Francesco Lin

Associate Professor of Mathematics at Columbia University. CV and MathGenealogy.

Office: Department of Mathematics, Room 613.

Email: flin-at-math.columbia.edu (research), fl2550-at-columbia.edu (teaching).

In Spring 2026, I am teaching Modern Analysis II.

I am interested in geometry, topology and analysis in low dimension (especially three), with a focus on Floer homology, hyperbolic geometry and spectral theory. I think a lot about the equations of Maxwell and Dirac.

My work is currently supported by the grants NSF DMS-2503714 and Simons Foundation TSM-00013131.

Publication list and preprints:

Dirac spectral flow and Floer theory of hyperbolic three-manifolds, with M. Lipnowski (spectral movie).
Adjunction inequalities and the Davis hyperbolic four-manifold, with B. Martelli.
Divergence-free framings of three-manifolds via eigenspinors, Documenta Mathematica.
Topology of the Dirac equation on spectrally large three-manifolds, Journal of the IMJ.

Materials for the topics course Gauge theory and topology: an introduction.

Syllabus and lecture notes:

1 - From magnetostatics to gauge theory.
2 - Curvature and characteristic classes.
3 - Yang-Mills theory on Riemannian manifolds.
4 - Instantons, monopoles, vortices.
5 - Elliptic operators and Sobolev spaces.
6 - The Hodge theorem and index theory.
7 - The vortex equations on a surface.
8 - Geometry of holomorphic line bundles.
9 - Symplectic manifolds and moment maps.
10 - Moduli spaces in gauge theory.
11 - Vortices and symmetric products.
12 - Seiberg-Witten theory and Floer homology.

Last updated 12/25.