Białynicki-Birula and Motivic Decompositions

Thursday, April 04th, 2013 | Author:

This is about Białynicki-Birula's paper from '72 on actions of reductive linear algebraic groups on non-singular varieties, in particular Gm-operations on smooth projective varieties. I give a proof sketch of Theorem 4.1 therein and explain a little bit how Brosnan applied these results in 2005 to get decompositions of the Chow motive of smooth projective varieties with Gm-operation. Wendt used these methods in 2010 to lift such a decomposition on the homotopy-level, to prove that smooth projective spherical varieties admit stable motivic cell decompositions. Most of this blogpost consists of an outline of the B-B paper.

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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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Homotopy Theory of Proofs?

Monday, March 25th, 2013 | Author:

This question is not suitable for MathOverflow, as it is inherently vague.

In short:
Is there a Homotopy Theory of Proofs?

Category: English, Mathematics | 7 Comments

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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From the Backlog 2010

Thursday, February 21st, 2013 | Author:

I have to admit that I try to read more than I actually do, in particular blog posts. In this post I give you some links from 2010 that I now read skimmed a few weeks ago and found to be fresh enough to be shared again.

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Invariants of projective space III: Motives

Monday, December 10th, 2012 | Author:

I want to explain a particularly easy example of a motivic cellular decomposition: That of $n$-dimensional projective space. The discussion started with cohomology (part 1), continued with bundles and cycles (part 2) and in this part 3, we discuss motivic stuff.

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