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Category: number theory

Je (ne) suis (pas) Mochizuki

Published January 12, 2015 by lieven

Apologies to Joachim Roncin, the guy who invented the slogan “Je suis Charlie”, for this silly abuse of his logo:

I had hoped the G+ post below of end december would have been the last I had to say on this (non)issue: (btw. embedded G+-post below, not visible in feeds)



A quick recap :

– in august 2012, Shinichi Mochizuki finishes the fourth of his papers on ‘inter-universal Teichmuller theory’ (IUTeich for the aficianados), claiming to contain a proof of the ABC-conjecture.

– in may 2013, Caroline Chen publishes The Paradox of the Proof, summing up the initial reactions of the mathematical world:

“The problem, as many mathematicians were discovering when they flocked to Mochizuki’s website, was that the proof was impossible to read. The first paper, entitled “Inter-universal Teichmuller Theory I: Construction of Hodge Theaters,” starts out by stating that the goal is “to establish an arithmetic version of Teichmuller theory for number fields equipped with an elliptic curve…by applying the theory of semi-graphs of anabelioids, Frobenioids, the etale theta function, and log-shells.”

[quote name=”Caroline Chen”]
This is not just gibberish to the average layman. It was gibberish to the math community as well.
[/quote]

“Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,” wrote Ellenberg on his blog.
“It’s very, very weird,” says Columbia University professor Johan de Jong, who works in a related field of mathematics.”

– at the time i found these reactions premature. It often happens that the first version of a proof is not the most elegant or shortest, and i was hoping that Mochizuki would soon come up with a streamlined version, more accessible to people working in arithmetic geometry. I spend a couple of weeks going through “The geometry of Frobenioids 1” and recorded my stumbling progress (being a non-expert) on Google+.

– i was even silly enough to feed almost each and every one of Mochizuki papers to Wordle and paste the resulting Word-clouds into a “Je suis Mochizuki”-support clip. However, in the process I noticed a subtle shift from word-clouds containing established mathematical terms to clouds containing mostly self-defined terms:

.

the situation, early 2015

In recent (comments to) Google+ posts, there seems to be a growing polarisation between believers and non-believers.

If you are a professional mathematician, you know all too well that the verification of a proof is a shared responsability of the author and the mathematical community. We all received a referee report once complaining that a certain proof was ‘unclear’ or even ‘opaque’?

The usual response to this is to rewrite the proof, make it crystal-clear, and resubmit it.

Few people would suggest the referee to spend a couple of years reading up on all their previous papers, and at the same time, complain to the editor that the referee is unqualified to deliver a verdict before (s)he has done so.

Mochizuki is one of these people.

His latest Progress Report reads more like a sectarian newsletter.

There’s no shortage of extremely clever people working in arithmetic geometry. Mochizuki should reach out to them and provide explanations in a language they are used to.

Let me give an example.

As far as i understand it, ‘Frobenioids 1’ is all about a categorification of Arakelov line bundles, not just over one particular number ring, but also over all its extensions, and the corresponding reconstruction result recovering the number ring from this category.

Such a one-line synopsis may help experts to either believe the result on the spot or to construct a counter-example. They do not have to wade through all of the 178 new definitions given in that paper.

Instead, all we are getting are these ‘one-line explanations’:

Is it just me, or is Mochizuki really sticking up his middle finger to the mathematical community.

RIMS is quickly becoming Mochizuki’s Lasserre.

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$\mathbf{Ext}(\mathbb{Q},\mathbb{Z})$ and the solenoid $\widehat{\mathbb{Q}}$

Published July 31, 2014 by lieven

Note to self: check Jack Morava’s arXiv notes on a more regular basis!

It started with the G+-post below by +David Roberts:

Suddenly I realised I hadn’t checked out Morava‘s “short preprints with ambitious ideas, but no proofs” lately.

A couple of years ago I had a brief email exchange with him on the Habiro topology on the roots of unity, and, in the process he send me a 3 page draft with ideas on how this could be relevant to higher dimensional topological QFT (If my memory doesn’t fail me, I can’t find anything remotely related in the arXiv-list).

Being in a number-theory phase lately (yes, I also have to give next year, for the first time, in the second semester, a master-course on Number Theory) the paper A topological group of extensions of $\mathbb{Q}$ by $\mathbb{Z}$ caught my eyes.

The extension group $Ext(\mathbb{Q},\mathbb{Z})$ classifies all short exact sequences of Abelian groups

$0 \rightarrow \mathbb{Z} \rightarrow A \rightarrow \mathbb{Q} \rightarrow 0$

upto equivalence, that is commuting sequences with end-maps being identities.

