A vector bundle is a morphism that looks locally on the target like a product of the target with a vector space.

We will call the target space the base and the space of definition the total space. The preimage of a point of the base is called the fiber.

Is that the correct mathematical definition? It doesn't mention what kind of spaces we look at, what kind of morphism I'm talking about, what the product is, locally in which sense, vector space over which field, do we allow infinite dimension, ... so it's not a mathematical definition in the pedantic sense. I will give you pedantic definitions in this article, just to satisfy my need to write down what I consider to be a good terminology.

Continue reading «What is ... a vector bundle?»

Nowadays it is common to use $x \mapsto f(x)$ to denote that an element $x \in X$ is mapped to an element $f(x) \in Y$ by the map(ping) $f : X \to Y$. In particular, the arrow $\rightarrow$ (in LaTeX: \rightarrow) denotes a map, or more generally a morphism, while $\mapsto$ (in LaTeX: \mapsto) denotes how particular elements or objects are mapped to other elements or objects.

Have you ever seen an arrow which has a triangle as head? Like those:

Continue reading «An arrow notation for annotations»

This short article is intended to be read by non-mathematicians who don't quite remember what the term "matrix" or "polynomial" refers to. I'll try to give you an intuitive idea of what a PhD student working on "Algebraic Topology" studies nonetheless.

First of all, studying or researching mathematics is not about remembering formula or calculating some really large numbers. That is a part of mathematics, but it's not what drives it, it is merely a tool that is more and more handed to computer systems.
So, what is mathematics instead? I am not competent to give an answer (and there have been many many different answers to that question in the past) but I can explain you my view on pure mathematics:

Pure Mathematics is the study (by any means) of the statements of which truth can be derived syntactically, i.e. by some computational process. (Yes, I'm quite a syntactic thinker).

Continue reading «What is Algebraic Topology?»

The new term has begun! And with it a new academic year in Freiburg.

In the past I used to put extensive notes online, for any talks I gave (mostly in local seminars at my university). I have changed this to posting blog posts instead, but sometimes I also have some teXed talk notes. The quality of the teXed notes is usually bad, as I didn't intend to publish them. This is a compilation of talk notes from the last academic year.

Goal for the next year: focus :-)

• Seminar on Systolic Geometry: Aspherical Manifolds and Essential Manifolds.
• Seminar on Perverse Sheaves: Introduction to 6 functor formalism. I also blogged some stuff even more elementary about 4 functors before.
• Seminar on Geometric Langlands: From Langlands for Function Fields to Geometric Langlands; I have blogged an introduction and also some of the talk on Geometric Langlands here.
• Seminar on Cobordism: Bordism, Cobordism and Spectra (there's a blog post). There are also slides on the cobordism spectrum, though I remember it was definitely on a blackboard.
• Seminar on Mapping Class Groups: Spectra and Infinite Loop Spaces (these notes are actually not that bad).
• Seminar on Galois Representations: Grothendieck's Monodromy Theorem. I have written a small note on supernatural numbers, which appear in Tate's proof. There are also (bad, uncorrected) talk notes on Weil-Deligne representations in german.
• Seminar on Multiple Zeta Values: Upper bounds on periods (link goes to full lecture notes, which are quite good now).
• Student's conference of the german math society: 20 minutes on A¹-fundamental groups of isotropic groups.

You might know the theorem of Lagrange from group theory: A finite group of order n can have a subgroup of order m only if m divides n. Here the order is just the number of elements, a natural number.

Recently I came across a generalization to profinite groups. How do you make sense of the order of an infinite group? How to say that the order of a subgroup divides the order of the group? The solution is a simple concept called supernatural numbers, which I will explain in this short article. The main part should in principle be accessible to non-mathematicians as well.

Continue reading «Lagrange and the Supernaturals»

You probably know the integers $\mathbb{Z}$ which can be defined as the Grothendieck group of the monoid of natural numbers $\mathbb{N}$ which exists by the axioms of Zermelo-Fraenkel set theory and probably in almost all other axiom systems as well. So the integers are a quite fundamental object in mathematics (did I really just argue for that? Well, now I did).

It is also natural to look what one can derive from the integers. In this short article I want to describe some rings that can be obtained from the integers by quotients, subgroups, products, and in general (co)limits and combinations thereof. The purpose might not be so clear, but it was a by-product of other investigations and I hope it could be interesting to say something about some concrete rings.

Continue reading «Some Interesting Rings»