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G+ recovery 4 : #IUTeich (2)

Here’s batch 2 of my old google+ posts on ‘Inter Universal Teichmuller theory’, or rather on the number theoretic examples of Frobenioids.

June 5th, 2013

Mochizuki’s categorical prime number sieve

And now for the interesting part of Frobenioids1: after replacing a bunch of arithmetic schemes and maps between them by a huge category, we will reconstruct this classical picture by purely categorical means.

Let’s start with the simplest case, that of the ‘baby arithmetic Frobenioid’ dismantling $\mathbf{Spec}(\mathbb{Z}) (that is, the collection of all prime numbers) and replacing it by the category having as its objects

all $(a)$ where $a$ is a strictly positive rational number

and morphisms labeled by triples $(n,r,z)$ where $n$ and $z$ are strictly positive integers and $r$ is a strictly positive rational number and connecting two objects

\[
(n,r,z) : (a) \rightarrow (b) \quad \text{ if and only if } \quad a^n.z=b.r \]

Composition of morphisms is well-defined and looks like $(m,s,v) \circ (n,r,u) = (m.n,r^m.s,u^m.v)$ as one quickly checks.

The challenge is to recover all prime numbers back from this ‘Frobenioid’. We would like to take an object $(a)$ and consider the maps $(1,1,p)$ from it for all prime numbers $p$, but cannot do this as categorically we have to drop all labels of objects and arrows. That is, we have to recognize the map $(1,1,p)$ among all maps starting from a given object.

We can identify all isomorphisms in the category and check that they are precisely the morphisms labeled $(1,r,1)$. In particular, this implies that all objects are isomorphic and that there is a natural correspondence between arrows leaving $(a)$ and arrows leaving $(b)$ by composing them with the iso $(1,b/a,1) : (b) \rightarrow (a)$.

Another class of arrows we can spot categorically are the ‘irreducibles’, which are maps $f$ which are not isos but have the property that in any factorization $f=g \circ h$ either $g$ or $h$ must be an iso. One easily verifies from the composition rule that these come in two flavours:

– those of Frobenius type : $(p,r,1)$ for any prime number $p$
– those of Order type : $(1,r,p)$ for any prime number $p$

We would like to color the froBs Blue and the oRders Red, but there seems to be no way to differentiate between the two classes by purely categorical means, until you spot Mochizuki’s clever little trick.

start with a Red say $(1,r,q)$ for a prime number $q$ and compose it with the Blue $(p,1,1)$, then you get the morphism $(p,r^p,q^p)$ which you can factor as a composition of $p+1$ irreducibles

\[
(p,r^p,q^p) = (1,r,q) \circ (1,r,q) \circ …. \circ (1,r,q)o(p,1,1) \]

and if $p$ grows, so will the number of factors in this composition.

On the other hand, if you start with a Blue and compose it with either a Red or a Blue irreducible, the obtained map cannot be factored in more irreducibles.

Thus, we can identify the Order-type morphisms as those irreducibles $f$ for which there exists an irreducible $g$ such that the composition $g \circ f$ can be factored as the composition in at least $n$ irreducibles, where we can take n arbitrarily large.

Finally we say that two Reds out of $(a)$ are equivalent iff one is obtained from the other by composing with an isomorphism and it is clear that the equivalence classes are exactly the arrows labeled $(1,r,p)$ for fixed prime number $p$.

So we do indeed recover all prime numbers from the category.

Similarly, we can see that equivalence classes of Frobs from $(a)$ are of the form $(p,r,1)$ for fixed prime $p$. An amusing fact is that we can recover the prime $p$ for a Frob by purely categorical ways using the above long factorization of a composition with a Red.

There seems to be no categorical way to determine the prime number associated to an equivalence class of Order-morphisms though… Or, am i missing something trivial?

June 7th, 2013

Mochizuki’s Frobenioids for the Working Category Theorist

Many of you, including +David Roberts  +Charles Wells  +John Baez  (and possibly others, i didn’t look at all comments left on all reshares of the past 4 posts in this MinuteMochizuki project) hoped that there might be a more elegant category theoretic description of Frobenioids, the buzz-word apparently being ‘Grothendieck fibration’ …

Hence this attempt to deconstruct Frobenioids. Two caveats though:

– i am not a category theorist (the few who know me IRL are by now ROFL)
– these categories are meant to include all arithmetic information of number fields, which is a messy business, so one should only expect clear cut fibrations in easy situation such as principal ideal domains (think of the integers $\mathbb{Z}$).

Okay, we will try to construct the Frobenioid associated to a number field $K$ (that is, a finite dimensional extension of the rationals $\mathbb{Q}$) with ring of integers $R$ (the integral closure of $\mathbb{Z}$ in $K$). For a concrete situation, look at the quadratic case.

The objects will be fractional ideals of K which are just the R-submodules $I$ of $K$ such that there in an $r$ in $R$ such that $I.r$ is a proper ideal of $R$. Dedekind showed that any such thing can be written uniquely as a product

\[
I = P_1^{a_1} … P_k^{a_k} \]

where the $P_i$ are prime ideals of $R$ and the $a_i$ are integers (if they are all natural numbers, I will be a proper ideal of $R$). Clearly, if one multiplies two fractional ideals $I$ and $J$, the result $I.J$ is again a fractional ideal, so they form a group and by Dedekind’s trick this group is the free Abelian group on all prime ideals of $R$.

Next, we define an equivalence relation on this set, calling two fractional ideals $I$ and $J$ equivalent if there is a $k$ in $K$ such that $I=J.k$ (or if you prefer, if they are isomorphic as $R$-modules).