The note by Boardman Some Common Tor and Ext Groups hs a subsection on this group/rational vector space, starting out like this:

“This subsection is strictly optional. The group $Ext(\mathbb{Q}, \mathbb{Z})$ is much more difficult to determine. It is easy to see that it is a rational vector space, simply from the presence of $\mathbb{Q}$, but harder to see what its dimension is. This group is not as mysterious as is sometimes claimed, but is related to adèle groups familiar to number theorists.”

Boardman goes on to show that this extension group can be identified with $\mathbb{A}^f_{\mathbb{Q}}/\mathbb{Q}$ where $\mathbb{A}^f_{\mathbb{Q}}$ is the ring of finite adèles, that is, sequence $(x_2,x_3,x_5,…)$ of $p$-adic numbers $x_p \in \widehat{\mathbb{Q}}_p$ with all but finitely many $x_p \in \widehat{\mathbb{Z}}_p$, and $\mathbb{Q}$ is the additive subgroup of constant sequences $(x,x,x,…)$.

Usually though, one considers the full adèle ring $\mathbb{A}_{\mathbb{Q}} = \mathbb{R} \times \mathbb{A}^f_{\mathbb{Q}}$ and one might ask for a similar interpretation of the adèle class-group $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}$.

This group is known to be isomorphic to the character group (or Pontrtrjagin dual) of the rational numbers, that is, to $\widehat{\mathbb{Q}}$ which are all group-morphisms $\mathbb{Q} \rightarrow S^1$ from the rational numbers to the unit circle. This group is sometimes called the ‘solenoid’ $\Sigma$, it is connected but not path connected and the path-component of the identity $\Sigma_0 = \mathbb{R}$.

A very nice and accessible account of the solenoid is given in the paper The character group of $\mathbb{Q}$ by Keith Conrad.

The point of Morava’s note is that he identifies the solenoid $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}$ with a larger group of ‘rigidified’ extensions $Ext_{\mathbb{Z}_0}(\mathbb{Q},\mathbb{Z})$.That is, one starts with a usual extension in $Ext_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})$ as above, but in addition, one fixes a splitting of the induced sequence

$0 \rightarrow \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{R} \rightarrow A \otimes_{\mathbb{Z}} \mathbb{R} \rightarrow \mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{R} \rightarrow 0$

Forgetting the splitting this gives the exact sequence

$0 \rightarrow \mathbb{R} \rightarrow Ext_{\mathbb{Z}_0}(\mathbb{Q},\mathbb{Z}) \rightarrow Ext_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z}) \rightarrow 0$

which is isomorphic to the sequence involving the path-component of the solenoid!

$0 \rightarrow \Sigma_0 = \mathbb{R} \rightarrow \Sigma=\widehat{Q} \rightarrow \mathbb{A}^f_{\mathbb{Q}}/\mathbb{Q} \rightarrow 0$

Morava ends with: “I suppose the proposition above has a natural reformulation
in Arakelov geometry; but I don’t know anything about Arakelov geometry”…

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Farey symbols in SAGE 5.0

Published May 31, 2012 by lieven

The sporadic second Janko group $J_2$ is generated by an element of order two and one of order three and hence is a quotient of the modular group $PSL_2(\mathbb{Z}) = C_2 \ast C_3$.

This Janko group has a 100-dimensional permutation representation and hence there is an index 100 subgroup $G$ of the modular group such that the fundamental domain $\mathbb{H}/G$ for the action of $G$ on the upper-half plane by Moebius transformations consists of 100 triangles in the Dedekind tessellation.

Four years ago i tried to depict this fundamental domain in the Farey symbols of sporadic groups-post using Chris Kurth’s kfarey package in Sage, but the result was rather disappointing.

Now, the kfarey-package has been greatly extended by Hartmut Monien of Bonn University and is integrated in the latest version of Sage, SAGE 5.0, released a few weeks ago.

Using the Farey symbol sage-documentation it is easy to repeat the calculations from four years ago and, this time, we do obtain this rather nice picture of the fundamental domain

But, there’s a lot more one can do with the new package. By combining the .fractions() with the .pairings() info it is now possible to get the corresponding Farey code which consists of 34 edges, starting off with



Perhaps surprisingly (?) $G$ turns out to be a genus zero modular subgroup. Naturally, i couldn’t resist drawing the fundamental domain for the 12-dimensional permutations representation of the Mathieu group $M_{12}$ and compare it with that of last time.

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