Homology cobordism and the geometry of hyperbolic three-manifolds - Francesco Lin

Stony Brook Mathematics Colloquium
October 12, 2023
Francesco Lin, Columbia University
Homology cobordism and the geometry of hyperbolic three-manifolds

The three-dimensional homology cobordism group is a fundamental object of study in low-dimensional topology. A major challenge in the study of its structure is to understand the interaction between hyperbolic geometry and homology cobordism. In this talk, after introducing the main protagonists and explaining the role they play in topology, I will discuss how monopole Floer homology (a subtle invariant of three-manifolds obtained by suitably counting solutions to the Seiberg-Witten equations) can be used to study some basic properties of certain subgroups of the homology cobordism group generated by hyperbolic homology spheres satisfying natural geometric constraints of Riemannian and spectral nature.