We have a set with an equivalence relation and hence a groupoid where these is a unique isomorphism between any two equivalent objects. This groupoid is precisely the groupoid of isomorphisms of the Frobenioid we’re after.

The number of equivalence classes is finite and these classes correspond to the element of a finite group $Cl(R)$ called the class group of $R$ which is the quotient group of ideals modulo principal ideals (so if your $R$ is a principal ideal domain there is just one component). The ‘groups’ corresponding to each connected component of the groupoid are all isomorphic to the quotient group of the units in $K$ by the units in $R$.

Next, we will add the other morphisms. By definition they are all compositions of irreducibles which come in 2 flavours:

– the order-morphisms $P$ for any prime ideal $P$ of $R$ sending $I$ to $I.P$. Typically, these maps switch between different equivalence classes (unless $P$ itself is principal). We can even explicitly compute small norm prime ideals which will generate all elements in the class group $Cl(R)$.
– the power-maps $[p]$ for any prime number $p$ which sends $I$ to $I^p$. The nature of these maps really depend on the order of the component in the finite group $Cl(R)$.

Well, that’s it basically for the layer of the Frobenioid corresponding to the number field $K$. (You have to repeat all this for any subfield between $\mathbb{Q}$ and $K$). A cute fact is that all endomorphism-monoids of objects in the layer of K are all isomorphic as abstract monoid to the skew-monoid

$\mathbb{N}^x_{>0} x Prin(R)$ of the multiplicative group of all strictly positive integers $n$ with the monoid of all principal ideals in R with multiplication defined by

\[
(n,Ra).(m,Rb)=(nm,Rab^n) \]

The only extra-type morphisms we still have to include are those between the different layers of the Frobenioid, the green ones which M calls the pull-back morphisms.
They are of the following form: if $R_1$ and $R_2$ are rings of integers in the fields $K_1$ contained in $K_2$, then for any ringmorphism $\sigma : R_1 \rightarrow R_2$ one can extend a fractional ideal $I$ of $R_1$ to $K_2$ by considering $R_2.\sigma(I)$. These then give the morphism $r_2\sigma(I) \rightarrow I$ and as we will see in a next instalment, they encode the splitting behaviour of prime ideals.

a question for category people

Take the simplest situation, that of the integers $\mathbb{Z}$. So, we have just a groupoid with extra morphisms generated by the order-maps $o_p$ and the power maps $f_p$. The endo-ring of any object is then isomorphic top the abstract group generated by the $f_p$ and $o_p$ and satisfying following relations

\[
o_p.o_q=o_q.o_p \]
\[
f_p.f_q=f_q.f_p \]
\[
f_p.o_q=o_q^p.f_p \]

My question now is: if for two different primes $p$ and $q$ i switch their role in the endo-ring of 1 object and propagate this via all isos to all morphisms, do i get a category equivalence? (or am i missing something?).  (tbc)

June 11th, 2013

my problem with Mochizuki’s Frobenioid1

Let us see how much arithmetic information can be reconstructed from an arithmetic Frobenioids. Recall that for a fixed finite Galois extension $L$ of $\mathbb{Q}$ this is a category with objects all fractional ideals in subfields of $L$, and maps generated by multiplication-maps with ideals in rings of integers, power-maps and Galois-extension maps.

When all objects and morphisms are labeled it is quite easy to reconstruct the Galois field $L$ from it as well as all maps between prime spectra of rings of integers in intermediate fields, which after all was the intended use of Frobenioids, to ‘dismantle’ these arithmetic schemes and endow them with extra structure given by the power-maps.

However, in this reconstruction process we are only allowed t use the category structure, so all objects and morphisms are unlabelled (the situation top left) and we want to reconstruct from it the different layers of the Frobenioid (corresponding to the different subfields) and divide all arrows according to their type (situation bottom left).

First we can look at all isomorphisms. They will divide the category in the dashed regions, some of them will be an entire layer (for example for $\mathbb{Q}$) but in general a finite number of these regions will make up the full layer of a subfield (the regions labeled by the elements of the ideal class group).

Another categorical notion we can use are ‘irreducible morphisms’, that is a morphism $f$ which is not an iso but having the property that in each factorisation $f = g \circ h$ either $g$ or $h$ must be an iso. If we remember the different types of morphisms in our Frobenioid we see that the irreducibles come in 3 flavours:

– oRder-maps (Red) : multiplication by a prime ideal $P$ of the ring of integers of the subfield
– froBenius or power-maps (Blue) sendingg a fractional ideal $I$ to $I^p$ for a prime number $p$
– Galois-maps (Green) extending ideals for a subfield $K$ to $K’$ having no intermediate field.

We would like to determine the colour of these irreducibles purely categorical. The idea is that reds have the property that they can be composed with another irreducible (in fact, of power type) such that the composition can again be decomposed in irreducibles and that there is no a priori bound on the number of these terms (this uses the fact that $[p] \circ Q = Q \circ Q \circ … Q \circ [p]$ and that there are infinitely many prime numbers $p$). One checks that compositions of order or Galois maps with irreducibles have factorisation with a bounded number of irreducibles.

The most interesting case is the composition of a Galois map with an order map $P$, this can be decomposed alternatively as order maps in the bigger field followed by a Galois map, the required order maps are the bigger primes $Q_i$ occurring in the decomposition of the extended ideal

\[
S.P = Q_1.Q_2….Q_k \]

but the number $k$ of this decomposition is bounded by the dimension of the bigger field over the smaller one. Summarizing:

– we can determine all the red maps, which will then give us also the different layers
– we can determine the green maps as they move between different layers
– to the remaining blue ones we can even associate their label $[p]$ by the observed property of composition with order maps.

