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Tag: Manin

Dessinflateurs

I’m trying to get into the latest Manin-Marcolli paper Quantum Statistical Mechanics of the Absolute Galois Group on how to create from Grothendieck’s dessins d’enfant a quantum system, generalising the Bost-Connes system to the non-Abelian part of the absolute Galois group Gal(Q/Q).

In doing so they want to extend the action of the multiplicative monoid N× by power maps on the roots of unity to the action of a larger monoid on all dessins d’enfants.

Here they use an idea, originally due to Jordan Ellenberg, worked out by Melanie Wood in her paper Belyi-extending maps and the Galois action on dessins d’enfants.



To grasp this, it’s best to remember what dessins have to do with Belyi maps, which are maps defined over Q
π:ΣP1
from a Riemann surface Σ to the complex projective line (aka the 2-sphere), ramified only in 0,1 and . The dessin determining π is the 2-coloured graph on the surface Σ with as black vertices the pre-images of 0, white vertices the pre-images of 1 and these vertices are joined by the lifts of the closed interval [0,1], so the number of edges is equal to the degree d of the map.

Wood considers a very special subclass of these maps, which she calls Belyi-extender maps, of the form
γ:P1P1
defined over Q with the additional property that γ maps {0,1,} into {0,1,}.

The upshot being that post-compositions of Belyi’s with Belyi-extenders γπ are again Belyi maps, and if two Belyi’s π and π lie in the same Galois orbit, then so must all γπ and γπ.

The crucial Ellenberg-Wood idea is then to construct “new Galois invariants” of dessins by checking existing and easily computable Galois invariants on the dessins of the Belyi’s γπ.

For this we need to know how to draw the dessin of γπ on Σ if we know the dessins of π and of the Belyi-extender γ. Here’s the procedure



Here, the middle dessin is that of the Belyi-extender γ (which in this case is the power map tt4) and the upper graph is the unmarked dessin of π.

One has to replace each of the black-white edges in the dessin of π by the dessin of the expander γ, but one must be very careful in respecting the orientations on the two dessins. In the upper picture just one edge is replaced and one has to do this for all edges in a compatible manner.

Thus, a Belyi-expander γ inflates the dessin π with factor the degree of γ. For this reason i prefer to call them dessinflateurs, a contraction of dessin+inflator.

In her paper, Melanie Wood says she can separate dessins for which all known Galois invariants were the same, such as these two dessins,



by inflating them with a suitable Belyi-extender and computing the monodromy group of the inflated dessin.

This monodromy group is the permutation group generated by two elements, the first one gives the permutation on the edges given by walking counter-clockwise around all black vertices, the second by walking around all white vertices.

For example, by labelling the edges of Δ, its monodromy is generated by the permutations (2,3,5,4)(1,6)(8,10,9) and (1,3,2)(4,7,5,8)(9,10) and GAP tells us that the order of this group is 1814400. For Ω the generating permutations are (1,2)(3,6,4,7)(8,9,10) and (1,2,4,3)(5,6)(7,9,8), giving an isomorphic group.

Let’s inflate these dessins using the Belyi-extender γ(t)=274(t3t2) with corresponding dessin



It took me a couple of attempts before I got the inflated dessins correct (as i knew from Wood that this simple extender would not separate the dessins). Inflated Ω on top:



Both dessins give a monodromy group of order 35838544379904000000.

Now we’re ready to do serious work.

Melanie Wood uses in her paper the extender ζ(t)=27t2(t1)24(t2t+1)3 with associated dessin



and says she can now separate the inflated dessins by the order of their monodromy groups. She gets for the inflated Δ the order 19752284160000 and for inflated Ω the order 214066877211724763979841536000000000000.

It’s very easy to make mistakes in these computations, so probably I did something horribly wrong but I get for both Δ and Ω that the order of the monodromy group of the inflated dessin is 214066877211724763979841536000000000000.

I’d be very happy when someone would be able to spot the error!

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Two lecture series on absolute geometry

Absolute geometry is the attempt to develop algebraic geometry over the elusive field with one element F1. The idea being that the set of all prime numbers is just too large for Spec(Z) to be a terminal object (as it is in the category of schemes).

So, one wants to view Spec(Z) as a geometric object over something ‘deeper’, the “absolute point” Spec(F1).

Starting with the paper by Bertrand Toen and Michel Vaquie, Under Spec(Z), topos theory entered this topic.

First there was the proposal by Jim Borger to view λ-rings as F1-algebras. More recently, Alain Connes and Katia Consani introduced the arithmetic site.

Now, there are lectures series on these two approaches, one by Yuri I. Manin, the other by Alain Connes.

