Fall 2026
Facundo Mémoli
Subtitle:
Gromov's Filling Radius and Persistent Homology
Course Description:
This course surveys the geometry and topology of metric spaces through the lens of Gromov’s (homological) filling radius and its connections to systolic inequalities and various notions of width. We will develop the basic definitions, prove foundational properties, and explore rigidity and stability phenomena that link Riemannian geometry, quantitative topology, and applied topology.
We begin with a review of the Gromov–Hausdorff distance, the topology it induces on the class of compact metric spaces, and Gromov’s precompactness theorem. We then develop the theory of Gromov’s filling radius $\mathrm{FillRad}(X)$ (via homological fillings in a Banach ambient space) and compare it to systolic invariants (e.g., $\operatorname{sys}_1$ on essential manifolds) and to width-type quantities (e.g., Urysohn width). Along the way we highlight sharp inequalities and extremal examples.
A core theme is rigidity. We study Wilhelm’s rigidity theorem for positively curved manifolds and view it as an analogue—at the level of filling radius—of Bonnet–Myers–type diameter/curvature bounds. We examine equality and near-equality cases and what they force about the underlying metric and topology.
In the second half, we connect classical filling ideas to persistent homology. Working mainly with Vietoris–Rips filtrations, we show how $\mathrm{FillRad}$ controls—and is reflected in—the birth–death scales of homology classes. We discuss interleaving stability, Gromov–Hausdorff continuity of barcodes, and quantitative estimates that relate filling radii, systoles, and Rips persistence. Examples include spheres, projective spaces, flat tori, and selected non-manifold metric spaces.
Text:
N/A
Prerequisites:
Some background in differential geometry at the level of basic Riemannian geometry, algebraic topology (homology/cohomology), and metric geometry basics. Familiarity with persistent homology is helpful but not required.
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Fall 2025
Guangbo Xu
Subtitle:
Floer homology in symplectic geometry
Course Description:
In 1965 Arnold posed his famous conjecture on the number of fixed points of Hamiltonian diffeomorphisms. Based on Gromov’s pseudoholomorphic curve (1985) and Witten’s QFT interpretation of Morse theory (1982), Floer (1988) invented the Floer homology and provided the most influential idea towards the Arnold conjecture. This idea soon lead to a series of landmark developments in different fields, including symplectic geometry, gauge theory, and low-dimensional topology. In this course, we will go through the rigorous construction of the Hamiltonian version of Floer homology. At the same time, you will learn the detailed techniques related to pseudoholomorphic curves, including Gromov compactness, transversality, and gluing.
Text:
Main reference: D. Salamon, Lectures on Floer homology. Supplementary reference: McDuff-Salamon, J-holomorphic curves and symplectic topology,
Prerequisites:
Differential Geometry 1 (532), Differential Topology (548), Partial Differential Equations I (517), or instructor’s approval.
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Spring 2023
Chris Woodward
Subtitle:
Mirror Symmetry
Course Description:
(To be determined based on the interests of participants) An introduction to the ideas involved in the mirror symmetry conjectures focused on the example of toric varieties. Symplectic manifolds, Lagrangia submanifolds pseudoholomorphic curves, Floer homology, toric varieties, sheaves cohomology.
Prerequisites:
Understanding of manifolds and basic algebraic topology.
Text:
None
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Spring 2021
Chris Woodward
Subtitle:
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.
Prerequisites:
Basics of manifolds and algebraic topology
Text:
None
Schedule of Sections:
16:640:534 Schedule of Classes
Previous Semesters:
- Spring 2021 Prof. Woodward
- Spring 2019 Prof. X.C. Rong
SPRING 2019
Xiaochun Rong
Subtitle:
Structures on Manifolds of Ricci Curvature Bounded Below
Prerequisites:
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.
Text:
No textbook. Will distribute notes.
Course Description:
CONTENTS
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