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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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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Linux, OCR and PDF: Scan to PDF/A

Friday, March 29th, 2013 | Author:

The (by far) most visited post on this blog is from 2010, about OCRing a PDF in GNU/Linux (Optical Character Recognition), and it contains a small shell script that has been improved by others several times. After having bought a new flatbed scanner, I re-investigated how to scan and OCR pdfs, how to produce DJVU files that are incredibly small and how to get metadata right. It turns out what I really ever wanted was to create PDF/A compliant documents (I just didn't know what PDF/A was before). But let me explain the details after presenting you the quick solution. At the end, I have a shell script that scans directly to PDF/A.

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Measuring Time

Wednesday, March 27th, 2013 | Author:

The units scientists use in their daily work are the SI units, and sometimes the equivalent cgs units. In these unit systems, everything is based upon the physical quantities length, mass, time (and some more). The part "physical" is only deserved if we mean something measurable by these "quantities". So, what does it mean to measure length, mass and time?

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Homotopy Theory of Proofs?

Monday, March 25th, 2013 | Author:

Diagram proving the properties of a simplicial group covering

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

In short:
Is there a Homotopy Theory of Proofs?

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Category: English, Mathematics | 7 Comments