This is a list of resources available on the web for research mathematicians (as opposed to teachers).
I intend to update it, as time passes by. The first version was published May 5, 2012.
The last update was on May 9, 2012.

Continue reading «Mathematics Resources»

This post will discuss the definition of the four functors "pushforward" $f_\ast$, "pullback" $f^\ast$, "pushforward with compact support" $f_{!}$ and "exceptional pullback" $f^{!}$ of sheaves of abelian groups, associated to a continuous morphism $f : X \to Y$ of topological spaces $X$ and $Y$. Then we will look at maps $f$ which are open immersions or closed immersions, and calculate in the example of $\mathbb{C}^\times \to \mathbb{C}$ and its closed complement $\{0\} \to \mathbb{C}$ exactly what happens. This is intended to give some intuition what the general four functor calculus is about.

Continue reading «The four functors of Grothendieck in examples»

These days I read Akhil Mathew's post on Vaserstein's proof of the Quillen-Suslin theorem, once known as Serre's Problem. This inspired the following.

Serre asked whether algebraic vector bundles over affine space are all trivial or not. Quillen and Suslin proved independently that they are, in fact, all trivial. This is some kind of analogue to the topological situation, where all vector bundles over n-dimensional complex affine space (or even n-dimensional real affine space) are trivial.

Continue reading «Is it possible to prove Serre's Problem (the Quillen-Suslin theorem) via Motivic Homotopy ...»

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.

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 $f_\ast([M]) \neq 0 \in H_n(K(\pi_1(M,\ast),1),\mathbb{Z})$.

Continue reading «Essential manifolds»