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Yuri I. Manin in Ghent

On Tuesday, February 3rd, Yuri I. Manin will give the inaugural lectures of the new F1-seminars at Ghent University, organised by Koen Thas.

Coffee will be served from 13.00 till 14.00 at the Department of Mathematics, Ghent University, Krijgslaan 281, Building S22 and from 14.00 till 16.30 there will be lectures in the Emmy Noether lecture room, Building S25:

14:00 – 14:25: Introduction (by K. Thas)
14:30 – 15:20: Lecture 1 (by Yu. I. Manin)
15:30 – 16:20: Lecture 2 (by Yu. I. Manin)

Recent work of Manin related to F1 includes:

Local zeta factors and geometries under Spec(Z)

Numbers as functions

Alain Connes on the Arithmetic Site

Until the beginning of march, Alain Connes will lecture every thursday afternoon from 14.00 till 17.30, in Salle 5 – Marcelin Berthelot at he College de France on The Arithmetic Site (hat tip Isar Stubbe).

Here’s a two minute excerpt, from a longer interview with Connes, on the arithmetic site, together with an attempt to provide subtitles:

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(50.36)

And,in this example, we saw the wonderful notion of a topos, developed by Grothendieck.

It was sufficient for me to open SGA4, a book written at the beginning of the 60ties or the late fifties.

It was sufficient for me to open SGA4 to see that all the things that I needed were there, say, how to construct a cohomology on this site, how to develop things, how to see that the category of sheaves of Abelian groups is an Abelian category, having sufficient injective objects, and so on … all those things were there.

This is really remarkable, because what does it mean?

It means that the average mathematician says: “topos = a generalised topological space and I will never need to use such things. Well, there is the etale cohomology and I can use it to make sense of simply connected spaces and, bon, there’s the chrystaline cohomology, which is already a bit more complicated, but I will never need it, so I can safely ignore it.”

And (s)he puts the notion of a topos in a certain category of things which are generalisations of things, developed only to be generalisations…

But in fact, reality is completely different!

In our work with Katia Consani we saw not only that there is this epicyclic topos, but in fact, this epicyclic topos lies over a site, which we call the arithmetic site, which itself is of a delirious simplicity.

It relies only on the natural numbers, viewed multiplicatively.

That is, one takes a small category consisting of just one object, having this monoid as its endomorphisms, and one considers the corresponding topos.

This appears well … infantile, but nevertheless, this object conceils many wonderful things.

And we would have never discovered those things, if we hadn’t had the general notion of what a topos is, of what a point of a topos is, in terms of flat functors, etc. etc.

(52.27)

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I will try to report here on Manin’s lectures in Ghent. If someone is able to attend Connes’ lectures in Paris, I’d love to receive updates!

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Manin’s three-space-2000

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, 59-101 and there’s a review of the paper available online in the Bulletin of the AMS written by Daniel Burns.

In the introduction Manin makes some highly speculative but inspiring conjectures. He considers the ring

Z[x1,,xm;ξ1,,ξn]

where Z are the integers, the ξi are the “odd” variables anti-commuting among themselves and commuting with the “even” variables xj. To this ring, Manin wants to associate a geometric object of dimension 1+m+n where 1 refers to the “arithmetic dimension”, m to the ordinary geometric dimensions (x1,,xm) and n to the new “odd dimensions” represented by the coordinates (ξ1,,ξn). Manin writes :

“Before the advent of ringed spaces in the fifties it would have been difficult to say precisely what me mean when we speak about this geometric object. Nowadays we simply define it as an “affine superscheme”, an object of the category of topological spaces locally ringed by a sheaf of Z2-graded supercommutative rings.”

Here’s my own image (based on Mumford’s depiction of Spec(Z[x])) of what Manin calls the three-space-2000, whose plain x-axis is supplemented by the set of primes and by the “black arrow”, corresponding to the odd dimension.

Manin speculates : “The message of the picture is intended to be the following metaphysics underlying certain recent developments in geometry: all three types of geometric dimensions are on an equal footing”.

Probably, by the addition “2000” Manin meant that by the year 2000 we would as easily switch between these three types of dimensions as we were able to draw arithmetic schemes in the mid-80ties. Quod non.

Twelve years into the new millenium we are only able to decode fragments of this. We know that symmetric algebras and exterior algebras (that is the “even” versus the “odd” dimensions) are related by Koszul duality, and that the precise relationship between the arithmetic axis and the geometric axis is the holy grail of geometry over the field with one element.

For aficionados of F1 there’s this gem by Manin to contemplate :

“Does there exist a group, mixing the arithmetic dimension with the (even) geometric ones?”

Way back in 1984 Manin conjectured : “There is no such group naively, but a ‘category of representations of this group’ may well exist. There may exist also certain correspondence rings (or their representations) between Spec(Z) and x.”

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