Taking an object in a layer, we get the set of prime ideals of the ring on integers in that field as the set of all red arrows leaving that object unto equivalence (by composing with an isomorphism), so we get the prime spectra $\mathbf{Spec}(S)$.

For a ring-extension  $R \rightarrow S$ we also can recover the cover map $\mathbf{Spec}(S) \rightarrow \mathbf{Spec}(R)$

Indeed, composing the composition of the Galois map with the order-map $P$ and decomposing it alternatively will give us the finite number of prime ideals $Q_i$ of $S$ lying over $P$. That is, we get all splitting behaviour of prime ideals in intermediate field-extensions.

Let $K$ be a Galois subfield of $L$ then we have a way to see how a prime ideals in $\mathbf{Spec}(\mathbb{Z})$ splits, ramifies or remains inert in $K$ and so by Chebotarev density this gives us the dimension of $K$ as well as the Galois group. And, if we could label the prime ideal by a prime number $p$, we could even reconstruct $K$ itself as $K$ is determined once we know all prime numbers which completely split.

The problem i have is that i do not see a categorical way to label the red arrows in $Frob(\mathbb{Z})$ by prime numbers. Mochizuki says we can do this in the proof of Thm 6.4(iii) by using the fact that the $log(p)$ are linearly independent over $\mathbb{Q}$.

This suggests that one might use the ‘Arakelov information’ (that is the archimidean valuations) to do this (the bit i left out so far), but i do not see this in the case of $\mathbb{Q}$ as there is just 1 extra (real) valuation determined by the values of the nonarchimidean valuations.

Probably i am missing something so all sorts of enlightenment re welcome!

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G+ recovery 3 : #IUTeich (1)

Perhaps the ‘hottest’ topic I got involved with on Google+ was Mochizuki’s (claimed) proof of the ABC-conjecture.

Rereading these posts in chronological order shows my changing attitude to this topic, from early skepticism, over attempts to understand at least one pre-IUTeich paper (Frobenioids 1) to a level of belief, to … resignation.

Here’s the first batch of #IUTeich posts (here IUTeich stands for Mochizuki’s ‘Inter Universal Teichmuller theory).

June 12th, 2012

Linked post: ABC conjecture rumor at the Secret Blogging Seminar.

My understanding is that Google+ is for spreading (or debunking) such rumours. Sure would love to have more details!

May 26th, 2013

The Mochizuki craze continues…
Now Ted Nelson claims Mochizuki must be none other than the mysterious Satoshi Nakamoto, inventor of BitCoin.
I’ve seen BitCoin explained in 60 second on YouTube. It would be great if someone did something similar and try to explain inter-universal Teichmuller theory in 60 seconds…

May 27th, 2013

in IUTeich the theta function corresponds to the gaze of the little girl into the “small house”

Yesterday i was hoping for a 60 second introduction to inter-universal Teichmuller theory. Today i learned that Mochizuki himself provided such a thing.

Well, sort of…

In a one-page post he gives an ‘explanation’ of IUTeich via Sokkuri-Animation (Sokkuri being the Japanese for mirror-image). 

IUTeich, he says, should be viewed as a sequence of nested universes, which are represented by ‘houses’ in Sokkuri-Animation.
Galois groups and arithmetic fundamental groups behave as though they are made of a “mysterious substance” and in Sokkuri this “mysterious substance” corresponds to the “mysterious stars” that form the link between the “small” and “large” houses.
And, most importantly, the bridge between these nested universes corresponds to the gaze of the little girl into the “small house”…

Well, that explains everything, i think.

There’s a small image in the post but the link no longer works as the service stopped in january (via Google-translate).

A post at Hacker news identifies the image as coming from the YouTube linked below (2/4/2019 : link lost) and  discovers a connection to the paper titled ‘The ABC Conjecture and the Last Judgment by Giotto’ 

Yesterday i seriously doubted Ted Nelson’s claim that Mochizuki might be the anonymous Hatoshi Nakamoto, designer of BitCoin, having found little or no evidence of CypherPunk-parlance in Mochizuki’s papers…

Today, i’m not that certain any more.

May 28th, 2013

MochizukiDenial

Starting head-on with the 4 papers on inter-universal Teichmuller theory (IUTeich for the fans) is probably not the smartest move to enter Mochizuki-territory.

If you glanced through any of these papers or one of his numerous talks, you’ll know he makes a point of ‘extending’ or even ‘partially dismantling’  scheme theory using the new notion of Frobenioids, which should be some generalisation of Galois categories. 

So if you ever feel like wasting some months trying to figure out what his claimed proof of the ABC-conjecture is all about, a more advisable route might be to start with his two papers on ‘the geometry of Frobenioids’.

Lots of people must have tried that entrance before, and some even started a blog to record their progress as did the person (or persons) behind MochizukiDenial. Here’s the idea:

“Mochizuki deniers by contrast believe that the claim is not serious. They believe that the body of Mochizuki’s work contains neither a proof outline nor ideas powerful enough to resolve the ABC conjecture. We might be wrong. How do we propose to determine whether or not we are. In contrast to Mochizuki boosters on the internet, we will do this by determining what it is that Mochizuki’s papers purport to do. Stay tuned.”

Unfortunately, the project was given up after 2 days and three posts…

Today i tried to acquaint myself with the 126 pages of Frobenioids1 and have a splitting headache because i miss an extra 2Gb RAM to remember the 173 (or more) new concepts he introduces. 

