Szego Seminar: "Generic Smooth Maps and Foliations"

Abstract: Morse theory is a suite of tools and techniques to analyze level sets of smooth real-valued functions on smooth manifolds. Morse theory and its infinite dimensional analogue have been central to many developments in low-dimensional topology both before and after the settling of the Poincaré conjecture. Foliation theory is very naturally connected with Morse theory, and has seen renewed interest among low dimensional topologists since Perelman settled the Geometrization and Poincaré conjectures in dimension 3. I’ll discuss Morse functions, a slight generalization of Morse functions to taking values in the Riemann sphere (where secretly something holomorphic is happening), what a foliation is, how these foliation and Morse objects interact, and if time permits how it relates to the above conjectures and beyond.

I won’t assume any knowledge of Morse functions or foliations. The kind of qual course knowledge one should have to come along for the whole ride would be that of a smooth atlas for a topological manifold, the fundamental group, a group action, and only for added flair the notion of differential forms.

Host: Nathan Wagner