Motivic Cell Structure of Toric Surfaces

Wednesday, April 17th, 2013 | Author:

In this post I'll do a few very explicit computations for motivic cell structures of smooth projective toric varieties coming from the Białynicki-Birula decomposition, namely \mathbb{P}^1, \mathbb{P}^1 \times \mathbb{P}^1, \mathbb{P}^2 and Hirzebruch surfaces. It is a bit lengthy but maybe helpful to anyone who wants to do some explicit calculations with BB-decompositions. I hope you're accustomed to toric varieties, but I won't do anything fancy. You can safely skip the motivic part of this post.

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All these fundamental groups!

Tuesday, April 02nd, 2013 | Author:

There are a lot of fundamental groups floating around in mathematics. This is an attempt to collect some of the most popular and sketch their relations to each other.

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Feynman Graphs and Motives

Wednesday, March 20th, 2013 | Author:

Being on a school about Feynman graphs and Motives, I just learned how these are related. It's a cute story! Actually, you don't need any physics to appreciate it, though physics might let you appreciate it even more.

A Feynman graph is just a (non-directed) graph with a finite number of vertices and a finite number of edges. Physicists are interested in computing certain integrals defined in terms of Feynman graphs, which they call amplitudes.

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Motives of Projective Bundles

Friday, December 14th, 2012 | Author:

Given a vector bundle E-->X of rank r+1 one can take the projective space of lines in each fiber, which results in a projective bundle P(E)-->X. A projective bundle formula for a functor F from spaces to rings tells us that F(P(E)) is a free F(X)-module of rank r.

In this post I look at some computations around projective bundle formulae for the Chow ring, the algebraic K-Theory and the (Chow) motive of some spaces, in particular flag varieties. We recover some results from the previous posts on cohomology, cycles & bundles and motive of projective space.

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Divisorial Jungle

Thursday, November 29th, 2012 | Author:

I'd like to compile a short list of definitions of Weil and Cartier Divisors, Line Bundles and Invertible Sheaves, Class Groups and Picard Groups, Cohomology, (higher) Chow Groups and K-theory for algebraic schemes and their relations. I intentionally omit proofs, but there are some ideas. I couldn't resist to jot down some properties of the objects which are important to me (homotopy invariance, existence of pullbacks and pushforwards).

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Model structures on simplicial presheaves

Friday, November 23rd, 2012 | Author:

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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