Update, March 14: Because of the ongoing coronavirus situation, we will continue the seminar in online form, probably through Zoom. We will have an organizational (online) meeting on March 16 at 3:15 pm. All further lectures have been postponed by one week.
In Spring 2020 Yuqing Shi, Mingcong Zeng, and I will be running a seminar for MSc students on exotic spheres. Here is the syllabus. We meet on Mondays, 3:15-5pm, in HFG 610.
The aim of the seminar is to understand (part of) the papers of Milnor and Kervaire-Milnor. We will spend roughly the first half of the seminar on learning about characteristic classes, which are a ubiquitous tool in algebraic and differential topology. Useful references for this material are the books of Milnor-Stasheff, Hatcher, and Bott-Tu.
Schedule
Feb 3, Gijs Heuts: Introduction and overview.
Feb 10, Coline Emprin: Stiefel-Whitney classes and their basic properties.
Feb 17, Sven van Nigtevecht: The classification of vector bundles. Grassmannians. Sven's notes.
Feb 24, Bram Buiting: Proof of the existence of Stiefel-Whitney classes.
Mar 2, Jaco Ruit: Euler classes and the Thom isomorphism.
Mar 9, Wilmer Smilde: Poincaré duality and the Euler characteristic. Wilmer's notes and exercises.
Mar 16: Organizational online meeting.
Mar 23, Max Blans: Chern classes and their relation to differential geometry. Max' exercises and notes.
Mar 30, Ryan Quinn: Chern numbers and Pontryagin numbers.
Apr 6, Martijn Waal: Oriented cobordism and the Pontryagin-Thom construction.
Apr 20, Leonard Tokic: Hirzebruch's signature theorem.
Apr 27, Max and Sven: On manifolds homeomorphic to the 7-sphere. Some extra notes.
May 4, Coline and Jack: A manifold which does not admit any differentiable structure. Coline's slides and Jack's slides.
May 11, Leonard and Ryan: Groups of homotopy spheres, part I.
May 18, Bram and Martijn: Groups of homotopy spheres, part II.
May 25, Jaco and Wilmer: Groups of homotopy spheres, part III.
June 2, Mingcong Zeng: The Kervaire invariant one problem.