This is a very short notice to memorize some of the various model structures on simplicial presheaves in a systematic way.

[UPDATE 2013-03-06] I gave a talk in our working group seminar about model structures on simplicial presheaves, homotopy sheaves and h-principles [/UPDATE]

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This is supposed to be a short intuitive introduction to model categories.

Suppose you have a category $\mathcal{C}$ and some class of morphisms $W$ which behave somewhat like isomorphisms (for example: Chain complexes and Quasi-isomorphisms, or topological spaces and homotopy equivalences, or simplicial sets and weak homotopy equivalences ...). We will call this class "weak equivalences". Then you can look at the localized category $[W^{-1}]\mathcal{C}$, where the morphisms in $W$ are made invertible. If you're lucky, not all objects are isomorphic to each other, and if you're really lucky, you can compute something.

But, as it turns out, usually you don't work with the localized category abstractly, but by some explicit construction of some special case (say, Verdier localization of triangulated categories in the homological setting or explicit homotopies in the topological setting).

Model categories (and its cousins, weak factorization systems, categories of (co)fibrant objects, homotopical categories, etc.) provide a framework to compute stuff in $[W^{-1}]\mathcal{C}$.

[UPDATE 2013-03-06] I gave a 30-Minute talk about model categories, with very little content. [/UPDATE]

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The new term has begun! And with it a new academic year in Freiburg.

In the past I used to put extensive notes online, for any talks I gave (mostly in local seminars at my university). I have changed this to posting blog posts instead, but sometimes I also have some teXed talk notes. The quality of the teXed notes is usually bad, as I didn't intend to publish them. This is a compilation of talk notes from the last academic year.

Goal for the next year: focus :-)

• Seminar on Systolic Geometry: Aspherical Manifolds and Essential Manifolds.
• Seminar on Perverse Sheaves: Introduction to 6 functor formalism. I also blogged some stuff even more elementary about 4 functors before.
• Seminar on Geometric Langlands: From Langlands for Function Fields to Geometric Langlands; I have blogged an introduction and also some of the talk on Geometric Langlands here.
• Seminar on Cobordism: Bordism, Cobordism and Spectra (there's a blog post). There are also slides on the cobordism spectrum, though I remember it was definitely on a blackboard.
• Seminar on Mapping Class Groups: Spectra and Infinite Loop Spaces (these notes are actually not that bad).
• Seminar on Galois Representations: Grothendieck's Monodromy Theorem. I have written a small note on supernatural numbers, which appear in Tate's proof. There are also (bad, uncorrected) talk notes on Weil-Deligne representations in german.
• Seminar on Multiple Zeta Values: Upper bounds on periods (link goes to full lecture notes, which are quite good now).
• Student's conference of the german math society: 20 minutes on A¹-fundamental groups of isotropic groups.

These days I read Akhil Mathew's post on Vaserstein's proof of the Quillen-Suslin theorem, once known as Serre's Problem. This inspired the following.

Serre asked whether algebraic vector bundles over affine space are all trivial or not. Quillen and Suslin proved independently that they are, in fact, all trivial. This is some kind of analogue to the topological situation, where all vector bundles over n-dimensional complex affine space (or even n-dimensional real affine space) are trivial.

Continue reading «Is it possible to prove Serre's Problem (the Quillen-Suslin theorem) via Motivic Homotopy ...»

I just re-discovered some notes (in german) for expository talks I gave at the University of Freiburg in student's seminars.

Here they are:

• Direktes Bild und Inverses Bild von D-Moduln
• Das Hilbertpolynom für Moduln über der Weylalgebra
• Erste Cech-Kohomologie und G-Hauptfaserbündel
• Postnikov-Türme, k-Invarianten und Hindernistheorie
• Derivierte Kategorien (part I)
• Derivierte Kategorien (part II)
• Zelluläre Approximation von Abbildungen

For the sake of completeness, here are links to blog posts discussing other old (german) expository talks:

• Bott-Periodizität in der komplexen K-Theorie
• Relative Entropie und der Satz von Bayes

And soon you might see some notes on the Weil conjectures on elliptic curves (I'm still preparing this..)

Today I'm attending the second Workshop on Personal Knowledge Management (PKM2010) at the Human-Computer-Interaction (HCI) conference "Mensch und Computer" in Duisburg (Germany).

I have absolutely no idea what to expect, so I expect to be surprised.

UPDATE: Now that the workshop is almost over (coffee break right now), maybe the most important for me:

It was fun!

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