Invariants of projective space II: Cycles and Bundles

Thursday, December 06th, 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) and in this part 2, we discuss intersection-theoretic and bundle-theoretic invariants. In part 3 we will see the motivic stuff.

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Invariants of projective space I: Cohomology

Tuesday, December 04th, 2012 | Author:

I want to explain a particularly easy example of a motivic cellular decomposition: That of n-dimensional projective space. We will have a look at the cohomology, the Chow groups and the algebraic K-theory of projective space -- a discussion probably interesting to non-motivic people as well. After these invariants, I will look at the motive and the A¹-homotopy type. Then I want to describe the decomposition of the motive (and the homotopy type) homotopy-theoretically, by means of cofiber sequences. (We will see that projective space is not isomorphic to a coproduct of motivic spheres with the same motive). Of course, nothing is new, I'm just working out exercises here.

In this part 1, I discuss only the cohomology of \mathbb{P}^n. Part 2 contains a discussion of the intersection theory and bundles and part 3 contains the motivic stuff. I intentionally left out usage of projective bundle formulas, as I will discuss them separately.

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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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Get your own LaTeX-enabled wiki in the cloud with Instiki on Heroku

Wednesday, November 21st, 2012 | Author:

I guess you all know what a WikiWikiWeb (short: wiki) is, it's a website where you can easily add new pages and modify existing ones. MathOverflow is some kind of hybrid between Q&A and a wiki, since users with enough reputation can edit other people's questions and answers. MathOverflow made the Markdown syntax very popular, and people got used to using LaTeX online. Some of my readers surely know the nLab, a collaborative wiki on n-categorical math(ematical physics) and stuff. The nLab runs on a software called Instiki, which is a wiki written in Ruby (an intepreted language similar to Python, and somewhat similar to Lisp, Perl and JavaScript; which is often used for web applications like wikis). The good thing about Instiki is that it supports editing pages in Markdown syntax with embedded LaTeX, so it is able to support your personal knowledge management needs. In addition, Instiki is small (thus not many bugs are to be expected), fast and the code is quite readable; something I wouldn't say about MediaWiki, the software behind Wikipedia.

In this post, I will tell you how to run your own wiki like the nLab. [UPDATED 2013-01-07; easier fix]

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Category: English, Mathematics, Not Mathematics | 2 Comments

What's a point of this?

Monday, November 19th, 2012 | Author:

I recently came across a paper using a "universal domain" to discuss "generic points" of a variety, using Weil's foundations of algebraic geometry instead of Grothendieck's. First I had to learn that stuff, then I wanted to translate it. This lead to a more systematic study of what it means to be a point of a variety or scheme, in the various different definitions.

So in this post I will explain closed points, generic points, points in general position, schematic points, generalized points, rational points, geometric points, and in particular, which of these notions can be considered a particular case of another of these. I will try to give you a hint why one wants to generalize the ordinary (closed) points of a variety that much, to answer the question in the title: "What's the point of this?".

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Category: English, Mathematics | 4 Comments