May 29th, 2013

a baby Arithmetic Frobenioid

probably i should start-up a YouTube channel MinuteMochizuki but until i do here’s what i learned today: Alexei Bondal once told me that some Russians start a paper at the end and work their way to the front when needed. Sound advice when approaching M-papers!

So today i did start with the last section ‘Some Motivating Examples’ from Frobenioids1 and worked out what the simplest possible ArithmeticFrobenioid might be, that associated to the integers $\mathbb{Z}$.

Brace yourself here it comes. It is the category $\mathbf{C}$ with objects $(a)$ where $a$ is a strictly positive rational number and morphisms
$(a) \rightarrow (b)$ given by a couple $(n,c)$ where $n$ is a strictly positive integer and $c$ a strictly positive rational number subject to the condition that $a^n$ divides $c.b$ (meaning that the quotient is a strictly positive integer).

Cute (trivial) fact: compositions exist in the sense that
\[
(m,d) \circ (n,c)=(n.m,c^m.d). \]
What makes $\mathbf{C}$ a frobenioid is that it comes with a functor $\mathbf{C} \rightarrow \mathbf{F}$ where $\mathbf{F}$ is the category with one object $*$ and morphisms elements of the noncommutative monoid consisting of all couples $(n,c)$ as before and multiplication as composition above, functor sending all $(a)$ to $*$ and morphisms to corresponding element of the monoid.

Mochizuki proves that we can recover $\mathbf{C}$ from the functor $\mathbf{C} \rightarrow \mathbf{F}$ (look at endomorphism-submonoids of the nc-monoid above) and that $\mathbf{C}$ contains enough info to reconstruct the scheme $\mathbf{Spec}(\mathbb{Z})$ from it (again, use the functor).

Next goal: bring in some Galois categories (tbc)…

May 31st, 2013

MinuteMochizuki 2 : a quadratic arithmetic Frobenioid

Let $m$ be a squarefree number not 1 mod 4 and say you want to recover the classical arithmetic scheme cover $\mathbf{Spec}(\mathbb{Z}[\sqrt{m}])$ onto $\mathbf{Spec}(\mathbb{Z})$ (top left hand corner for m=3).
Prime ideals of $\mathbb{Z}$ may remain prime in $\mathbb{Z}[\sqrt{m}]$ (e.g. (5) and (7)) or split into two prime ideals (e.g. (11)) or ramify (only (2) and (3)), all this governed by the Kronecker symbol.
If it remains prime then the quotient $\mathbb{Z}[\sqrt{m}]/(p)$ is the finite field on $p^2$ elements so has a non-trivial Frobenius morphism, which cannot be lifted to an auto of $\mathbb{Z}[\sqrt{m}]$ to cannot be expressed in scheme language.

For this reason (i think) Mochizuki introduced Frobenioids which are categories allowing one to recover the classical scheme cover but also containing info on the power maps $x \rightarrow x^n$ for all $n$.

In the quadratic case, the objects of the Frobenioid for $\mathbb{Z}[\sqrt{m}]$ is the union of all fractional ideals (that is submodules of $\mathbb{Q}(\sqrt{m})$ of the form $I.q$ where $I$ is an ideal and $q$ non-zero in $\mathbb{Q}(\sqrt{m}))$ and all fractional ideals of $\mathbb{Z}$ (similar defined). Dedekind already knew they correspond to elements of the free Abelian groups on the set of prime ideals and hence have a natural poset-structure.

Now, there are monoid actions on these two posets, giving another set of arrows (for fractional ideals of $\mathbb{Z}[\sqrt{m}]$ the black arrows indicate the poset and the red arrow is the action. Now, the morphisms in the Frobenioid $\mathbf{C}(m)$ are all compositions of an action arrow followed by a poset-arrow (the green ones in the $\mathbb{Z}[\sqrt{m}]$ part, the red ones in the $\mathbb{Z}$-part. Then there is a third set of arrows encoding the Galois-covering info (the black arrows between the two parts). Again, one verifies that compositions exist.

A very special case of Mochizuki’s first Frobenioid paper is that $\mathbf{C}(m)$ contains enough info to recover the scheme cover and even the quadratic field $\mathbb{Q}(\sqrt{m})$. That is, if $\mathbf{C}(m)$ is equivalent to another such one $\mathbf{C}(m’)$ then $m=m’$. Also note that $\mathbf{C}(m)$ contains info on all power maps, so we have somehow ‘lifted’ the Frobenius-maps from the quotients to $\mathbb{Z}[\sqrt{m}]$ at the expense of ‘partially dismantling scheme theory’ (M’s words).

Another neat fact about $\mathbf{C}(m)$ is that all arrows are epimorphisms (compare this to groupoids where all morphisms are isomorphisms). Of course, this only becomes important for more complicated Galois-settings, ideally for the algebraic closure of $\mathbb{Q}$.

June 1st, 2013

Should I stay or should I blog now?

+Alex Nelson commented on +John Baez reshare of my last post on Mochizuki’s Frobenioids:
 
“I just wish he’d blog these, to make it a wee bit easier to read…and print out…”

Valid point, so i did spend some time to make blog-versions of my two G+ posts on this, the last one is linked below, the first one is here (2/4/2019 : lost post).

I’m not sure whether i should continue with this cross-posting. I kinda liked the quick-and-dirty approach of instant-uploading snaps-shots of doodles here. Writing a blogpost  takes more time.

Let me know if it does make a difference to you.

June 3rd, 2013

Mochizuki’s menagerie of morphisms

After zillions of definitions, Mochizuki almost shows empathy with the reader (on page 26 of Frobenioids1) and remarks  that it may be useful to draw a “Chart of Types of Morphisms in a Frobenioid”  (and does this on page 124).

