Exotic Spheres and Singularities - Math Associates Seminar
Channel: ICTP Mathematics
Date: 2021-03-10

Speaker: José Cisneros-Molina (UNAM, Mexico)


 Abstract: Milnor's fibration theorem is a landmark in singularity theory, it allows to deepen the study of the geometry and topology of analytic maps near their critical points. To each singular point of a complex hypersurface it associates a fibre bundle, known as the Milnor Fibration of the singularity.
The birth of Milnor's Fibration Theorem is a curious story, since its origin is not in Singularity Theory. It begins with the discovery by Milnor in 1956 of the first exotic spheres": smooth 7-manifolds homeomorphic to the standard 7-sphere, but with non-equivalent differentiable structures. Later, work of some great  mathematicians like Brieskorn, Jänich, Hirzebruch and Pham, led to the discovery of some examples of complex hypersurface singularities whose links are exotic spheres. This motivated Milnor to determine when the link of a complex hypersurface singularity is a homotopy sphere, culminating in his now famous book "Singular points of complex hypersurfaces". One of the main results in this book is precisely Milnor's Fibration Theorem.
The aim of this non-technical talk is to tell this interesting story.