number theory – neverendingbooks

Leila Schneps on Grothendieck

Published August 20, 2022 by lieven

If you have neither the time nor energy to watch more than one interview or talk about Grothendieck’s life and mathematics, may I suggest to spare that privilege for Leila Schneps’ talk on ‘Le génie de Grothendieck’ in the ‘Thé & Sciences’ series at the Salon Nun in Paris.

I was going to add some ‘relevant’ time slots after the embedded YouTube-clip below, but I really think it is better to watch Leila’s interview in its entirety. Enjoy!

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The hype cycle of an idea

Published February 13, 2022 by lieven

These three ideas (re)surfaced over the last two decades, claiming to have potential applications to major open problems:

(2000) $F_{1}$ -geometry tries to view $Spec (Z)$ as a curve over the field with one element, and mimic Weil’s proof of RH for curves over finite fields to prove the Riemann hypothesis.
(2012) IUTT, for Inter Universal Teichmuller Theory, the machinery behind Mochizuki’s claimed proof of the ABC-conjecture.
(2014) topos theory : Connes and Consani redirected their RH-attack using arithmetic sites, while Lafforgue advocated the use of Caramello’s bridges for unification, in particular the Langlands programme.

It is difficult to voice an opinion about the (presumed) current state of such projects without being accused of being either a believer or a skeptic, resorting to group-think or being overly critical.

We lack the vocabulary to talk about the different phases a mathematical idea might be in.

Such a vocabulary exists in (information) technology, the five phases of the Gartner hype cycle to represent the maturity, adoption, and social application of a certain technology :

Technology Trigger
Peak of Inflated Expectations
Trough of Disillusionment
Slope of Enlightenment
Plateau of Productivity

This model can then be used to gauge in which phase several emerging technologies are, and to estimate the time it will take them to reach the stable plateau of productivity. Here’s Gartner’s recent Hype Cycle for emerging Artificial Intelligence technologies.

Picture from Gartner Hype Cycle for AI 2021

What might these phases be in the hype cycle of a mathematical idea?

Technology Trigger: a new idea or analogy is dreamed up, marketed to be the new approach to that problem. A small group of enthusiasts embraces the idea, and tries to supply proper definitions and the very first results.
Peak of Inflated Expectations: the idea spreads via talks, blogposts, mathoverflow and twitter, and now has enough visibility to justify the first conferences devoted to it. However, all this activity does not result in major breakthroughs and doubt creeps in.
Trough of Disillusionment: the project ran out of steam. It becomes clear that existing theories will not lead to a solution of the motivating problem. Attempts by key people to keep the idea alive (by lengthy papers, regular meetings or seminars) no longer attract new people to the field.
Slope of Enlightenment: the optimistic scenario. One abandons the original aim, ditches the myriad of theories leading nowhere, regroups and focusses on the better ideas the project delivered.
A negative scenario is equally possible. Apart for a few die-hards the idea is abandoned, and on its way to the graveyard of forgotten ideas.
Plateau of Productivity: the polished surviving theory has applications in other branches and becomes a solid tool in mathematics.

It would be fun so see more knowledgable people draw such a hype cycle graph for recent trends in mathematics.

Here’s my own (feeble) attempt to gauge where the three ideas mentioned at the start are in their cycles, and here’s why:

IUTT: recent work of Kirti Joshi, for example this, and this, and that, draws from IUTT while using conventional language and not making exaggerated claims.
$F_{1}$ : the preliminary programme of their seminar shows little evidence the $F_{1}$ -community learned from the past 20 years.
Topos: Developing more general theory is not the way ahead, but concrete examples may carry surprises, even though Gabriel’s topos will remain elusive.

Clearly, you don’t agree, and that’s fine. We now have a common terminology, and you can point me to results or events I must have missed, forcing me to redraw my graph.