Extremely useful indeed. Try to figure out what a morphism of Frobenius-type might be. It starts like this: it is an LB-invertible base-isomorphism. An LB-invertible map itself is a co-angular and isometric map. A co-angular map itself is defined by the property that for any factorization aoboc of it, where a is linear, b is an isometric pre-step and either a or c are base-isomorphisms, it follows that b is an isomorphism. A pre-step itself is …. etc.etc.

Today i tried to work out what all this means in the case of an arithmetic Frobenioid. Recall that this is a category with objects the elements of layers of posets, each layer (meant to represent the ‘dismantled’ prime-spectrum of a ring of integers $R$ in a number field $K$) the set of fractional ideals in the number field. In each layer we have operations of three types

– Frobenius-type : sending a fractional ideal $I$ to an n-th power $I^n$ (the blue morphisms)
– Poset-type : given by multiplying $I$ with an ideal $K$ of $R$ (the red morphisms, the ‘steps’ in M-parlance)
– Isomorphisms (the blacks, ‘isometric pre-steps’ according to M) which are either isos of fractional ideals (given by multiplying with a non-zero element $q$ of $K$) or Galois action maps on fractional ideals.

The relevant operation between different layers is that of extension of fractional ideals. So, let $\sigma : R \rightarrow S$ be a ring-map induced by a field-iso on $L$, the number-field of $S$, then this operation sends a fractional ideal $J$ in $K$ to the extended fractional ideal $S.\sigma(J)$ in $L$.

What Mochizuki shows is that any arrow in the arithmetic Frobenioid has an essentially unique factorisation into these four types of morphisms (essentially meaning unto irrelevant  isos trown in at one place and compensated by the inverse at the next place).

Let’s work this out for the baby arithmetic Frobenioid corresponding to the integers.

Objects correspond to strictly positive rational numbers $a$, and morphisms are of the form $\phi=(n,q,z) : a \rightarrow b$  where $n$ and $z$ are strictly positive integers and $q$ a strictly positive rational number satisfying $a^n.z=b.q$.

This map can then be factored as a Frobenius-type map $(n,1,1)$ sending $a$ to $a^n$, followed by a poset-map $(1,1,z)$ sending $a^n$ to $a^n.z$ followed by an isomorphism $(1,q^{-1},1)$ sending $a^n.z$ to $a^n.z.q^{-1}=b$.

These different types of morphisms will become important when we want to reconstruct the arithmetic schemes (and covers) from the category structure of the Frobenioid. The essential trick will be to classify and distinguish the irreducible morphisms which are non-isomorphisms phi such that in any factorisation $\phi = \alpha \circ \beta$, either $\alpha$ or $\beta$ must be an isomorphism. (tbc)

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G+ recovery 2 : Grothendieck

My Google+ account is going away on April 2, 2019, so i’ll try to rescue here some posts, in chronological order and around one theme. Here’s Grothendieck-stuff, part two.

March 18th, 2014

crowd-funding Grothendiecks biography?  

+John Baez has a post out at the n-cat-cafe on Leila Schneps’s quest to raise $6000 to translate Scharlau’s 3-volume biography of Grothendieck.

If you care to contribute : go here.

Lots of good stuff in volume 3 on Groths hippy/eco/weirdo years. I’ve plundered Scharlau’s text last year trying to pinpoint the location of Groths hideout in the French Pyrenees.

As far as i know, part 2 (the most interesting part on Groths mathematical years) is still under construction and will be compiled by the jolly group called the “Grothendieck circle”.

There’s a nice series of G-recollections out here (a.o. by Illusie, Karoubi, Cartier, Raynaud, Mumford, Hartshorne, Murre, Oort, Manin, Cartier).

I’m pretty sure Groth himself would prefer we’d try to get his Recoltes et Semailles translated into English, or La Clef des Songes.

November 18th, 2014

Grothendieck’s last hideout

The past ten days I’ve been up in the French mountains (without internet access), not that far from the Ariege, so I’m just now catching up with all (blog)posts related to Grothendieck’s death.

At our place, the morning of thursday november 13th was glorious!

Even though FranceInter kept telling horror stories about flooding in more southern departements, I can only hope that Grothendieck passed away in that morning sun.
About a year ago, on the occasion of Groth’s 85th birthday, I ran a series of posts on places where he used to live, ending with his last known hideout

At the time I didn’t include the precise location of his house, but now that pictures of it are in the French press I feel free to suggest (if you are interested to know where Grothendieck spend his later years) to point your Google-earth or Google-maps (in streetview!) to:
lat 43.068254  lon 1.169080

November 18th, 2014

Mormoiron and Lasserre acknowledge Grothendieck

In the series of post on Grothendieck-places I wrote a year ago (see here and links at the end) I tried to convince these French villages to update their Wikipedia page to acknowledge the existence of Grothendieck under the heading ‘Personnalités liées à la commune’, without much success.

Today it is nice to see that Lasserre added Grothendieck to their page:

“Alexandre Grothendieck (1928-2014), considéré comme un des plus grands mathématiciens du xxe siècle, y a vécu en quasi-ermite de 1990 à sa mort.”

Also Mormoiron, where Grothendieck lived in the 80ties (see picture below) has updated its page:

“Alexandre Grothendieck a habité temporairement à Mormoiron (“Les Aumettes”)”

French villages who still have to follow suit:
Vendargues
Massy
Olmet-et-Villecun

November 19th, 2014

Please keep an eye on the GrothendieckCircle for updates

+Leila Schneps invested a lot of time over the years setting up the Grothendieck Circle website.

