The aim of this course is to introduce the tools used to construct, deconstruct and distinguish 3-dimensional manifolds.
Dimension 3 finds itself at a sweet spot: complicated enough to host a lot of interesting mathematics, but not so complicated as to be intractable. Therefore, it has been a fertile ground of research since the early 20th century. Landmark work of Thurston in the late 70s revealed that the topology of 3-manifolds is inextricably linked to their geometry. This insight has led to many outstanding results, chief among them Perelman's proof of the Poincaré conjecture. On a (seemingly) unrelated path, Witten has shown a surprising connection between 3-manifolds, Chern-Simons theory and Topological Quantum Field Theories (TQFT).
Some of the topics that we will discuss are Heegaard splittings, Dehn fillings, the sphere and torus decomposition of a 3-manifold, Thurston's 8 geometries and Perelman's Geometrization theorem (of course without any mention of a proof!), and quantum invariants of 3-manifolds coming from TQFTs.
The target audience for this class are Master's students in Mathematics or Master's students in Physics with a strong mathematical background. Prerequisites of the class are a basic knowledge of differential topology (such as the definitions of smooth manifolds and vector bundles), Riemannian geometry (such as the definition of Riemannian manifold, isometries and curvature) and algebraic topology (such as the fundamental group and singular homology).
Tuesdays from 16 to 18
Wednesdays from 16 to 18
The class will be held in English on Zoom. Video recordings of the lecture will be available on Mampf. Contact me if you want access to the Zoom room.
W.B. Raymond Lickorish, An Introduction to Knot Theory