One of the more surprising analogies around is that prime numbers can be viewed as knots in the 3-sphere $S^3$. The motivation behind it is that the (etale) fundamental group of $\pmb{spec}(\mathbb{Z}/(p))$ is equal to (the completion) of the fundamental group of a circle $S^1$ and that the embedding $\pmb{spec}(\mathbb{Z}/(p)) \subset \pmb{spec}(\mathbb{Z})$ embeds this circle as a knot in a… Read more →

# Tag Archive for Kapranov

# Mazur’s knotty dictionary

The algebraic fundamental group of a scheme gives the Mazur-Kapranov-Reznikov dictionary between primes in number fields and knots in 3-manifolds. Read more →

# Andre Weil on the Riemann hypothesis

Some quotes of Andre Weil on the Riemann hypothesis. Read more →

# F_un and braid groups

Recall that an n-braid consists of n strictly descending elastic strings connecting n inputs at the top (named 1,2,…,n) to n outputs at the bottom (labeled 1,2,…,n) upto isotopy (meaning that we may pull and rearrange the strings in any way possible within 3-dimensional space). We can always change the braid slightly such that we can divide the interval between… Read more →

# Absolute linear algebra

Today we will define some basic linear algebra over the absolute fields $\mathbb{F}_{1^n} $ following the Kapranov-Smirnov document. Recall from last time that $\mathbb{F}_{1^n} = \mu_n^{\bullet} $ and that a d-dimensional vectorspace over this field is a pointed set $V^{\bullet} $ where $V $ is a free $\mu_n $-set consisting of n.d elements. Note that in absolute linear algebra we… Read more →

# The F_un folklore

All esoteric subjects have their own secret (sacred) texts. If you opened the Da Vinci Code (or even better, the original The Holy blood and the Holy grail) you will known about a mysterious collection of documents, known as the “Dossiers secrets“, deposited in the Bibliothèque nationale de France on 27 April 1967, which is rumoured to contain the mysteries… Read more →

# noncommutative Fourier transform

At the noncommutative algebra program in MSRI 1999/2000, Mikhail Kapranov gave an intriguing talk Noncommutative neighborhoods and noncommutative Fourier transform and over the years I’ve watched the video of this talk a number of times. The first part of the talk is about his work on Noncommutative geometry based on commutator expansions and as I’ve once worked through it this… Read more →

# nog course outline

Now that the preparation for my undergraduate courses in the first semester is more or less finished, I can begin to think about the courses I’ll give this year in the master class non-commutative geometry. For a change I’d like to introduce the main ideas and concepts by a very concrete example : Ginzburg’s coadjoint-orbit result for the Calogero-Moser space… Read more →