Some material had to be removed a few years ago as per Groth’s request.

I’m sure many of you will be as thrilled as I was to get this message from Leila:

“I have already started modifying the Grothendieck circle website and it will of course eventually return completely.  Plus many things will be added, as we will now have access to Grothendieck’s correspondence and many other papers.”

Leila already began to update the site, for example there’s this new page on Groth’s life in Lasserre.

I understand Leila is traveling to Lasserre tomorrow, presumably for Grothendieck’s funeral. Hopefully she will eventually post something about it on the GrothendieckCircle (or, why not here on G+).

December 4th, 2014

 
Nicolas Bourbaki is temporarily resurrected to announce the death of Grothendieck in the French newspaper Le Monde.

You may recall that Bourbaki passed away on november 11th 1968, see +Peter Luschny’s post on his death announcement.

December 6th, 2014

The ‘avis de décès’ released by Grothendieck’s family and friends, published in the local French newspaper ‘La Depeche’, on saturday november 15th.

It announces Grothendieck’s cremation, on november 17th at 11.30h in the village of Pamiers, bordering the ‘Camp du Vernet’, where Grothendieck’s father Sasha was imprisoned, before being deported to Auschwitz and murdered by the Nazis in 1942. 

June 12th, 2015

Grothendieck’s later writings  

Next week there’s a Grothendieck conference at Montpellier. George Maltsiniotis will give a talk thursday afternoon with the  exciting title “Grothendieck’s manuscripts in Lasserre” (hat tip +Pieter Belmans ).

You may recall that G’s last hideout was in the Pyrenean village of Lasserre.

After a bit of sleuthing around I’ve heard some great news.

Grothendieck’s family have donated all of his later writings (apart from his correspondences and other family-related stuff) to the Bibliotheque Nationale. The BNF have expressed their intention of scanning all this material (thousands of pages it seems) and making them (eventually) available online!

Rumour has it that the donation consists of 41 large folders containing G’s reflections, kept in the form of a diary (a bit like ‘Clef des Songes’), on G’s usual suspects (evil, Satan, the cosmos), but 2 or 3 of these folders contain mathematics (of sorts).

Probably, Maltsiniotis will give a preview on this material. To anyone lucky enough to be able to go down south next week and to attend his talk, please keep me in the loop…

June 19th, 2015

Maltsiniotis’ talk on Grothendieck’s Lasserre-gribouillis

Yesterday, George Maltsiniotis gave a talk at the Gothendieck conference in Montpellier with title “Grothendieck’s manuscripts in Lasserre”.

This morning, +David Roberts  asked for more information on its content, and earlier i gave a short reply on what i learned, but perhaps this matter deserves a more careful write-up.

+Damien Calaque  attended George’s talk and all info below is based on his recollections. Damien stresses that he didn’t take notes so there might be minor errors in the titles and order of the parts mentioned below.

EDIT: based on info i got from +Pieter Belmans  in the comments below (followed up by the picture he got via +Adeel Khan  taken by Edouard Balzin) i’ve corrected the order and added additional info.

The talk was videotaped and should become public soon.

As i mentioned last week Grothendieck’s family has handed over all non-family related material to the Bibliotheque Nationale. Two days ago, Le Monde wrote that the legacy consists of some 50.000 pages.

Maltsiniotis insisted that the BNF wants to make these notes available to the academic community, after they made an inventory (which may take some time).

I guess from the blackboard-picture i got from Pieter, the person responsible at the BNF is Isabelle le Masme de Chermont.

The Lasserre-griboillis themselves consists of 5 parts:

1. Géométrie élémentaire schématique. (August 1992)
This is about quadratic forms and seems to be really elementary.

2. Structure de la psyché. (12/10/1992-28/09/1993) 3600 pages
This one is about some combinatorics of oriented graphs with extra-structure (part of the structure are successor and predecessor operators on the set of arrows).

3. Psyché et structures (26/03/93-20/06/93) 700 pages
This one is non-mathematical.

4. Maxwell equations.
Maltsiniotis mentioned that he was surprised to see that there was at best one mathematics book in G’s home, but plenty of physics books.

5. Le problème du mal. (1993-1998) 
This one is huge (30.000 pages) and is non-mathematical.

Note that also the Mormoiron-gribouillis will be made public by the University of Montpellier, or if you prefer video.

Finally, is the photo below what you think it is? Yep!

January 20th, 2016

where are the videos of the Grothendieck conference?

Mid june 2015 a conference “Mathematics of the 21st century: the vision of Alexander Grothendieck” was held in Montpellier. In a comment to a post here on Maltsiniotis’ talk i mentioned that most of the talks were video-taped and that they would soon be made public.

When they failed to surface on the Montpellier website, i asked +Damien Calaque  for more information. Some months ago Damien told me the strange (and worrying) tale of their fate.

At that moment Damien was in a process of trying to recover the videos. Two weeks ago he told me things were looking good, so i now feel free to post about it.

Michael Wright is the head of the Archive for Mathematical Sciences & Philosophy. He arranged with the organizers of the conference that he would send someone over to video-tape the lectures and that he would make them available on his Archive. He also promised to send a copy of the videos to Montpellier, but he never did. Nor did the tapes appear on his site.

Damien Calague emailed Wright asking for more information and eventually got a reply. It appears that Wright will not be able to edit the videos nor put them online in a reasonable time.

They agreed that Damien would send him a large capacity USB-drive. Wright would copy the videos on it and send it back. Damien will arrange for the videos to be edited and the University of Montpellier will put them online. Hopefully everything will work out smoothly.

