricci curvature comparison: Laplacian and volume.

1.2. Maximal principle and splitting rigidity

1.3. Gromov-Haudorff topology

1.4. Warp product metric rigidity and quantitative rigidity

1.5. Structures on Ricci limit spaces

II. Nilpotent structures on Collapsed Manifolds with Bounded Ricci

2.1. Maximally collapsed manifolds with local rewinding volume bounded below

2.2. Nilpotent structures on collapsed manifolds with local rewinding Reifenberg points

2.3. Collapsed manifolds with local Ricci bounded covering geometry

2.4. Collapsed Einstein manifolds