Ich mach mal eine Liste von den Grundlagen, und woher ich sie gelernt habe (=was ich als brauchbare Lektüre empfinde, also meine Lieblingsliteratur)
Continue reading «Gute Mathematik Skripten und Bücher»

Here is the second part of my walk-through to Voevodskys A¹-homotopy theory:

On page 48, the first Lemma is shown. Without proof - so I will try to illuminate things a little bit by giving the proof. This lemma isn't used until section 3, so you can skip it, if you want to. I suggest not to do so, if you are intimidated by the diagram, because it isn't that hard, and it's a nice exercise to get the concepts in your head right.

Continue reading «Walk-through to Morel-Voevodsky A¹-homotopy theory part II (page 48, Lemma 1.1)»

On Math Overflow, someone asked for "A single paper everyone should read?"
and some answers were particularly nice to read for me, so I repeat it for you, ordered by how much math is needed (from none up to little):

• Paul Lockhart: "A Mathematician's Lament" shares my opinion about the math eduction disaster in schools. I think you should read this if you disliked your math classes in school or if you will ever have children (who will have to take a math class, then).
• Terry Tao: "What is good mathematics?" which is a short (10 pages) paper about the benefit we have from mathematicians different tastes and approaches. I recommend to every scientist reading the first 3 pages (the other 7 pages are only understandable with some background in mathematics).
• Freeman Dyson: "Birds and Frogs" which is a must-read for anyone interested in history and/or progress of mathematics.
• Misha Gromov: "Spaces and Questions" which is readable with almost no background, although might be funnier if you know basic differential geometry. It tells a dense story of geometric ideas and their development in history. And it doesn't take much time to read/skim it.
• Timothy Chow: A beginner's guide to forcing is a really gentle introduction to forcing.

Math Overflow is a new community website where mathematicians can discuss research problems. It is based on Stack Exchange, the software powering Stack Overflow, which does the same for computer science.

In this post, I will sketch a classification of Riemann surfaces.

For those who haven't heard about the subject before, there is an introduction. For the impatient, look at the bottom of the post, where I have written a very short summary.

Table of contents:

• Definitions
• Examples and counter-examples
• Topological classification, the genus
• Digression about algebraic field extensions
• Biholomorphic classification via universal covering deck transformations
• Brief discussion of Gaussian curvature and conformal geometry
• Summary

Continue reading «Classifying Riemann surfaces»

As I'm currently reading Morel&Voevodskys paper on A¹-homotopy theory of schemes, I will by and by write a little "walk-through" with hints & comments on how to find additional information to better (or even start to) understand the paper. Maybe for a newcomer to the subject (like me) it's difficult at first to stick together all these concepts like model theory, simplicial objects, Grothendieck topologies, algebraic geometry and so on. I will try to provide helpful comments to make the paper more accessible to those who are not that familiar with these notions above.

Continue reading «Walk-through to Morel-Voevodsky: A¹-homotopy theory of schemes (1999)»

Gegeben eine holomorphe Abbildung $f : X \rightarrow Y$ zwischen Riemannschen Flächen $X,\ Y$, können wir uns fragen: Ist $f$ eine Überlagerung? Unter welchen hinreichenden oder notwendigen Bedingungen?

Continue reading «Holomorphe Abbildungen sind manchmal schon Überlagerungen»