So please keep an eye on the website of l’Institut Montpelliérain Alexander Grothendieck

May 6th, 2017

Grothendieck’s Montpellier notes will hit the net May 10th

At last there is an agreement between the university at Montpellier and Grothendieck’s children to release the ‘Montpellier gribouillis’ (about 28000 pages will hit the net soon).
Another 65000 pages, found at Lasserre after Grothendieck’s death, might one day end up at the IHES or the Bibliotheque Nationale.
If you are interested in the history of Grothendieck’s notes, there is this old post on my blog.

(h/t Theo Raedschelders for the Liberation link)

May 11th, 2017

Buy a Grothendieck painting to get the Lasserre notes online!

As of yesterday, most of Grothendieck’s Montpellier notes are freely available at this site.

There’s much to say about the presentation (eg. It is not possible to link directly to a given page/article, it is scanned at only 400 dpi etc. etc.) but hey, here they are at last, for everyone to study.

By far the most colourful (in my first browsing of the archive) is cote No. 154, on ‘systeme de pseudo-droites’. You can download it in full (a mere 173 Mb).

As you know, the Montpellier notes are only a fraction of the material Grothendieck left behind. By far the largest (though probably not the most interesting, mathematically) are the Lasserre notes, which to the best of my knowledge are in the care of a Parisian bookseller.

Here’s an idea:
almost every page of No. 154 (written on ancient computer-output) looks like a painting. No doubt, most math departments in the world would love to acquire one framed page of it. Perhaps this can raise enough money to safeguard the Lasserre notes…

July 13th, 2017

le Tour de France in Grothendieck’s backyard

If you want to see the scenery Grothendieck enjoyed in his later years, watch the Tour de France tomorrow.

It starts in Saint-Girons where he went to the weekly market (and died in hospital, november 13th 2014), ending in Foix with 3 category 1 climbs along the way (familiar to anyone familiar with Julia Stagg’s expat-lit set at ‘Fogas’ or you can read my own post on Fogas).

It will not pass through Lasserre (where G spend the final 20 years of his life) which is just to the north of Saint-Girons.

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G+ recovery 1 : Grothendieck

My Google+ account is going away on April 2, 2019, and all attempts to automatically backup my G+ posts seem to fail so far. So i’ll try to rescue here some of them, in chronological order and around one theme. Today, Grothendieck-stuff, part one.

May 30th, 2013

Recordings of a 1972 talk by Grothendieck at Cern “Réflexions sur la science- responsabilité du savant”.
If you don’t have time to listen to all 138 minutes, try to grab from part1 the fragment 29:10 – 30:40 on “the strange ritual of inviting experts to give a talk on some esoteric subject for an audience of 50 to 100 people, one or two of whom will perhaps be able to painfully understand a few bits and pieces, and all others find themselves in a position of humiliation, as they gave in to social pressure to be there, even though the topic itself didn’t interest them at all” (poor translation on my part)
These recordings are illustrative for Grothendieck’s talks in his ‘Survivre’ period, early 70ties. 
(h/t Matilde Marcolli on FB)

June 8th, 2013

Grothendieck’s christmas tree

In the pdf-version of “Recoltes et Semailles” Grothendieck writes on page 463 in the Yin-Yang chapter:

“j’ai fini par aboutir à un diagramme, vaguement en forme d’arbre de Noël”

Here’s the actual diagram, from the original typescript of “Les portes sur l’univers”, the appendix to the ‘Clef du Yin et du Yang’.

Sadly, this appendix (and the many drawings contained in it) didn’t make it into the pdf-release of RecS…

June 9th, 2013

Grothendieck’s yin-yang sunflower

Grothendieck’s ‘Les Portes sur l’Univers’ (Gateways to the Universe(?)) is a truly fascinating text, containing several mysterious drawings (and even a bit of icosahedral-math towards the end).

On PU46, he draws the sunflower of yin and yang, having 12 leafs (he claims, corresponding to 12 yin-terms on the inner circle, 12 yang-terms on the outer circle, as well as to the 12 signs of the zodiac…).

He continues: “On l’appellera, au choix, l’accordeon cosmique, ou l’harmonica cosmique, ou (pour mettre tout le monde d’accord) l’harmonium cosmique”.
(One might call it, as one prefers, the cosmic accordion, or the cosmic harmonica, or (in order to seek general consensus) the *cosmic harmony*).

June 10th, 2013

Grothendieck’s icosahedral theorem

On april 12th 1986, Grothendieck decides to add a mathematical annexe to his esoteric text ‘Les portes sur l’univers’. 

“Par contre, c’est peu pour mon ardeur de mathématicien, laquelle s’est a nouveau réveillée ces jours derniers – et voila repartie ma réflexion sur l’icosaèdre, cet amour mathématique de mon âge mur! Je vais donc peut-être rajouter a ces notes quelques compléments sur la combinatoire de l’icosaèdre et sur la géométrie des ensembles a six éléments…”

He starts with a set S of 6 elements (the vertices), any pair of elements is an edge and any triple a triangle. He then calls a set of triangles F an *icosahedral structure* provided every edge is contained in exactly two triangles in F.

His main result is that all such icosahedral structures are isomorphic (and has exactly 60 isomorphisms), an icosahedral structure consist of exactly 10 triangles and a choice of triangle determines the structure uniquely. Moreover, there are exactly 12 different such octahedral structures and there is an involution on this set coming from ‘complementary’ structures.

