Here is my mind-map for first-course complex analysis. It contains some well-known theorems and "arrows" between them.

Here it is, and of course you can download it as a PDF or as a SVG (vector graphics) as well (click on the image to enlarge it):

The license is CC-BY-NC-SA (if you redistribute, put my name on it, don't make profit, share alike).

There are some aspects which require an explanation:

Continue reading «Mindmap on complex analysis in one variable»

To prepare for my oral exams in algebraic geometry (covering Hartshorne's book "Algebraic Geometry" Chapter II and III) I sketched an overview diagram of morphism properties in the category of noetherian schemes. Maybe this is a good cheat sheet to keep with you while reading the book for the first or second time (ok, and I dropped a "Nisnevich" for no good reason, you can ignore it).

You can get a PDF version of the image or click on it to get a readable version.

I'm still in the process of writing down examples and counter-examples to these properties, maybe that list will be online some day (another kind of "counterexamples in algebraic geometry").

As always, I'm happy to hear any comments (did I miss an important arrow, did I get anything wrong) -- but I should stress that the diagram works in Hartshorne-world, not in EGA-terms (this kind of confusion cost me almost one entire day trying to prove wrong statements..)

UPDATE (2011-11-18): improved diagram (more information, less colour) and higher quality PNG file.

If you happened to buy books from Amazon.com (or, in my case, Amazon.de) and maybe used the recommendation engine and the wishlist (and and and ...) then there will be lots of data about your books on the Amazon website. Have you ever thought about organizing your library with a different tool? May it be Google Books or LibraryThing or Shelfari, you will have to export this precious big amount of data from Amazon to the other service. Luckily, some intelligent people invented ISBN, so you basically need to extract a list of ISBNs to identify the books (neglecting your reviews and tags for now). Not that luckily, Amazon doesn't offer such export functionality to the layman. Searching the internet yields a Greasemonkey script that enables you to export wishlist content - but no ISBNs, so import into other services is not so easy.

The solution is to save each website of "your purchases" (or other such lists of books) as HTML file and let a smart script do the extraction work. This way, you're not violating Amazon's terms of service (which most likely don't allow any robots scraping the website) and on the positive side, it works.

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Now I'll explain a little bit what essential manifolds are and what they're good for.

Definition
A (connected closed orientable topological) n-manifold $M$ is called essential, if there exists a continuous map $f : M \to K(\pi_1(M,\ast),1)$ such that the induced morphism on the top homology $f_\ast : H_n(M,\mathbb{Z}) \to H_n(K(\pi_1(M,\ast),1),\mathbb{Z})$ maps the fundamental class $[M] \in H_n(M,\mathbb{Z})$ to some non-zero element $f_\ast([M]) \neq 0 \in H_n(K(\pi_1(M,\ast),1),\mathbb{Z})$.

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In this post I want to sketch the idea of aspherical manifolds - manifolds which don't admit higher homotopically non-trivial spheres - and the related concepts of Eilenberg-MacLane-spaces and classifying spaces for groups.

Definition
A topological space $M$ is called aspherical if all higher homotopy groups vanish, i.e. $\pi_n(M,m_0) = 0 \quad \forall n > 1$ where $m_0 \in M$ is an arbitrary basepoint and $M$ is assumed to be connected.

Since manifolds admit universal covers, you could equivalently define a manifold to be aspherical if and only if its universal cover is contractible.

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Now this is a slightly corrected (although still somewhat messy) version of my diploma thesis - in german:
Matsumotos Satz und A¹-Homotopietheorie.

You can read something about the content in this blog post, containing an extended abstract in english.

Continue reading «Diploma thesis (in german)»