All these fundamental groups!

Tuesday, April 02nd, 2013 | Author:

There are a lot of fundamental groups floating around in mathematics. This is an attempt to collect some of the most popular and sketch their relations to each other.

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Lagrange and the Supernaturals

Friday, August 03rd, 2012 | Author:

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.

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Some Interesting Rings

Monday, July 30th, 2012 | Author:

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.

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Zeta of x²+y²=1 over finite fields

Wednesday, July 18th, 2012 | Author:

I must admit that I have a weak background in elementary number theory. A few weeks ago I wanted to count the number of points of the affine circle x^2+y^2=1 over finite fields, by counting the point of the completed curve and the points at infinity. There one needs to decide whether a finite field with p^k elements contains roots of -1 or not.

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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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