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Smirnov on $F_{1}$ and the RH

Published January 21, 2022 by lieven

Wednesday, Alexander Smirnov (Steklov Institute) gave the first talk in the $F_{1}$ world seminar. Here’s his title and abstract:

Title: The 10th Discriminant and Tensor Powers of $Z$

“We plan to discuss very shortly certain achievements and disappointments of the $F_{1}$ -approach. In addition, we will consider a possibility to apply noncommutative tensor powers of $Z$ to the Riemann Hypothesis.”

Here’s his talk, and part of the comments section:

Smirnov urged us to pay attention to a 1933 result by Max Deuring in Imaginäre quadratische Zahlkörper mit der Klassenzahl 1:

“If there are infinitely many imaginary quadratic fields with class number one, then the RH follows.”

Of course, we now know that there are exactly nine such fields (whence there is no ‘tenth discriminant’ as in the title of the talk), and one can deduce anything from a false statement.

Deuring’s argument, of course, was different:

The zeta function $ζ_{Q \sqrt{- d}} (s)$ of a quadratic field $Q \sqrt{- d}$ , counts the number of ideals $a$ in the ring of integers of norm $n$ , that is
$\sum_{n} # (a : N (a) = n) n^{- s}$
It is equal to $ζ (s) . L (s, χ_{d})$ where $ζ (s)$ is the usual Riemann function and $L (s, χ_{d})$ the $L$ -function of the character $χ_{d} (n) = (\frac{- 4 d}{n})$ .

Now, if the class number of $Q \sqrt{- d}$ is one (that is, its ring of integers is a principal ideal domain) then Deuring was able to relate $ζ_{Q \sqrt{- d}} (s)$ to $ζ (2 s)$ with an error term, depending on $d$ , and if we could run $d \to \infty$ the error term vanishes.

So, if there were infinitely many imaginary quadratic fields with class number one we would have the equality
$ζ (s) . \underset{\to}{l i m} L (s, χ_{d}) = ζ (2 s)$
Now, take a complex number $s \neq 1$ with real part strictly greater that $\frac{1}{2}$ , then $ζ (2 s) \neq 0$ . But then, from the equality, it follows that $ζ (s) \neq 0$ , which is the RH.

To extend (a version of) the Deuring-argument to the $F_{1}$ -world, Smirnov wants to have many examples of commutative rings $A$ whose multiplicative monoid $A^{\times}$ is isomorphic to $Z^{\times}$ , the multiplicative monoid of the integers.

What properties must $A$ have?

Well, it can only have two units, it must be a unique factorisation domain, and have countably many irreducible elements. For example, $F_{3} [x_{1}, \dots, x_{n}]$ will do!

(Note to self: contemplate the fact that all such rings share the same arithmetic site.)

Each such ring $A$ becomes a $Z$ -module by defining a new addition $+_{n e w}$ on it via
$a +_{n e w} b = σ^{- 1} (σ (a) +_{Z} σ (b))$
where $σ : A^{\times} \to Z^{\times}$ is the isomorphism of multiplicative monoids, and on the right hand side we have the usual addition on $Z$ .

But then, any pair $(A, A^{'})$ of such rings will give us a module over the ring $Z ⊠_{Z^{\times}} Z$ .

It was not so clear to me what this ring is (if you know, please drop a comment), but I guess it must be a commutative ring having all these properties, and being a quotient of the ring $Z ⊠_{F_{1}} Z$ , the coordinate ring of the elusive arithmetic plane
$Spec (Z) \times_{Spec (F_{1})} Spec (Z)$

Smirnov’s hope is that someone can use a Deuring-type argument to prove:

“If $Z ⊠_{Z^{\times}} Z$ is ‘sufficiently complicated’, then the RH follows.”

If you want to attend the seminar when it happens, please register for the seminar’s mailing list.

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Category: number theory

Leila Schneps on Grothendieck

The hype cycle of an idea

Smirnov on F1 and the RH

Smirnov on $F_{1}$ and the RH