At a first glance, Grothendieck’s result appears to be closely related to one of the surprises in finite group theory: the outer automorphism of the symmetric group on 6 letters.

For more on this and related mathematical mysteries of the octahedron, try:

+John Baez  ‘Some Thoughts on the Number 6’  

+Noah Snyder  ‘The Outer Automorphism of S_6’

my own ‘Klein’s dessins d’enfant and the buckyball’

December 18th, 2013

for Grothendieck aficionados

a chance discovery last month en route from Les Vans – Lablachere (in the Ardeche region), a ‘ferronnerie d’art’ (a wrought-iron workshop) called ‘La Clef des Songes’.

All 315 pages of this Grothendieck meditation from 1987 can be found here.

The 691 pages of ‘Notes pour la clef des songes’ are a bit harder to get. Fortunately, the mysterious website ‘l’astree’ offers them as a series of 23 pdfs here. Enjoy the read!

January 3rd, 2014

Why did Grothendieck quit mathematics?

After yesterday’s post on the striking similarities between the lives of Grothendieck and JD Salinger it sure felt weird to stumble upon this footnote in “La Clef des Songes”  

Probably I’m reading way too much into it, but it appears to indicate that Grothendieck stopped doing mathematics to become … a writer!

April 23rd, 2014

Grothendieck documentary available on DVD

+catherine aira and Yves Le Pestipon made a 90 minute long documentary “Alexander Grothendieck, sur les routes d’un genie” which had successful showings in universities, at the Novela science festival, on Toulouse television, and elsewhere. It will be shown in Nantes, Toulouse, Montpellier, and Montreal.

Yves Le Pestipon is one of the people behind the mysterious website lastree.net which has (among many other things) posts on Grothendieck containing hints to his present whereabouts…

Here are some YouTube clips:
clip1

clip2

Here’s the tumblr page of the project:

All of us who cannot attend the viewings can still order the DVD for 25 Euros (20 Euros in France) by sending an email to catherine.aira@gmail.com.

A new release of the DVD, containing English subtitles, will be available soon.

Thanks to +Adeel Khan Yusufzai +David Roberts and +catherine aira 

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RH and the Ishango bone

“She simply walked into the pond in Kensington Gardens Sunday morning and drowned herself in three feet of water.”

This is the opening sentence of The Ishango Bone, a novel by Paul Hastings Wilson. It (re)tells the story of a young mathematician at Cambridge, Amiele, who (dis)proves the Riemann Hypothesis at the age of 26, is denied the Fields medal, and commits suicide.

In his review of the novel on MathFiction, Alex Kasman casts he story in the 1970ties, based on the admission of the first female students to Trinity.

More likely, the correct time frame is in the first decade of this century. On page 121 Amiele meets Alain Connes, said to be a “past winner of the Crafoord Prize”, which Alain obtained in 2001. In fact, noncommutative geometry and its interaction with quantum physics plays a crucial role in her ‘proof’.



The Ishango artefact only appears in the Coda to the book. There are a number of theories on the nature and grouping of the scorings on the bone. In one column some people recognise the numbers 11, 13, 17 and 19 (the primes between 10 and 20).

In the book, Amiele remarks that the total number of lines scored on the bone (168) “happened to be the exact total of all the primes between 1 and 1000” and “if she multiplied 60, the total number of lines in one side column, by 168, the grand total of lines, she’d get 10080,…,not such a far guess from 9592, the actual total of primes between 1 and 100000.” (page 139-140)

The bone is believed to be more than 20000 years old, prime numbers were probably not understood until about 500 BC…



More interesting than these speculations on the nature of the Ishango bone is the description of the tools Amiele thinks to need to tackle the Riemann Hypothesis:

“These included algebraic geometry (which combines commutative algebra with the language and problems of geometry); noncommutative geometry (concerned with the geometric approach to associative algebras, in which multiplication is not commutative, that is, for which $x$ times $y$ does not always equal $y$ times $x$); quantum field theory on noncommutative spacetime, and mathematical aspects of quantum models of consciousness, to name a few.” (page 115)

The breakthrough came two years later when Amiele was giving a lecture on Grothendieck’s dessins d’enfant.

“Dessin d’enfant, or ‘child’s drawing’, which Amiele had discovered in Grothendieck’s work, is a type of graph drawing that seemed technically simple, but had a very strong impression on her, partly due to the familiar nature of the objects considered. (…) Amiele found subtle arithmetic invariants associated with these dessins, which were completely transformed, again, as soon as another stroke was added.” (page 116)

Amiele’s ‘disproof’ of RH is outlined on pages 122-124 of “The Ishango Bone” and is a mixture of recognisable concepts and ill-defined terms.

“Her final result proved that Riemann’s Hypothesis was false, a zero must fall to the east of Riemann’s critical line whenever the zeta function of point $q$ with momentum $p$ approached the aelotropic state-vector (this is a simplification, of course).” (page 123)

More details are given in a footnote:

“(…) a zero must fall to the east of Riemann’s critical line whenever:

\[
\zeta(q_p) = \frac{( | \uparrow \rangle + \Psi) + \frac{1}{2}(1+cos(\Theta))\frac{\hbar}{\pi}}{\int(\Delta_p)} \]

(…) The intrepid are invited to try the equation for themselves.” (page 124)

Wilson’s “The Ishango Bone” was published in 2012. A fair number of topics covered (the Ishango bone, dessin d’enfant, Riemann hypothesis, quantum theory) also play a prominent role in the 2015 paper/story by Michel Planat “A moonshine dialogue in mathematical physics”, but this time with additional story-line: monstrous moonshine

Such a paper surely deserves a separate post.



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