This is a crash course in the form of a seminar on Motivic Homotopy Theory (MHT) to prepare ourselves for the lectures on $\mathbb{A}^1$-homotopy theory in the upcoming GQT-colloquium in the first week of June.
We follow the book by Dundas, e.a., Motivic Homotopy Theory. Universitext 2007.
MHT is a mix of algebraic topology and algebraic geometry. Since there is not that much time, the pace will be rather high. Therefore the algebraic topologists will give talks on topics that they are familiar with, whilst the algebraic geometers will talk about topics in their field. In other words, the goal is not so much to learn a lot from your own talk (as might be the case in other seminars), but to enlighten fellow mathematicians from a nearby field.
| Date | Room | Talks |
|---|---|---|
| Wed, Apr 15 | 00.308 | Johan Commelin —
Intro to schemes. Notes
Joost Nuiten — Simplicial sets, etc. |
| Wed, Apr 22 | 00.308 | Ben Moonen — Examples; Groups schemes; Functor of points.
Javier Gutiérrez — Model categories. |
| Fri, May 1 | 00.065 | Milan Lopuhaä — Finite; proper; flat.
Matan Prasma — Constructions in model categories. |
| Wed, May 6 | 03.632 | Johan Commelin — Smooth and étale morphisms. Notes
Simon Henry — Pretopologies, étale site, Nisnevich site. |
| Tue, May 12 | 00.086 | Giovanni Caviglia — Bousfield localization; simplicial model categories.
Joost Nuiten — Model structure on simplicial presheaves; $\mathbb{A}^{1}$-model structure. |
| Wed, May 20 | 00.308 | Milan Lopuhaä — Tate object.
Javier Gutiérrez — Spectra. |
| Wed, May 27 | 00.086 | Joost Nuiten — Model structure for motivic spectra.
Johan Commelin — Motivic cohomology. Notes |
The main source is the aforementioned book by Dundas, e.a., Motivic Homotopy Theory. Universitext 2007.
Brian Osserman has created two cheat sheets about properties of morphisms and properties of properties of morphisms.
For more information about weights (and Tate objects) in the cohomology of algebraic varieties, see the 6 page long article by P. Deligne, Theorie de Hodge I, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 425–430. Gauthier-Villars, Paris, 1971. (Also available online via his list of publications on the IAS website.)