Divisorial Jungle

Thursday, November 29th, 2012 | Author:

Arithmetic Geometry

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).

Continue reading «Divisorial Jungle»

Category: English, Mathematics | Leave a Comment

Model structures on simplicial presheaves

Friday, November 23rd, 2012 | Author:

A^1

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]

Continue reading «Model structures on simplicial presheaves»

Category: English, Mathematics | Leave a Comment

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

Wednesday, November 21st, 2012 | Author:

Old Computer

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]

Continue reading «Get your own LaTeX-enabled wiki in the cloud with Instiki on Heroku»

Category: English, Mathematics, Not Mathematics | 2 Comments

What's a point of this?

Monday, November 19th, 2012 | Author:

Arithmetic Geometry

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?".

Continue reading «What's a point of this?»

Category: English, Mathematics | One Comment

Cellular objects in the motivic model category

Friday, November 16th, 2012 | Author:

Cellular objects

This article explains the notion of cellular objects in the motivic model category, which are like CW complexes for the algebraic/motivic world.

(last edit on 2014-04-08, added a remark on realizations and a Thom construction)

Continue reading «Cellular objects in the motivic model category»

Category: English, Mathematics | Leave a Comment

Cellular objects: CW complexes

Wednesday, November 14th, 2012 | Author:

Cellular objects

We will investigate the notion of cellular objects in a model category; today: the classical case of CW-complexes in the model category of topological spaces with Serre-fibrations as fibrations.

A CW complex is a certain kind of topological space, together with a CW structure, which is a description how to glue the space from spheres (or from affine spaces, if you prefer), called the cells. The acronym CW stands for "closure finite, weak topology", which I will explain soon. CW complexes are a class of spaces broader than simplicial complexes, and they are still combinatorial in nature.

Continue reading «Cellular objects: CW complexes»

Category: English, Mathematics | Leave a Comment