# What is ... a vector bundle?

Thursday, November 01st, 2012 | Author:

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.

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# From the Langlands Correspondence for Function Fields to the Geometric Langlands Correspondence II

Tuesday, June 05th, 2012 | Author:

This post is the continuation of "From the Langlands Correspondence for Function Fields to the Geometric Langlands Correspondence I", and explains how to translate the Langlands Correspondence for function fields to a geometric question.

This post grew out of the preparation for a seminar talk on this topic and is separated in two parts, this being the second, and last part.

To repeat briefly, the Langlands Correspondence for a function field $F = \mathbb{F}_q(X)$ of a smooth projective curve $X$ states that certain n-dimensional irreducible l-adic Galois representations correspond (1:1) to irreducible cuspidal automorphic representations of $GL_n(\mathbb{A}_F)$. Furthermore, the L-functions of Galois representations and automorphic representations coincide.

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# From the Langlands Correspondence for Function Fields to the Geometric Langlands Correspondence I

Saturday, May 26th, 2012 | Author:

The (conjectural) Langlands correspondence for number fields gave rise to a Langlands correspondence for function fields (proved by Drinfeld and Lafforgue), where the most important difference is the absence of the infinite place which simplifies things in the latter. This, in turn, can be translated to a "geometric" Langlands correspondence for curves over fields, but there are certain differences.

First, I'm going to explain what the Langlands correspondence for function fields says, with the assumption in mind that you have been exposed to some algebra before. After that, I'm going to sketch how to go to the geometric Langlands correspondence, following Frenkel's storytelling in chapter 3 of his article "Langlands and conformal field theory". Of course, this being a blogpost, I won't repeat what Frenkel says (nor delve deeper) but try to summarise, to give an overview.

This post grew out of the preparation for a seminar talk on this topic and is separated in two parts, this being the first part. The continuation is here, discussing the geometrization.

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# Mindmap on complex analysis in one variable

Monday, November 14th, 2011 | Author:

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:

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# Classifying Riemann surfaces

Wednesday, October 21st, 2009 | Author:

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.

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?