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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If you don't have a solid education in linear algebraic groups, you might nevertheless encounter the term "reductive groups" now and then. People keep telling you to think about $GL_n$, as an algebraic group, and that's a good first approximation. If you want to go a step further, some confusion can happen, since the definition of reductive groups can be given for groups over the complex numbers in two quite different ways and only one of them generalizes to reductive groups over other fields, and if one wants to do non-perfect fields, it gets even more complicated.

I want to give some very short explanations on how to think about reductive groups and semisimple algebraic groups.

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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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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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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 ...»

Hi, I was reading in

Jet Nestruev: Smooth Manifolds and Observables, Springer, 2003

about a month ago (after I stumbled over a question on MO) and there was an exercise that resisted solution for more than a week.

Well.... now I found out that I have just misread the exercise. However, this way I basically did several exercises at once. Here comes the problem and its solution:

The problem

(inspired by page 28, chapter 3, exercise 3.17.5 in Nestruev)

Find a smooth (real) manifold $M$ such that its algebra of smooth functions $C^\infty(M,\mathbb R)$ is isomorphic to the algebra of all smooth functions $f : \mathbb R \to \mathbb R$ that have some rational period $\tau$ (i.e. there exists $\tau \in \mathbb Q$ such that $f(x)=f(x+\tau)$ for all x). Note that we don't fix a period $\tau$ here. Let's call the algebra in question (smooth functions on the real line with some rational period) $A$.

You might want to stop reading here and think for a second (or minutes) about the solution or similar problems that have easier solutions. A more vague problem would be

Find a space $M$ such that the functions $M \to \mathbb R$ correspond to functions $\mathbb R \to \mathbb R$ that are periodic with some rational period.

Continue reading «A manifold whose functions are the smooth functions on the real line with rational period»