
In several of his talks on #IUTeich, Mochizuki argues that usual scheme theory over $\mathbb{Z}$ is not suited to tackle problems such as the ABCconjecture. The idea appears to be that ABC involves both the additive and multiplicative nature of integers, making rings into ‘2dimensional objects’ (and clearly we use both ‘dimensions’ in the theory… Read more »

Some weeks ago, Robert Kucharczyk and Peter Scholze found a topological realisation of the ‘hopeless’ part of the absolute Galois group $\mathbf{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. That is, they constructed a compact connected space $M_{cyc}$ such that etale covers of it correspond to Galois extensions of the cyclotomic field $\mathbb{Q}_{cyc}$. This gives, at least in theory, a handle on… Read more »

We know embarrassingly little about the symmetries of the roots of all polynomials with rational coefficients, or if you prefer, the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. In the title picture the roots of polynomials of degree $\leq 4$ with small coefficients are plotted and coloured by degree: blue=4, cyan=3, red=2, green=1. Sums and products of roots… Read more »

absolute, geometry, number theory, stories
The Log Lady and the Frobenioid of $\mathbb{Z}$
Posted on by lievenlb“Sometimes ideas, like men, jump up and say ‘hello’. They introduce themselves, these ideas, with words. Are they words? These ideas speak so strangely.” “All that we see in this world is based on someone’s ideas. Some ideas are destructive, some are constructive. Some ideas can arrive in the form of a dream. I can… Read more »

Absolute geometry is the attempt to develop algebraic geometry over the elusive field with one element $\mathbb{F}_1$. The idea being that the set of all prime numbers is just too large for $\mathbf{Spec}(\mathbb{Z})$ to be a terminal object (as it is in the category of schemes). So, one wants to view $\mathbf{Spec}(\mathbb{Z})$ as a geometric… Read more »

absolute, math, number theory
Quiver Grassmannians and $\mathbb{F}_1$geometry
Posted on by lievenlbReineke’s observation that any projective variety can be realized as a quiver Grassmannian is bad news: we will have to look at special representations and/or dimension vectors if we want the Grassmannian to have desirable properties. Some people still see a silver lining: it can be used to define a larger class of geometric objects… Read more »

Almost three decades ago, Yuri Manin submitted the paper “New dimensions in geometry” to the 25th Arbeitstagung, Bonn 1984. It is published in its proceedings, Springer Lecture Notes in Mathematics 1111, 59101 and there’s a review of the paper available online in the Bulletin of the AMS written by Daniel Burns. In the introduction Manin… Read more »

Here are the scans of my crude prepnotes for some of the later seminartalks. These notes still contain mistakes, most of them were corrected during the talks. So, please, read these notes with both mercy are caution! Hurwitz formula imples ABC : The proof of Smirnov’s argument, but modified so that one doesn’t require an… Read more »

October 21st : Dear Diary, today’s seminar was fun, though a bit unconventional. The intention was to explain faithfully flat descent, but at the last moment i had the crazy idea to let students discover the main idea themselves (in the easiest of examples) by means of this thought experiment : “I am an alien,… Read more »

We’ve had three seminarsessions so far, and the seminarblog ‘angs+’ contains already 20 posts and counting. As blogging is not a linear activity, I will try to post here at regular intervals to report on the ground we’ve covered in the seminar, providing links to the original angs+ posts. This year’s goal is to obtain… Read more »

F_un Mathematics Hardly a ‘new’ blog, but one that is getting a new life! On its old homepage you’ll find a diagonal banner stating ‘This site has moved’ and clicking on it will guide you to its new location : cage.ugent.be/~kthas/Fun. From now on, this site will be hosted at the University of Ghent and… Read more »

Note to students following this year’s ‘seminar noncommutative geometry’ : the seminar starts friday september 30th at 13h in room G 0.16. However, if you have another good reason to be in Ghent on thursday september 22nd, consider attending the inaugural lecture of Koen Thas at 17h in auditorium Emmy Noether, campus De Sterre, Krijgslaan… Read more »

In preparing for next year’s ‘seminar noncommutative geometry’ I’ve converted about 30 posts to LaTeX, centering loosely around the topics students have asked me to cover : noncommutative geometry, the absolute point (aka the field with one element), and their relation to the Riemann hypothesis. The idea being to edit these posts thoroughly, add much… Read more »

Last time we did recall Manin’s comparisons between some approaches to geometry over the absolute point $\pmb{spec}(\mathbb{F}_1)$ and trends in the history of art. In the comments to that post, Javier LopezPena wrote that he and Oliver Lorscheid briefly contemplated the idea of extending Manin’s artsydictionary to all approaches they did draw on their Map… Read more »

In his paper Cyclotomy and analytic geometry over $\mathbb{F}_1$ Yuri I. Manin sketches and compares four approaches to the definition of a geometry over $\mathbb{F}_1$, the elusive field with one element. He writes : “Preparing a colloquium talk in Paris, I have succumbed to the temptation to associate them with some dominant trends in the… Read more »

The odd Knight of the round table problem asks for a consistent placement of the nth Knight in the row at an odd root of unity, compatible with the two different realizations of the algebraic closure of the field with two elements. The first identifies the multiplicative group of its nonzero elements with the group… Read more »

In 1956, Alexander Grothendieck (middle) introduced $\lambda $rings in an algebraicgeometric context to be commutative rings A equipped with a bunch of operations $\lambda^i $ (for all numbers $i \in \mathbb{N}_+ $) satisfying a list of rather obscure identities. From the easier ones, such as $\lambda^0(x)=1, \lambda^1(x)=x, \lambda^n(x+y) = \sum_i \lambda^i(x) \lambda^{ni}(y) $ to those… Read more »

Next time you visit your mathlibrary, please have a look whether these books are still on the shelves : Michiel Hazewinkel‘s Formal groups and applications, William Fulton’s and Serge Lange’s RiemannRoch algebra and Donald Knutson’s lambdarings and the representation theory of the symmetric group. I wouldn’t be surprised if one or more of these books… Read more »

Today, Alain Connes and Caterina Consani arXived their new paper Schemes over $ \mathbb{F}_1$ and zeta functions. It is a followup to their paper On the notion of geometry over $ \mathbb{F}_1$, which I’ve tried to explain in a series of posts starting here. As Javier noted already last week when they updated their first… Read more »

The algebraic fundamental group of a scheme gives the MazurKapranovReznikov dictionary between primes in number fields and knots in 3manifolds.

Manin proposes the idea of projecting spec(Z[x]) not only onto spec(Z), but also to a geometric axis by considering the integers as an algebra over the field with one element.

We propose to extend the ConnesConsani definition to noncommuntative F_un varieties.

Some links to posts on Soule’s algebraic geometry over the field with one element.

At the MaxPlanck Institute in Bonn Yuri Manin gave a talk about the field of one element, $\mathbb{F}_1 $ earlier this week entitled “Algebraic and analytic geometry over the field F_1”. Moreover, Javier LopezPena and Bram Mesland will organize a weekly “F_un Study Seminar” starting next tuesday. Over at Noncommutative Geometry there is an Update… Read more »

A couple of weeks ago, Alain Connes and Katia Consani arXived their paper “On the notion of geometry over $\mathbb{F}_1 $”. Their subtle definition is phrased entirely in Grothendieck‘s schemetheoretic language of representable functors and may be somewhat hard to get through if you only had a few years of mathematics. I’ll try to give… Read more »
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