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# Category: rants

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

• (2000) $\mathbb{F}_1$-geometry tries to view $\mathbf{Spec}(\mathbb{Z})$ 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 :

1. Technology Trigger
2. Peak of Inflated Expectations
3. Trough of Disillusionment
4. Slope of Enlightenment
5. 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?

1. 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.
2. 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.
3. 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.
4. 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.

5. 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.
• $\mathbb{F}_1$: the preliminary programme of their seminar shows little evidence the $\mathbb{F}_1$-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.

January 13th, Gallimard published Grothendieck’s text Recoltes et Semailles in a fancy box containing two books.

Here’s a G-translation of Gallimard’s blurb:

“Considered the mathematical genius of the second half of the 20th century, Alexandre Grothendieck is the author of Récoltes et semailles, a kind of “monster” of more than a thousand pages, according to his own words. The mythical typescript, which opens with a sharp criticism of the ethics of mathematicians, will take the reader into the intimate territories of a spiritual experience after having initiated him into radical ecology.

In this literary braid, several stories intertwine, “a journey to discover a past; a meditation on existence; a picture of the mores of a milieu and an era (or the picture of the insidious and implacable shift from one era to another…); an investigation (almost police at times, and at others bordering on the swashbuckling novel in the depths of the mathematical megapolis…); a vast mathematical digression (which will sow more than one…); […] a diary ; a psychology of discovery and creation; an indictment (ruthless, as it should be…), even a settling of accounts in “the beautiful mathematical world” (and without giving gifts…)”.”

All literary events, great or small, are cause for the French to fill a radio show.

January 21st, ‘Le grand entretien’ on France Inter invited Cedric Villani and Jean-Pierre Bourguignon to talk about Grothendieck’s influence on mathematics (h/t Isar Stubbe).

The embedded YouTube above starts at 12:06, when Bourguignon describes Grothendieck’s main achievements.

Clearly, he starts off with the notion of schemes which, he says, proved to be decisive in the further development of algebraic geometry. Five years ago, I guess he would have continued mentioning FLT and other striking results, impossible to prove without scheme theory.

Now, he goes on saying that Grothendieck laid the basis of topos theory (“to define it, I would need not one minute and a half but a year and a half”), which is only now showing its first applications.

Grothendieck, Bourguignon goes on, was the first to envision the true potential of this theory, which we should take very seriously according to people like Lafforgue and Connes, and which will have applications in fields far from algebraic geometry.

Topos20 is spreading rapidly among French mathematicians. We’ll have to await further results before Topos20 will become a pandemic.

Another interesting fragment starts at 16:19 and concerns Grothendieck’s gribouillis, the 50.000 pages of scribblings found in Lasserre after his death.

Bourguignon had the opportunity to see them some time ago, and when asked to describe them he tells they are in ‘caisses’ stacked in a ‘libraire’.

Here’s a picture of these crates taken by Leila Schneps in Lasserre around the time of Grothendieck’s funeral.

If you want to know what’s in these notes, and how they ended up at that place in Paris, you might want to read this and that post.

If Bourguignon had to consult these notes at the Librairie Alain Brieux, it seems that there is no progress in the negotiations with Grothendieck’s children to make them public, or at least accessible.

No, this is not another timely post about the British Royal family.

It’s about Richard Borcherds’ “teapot test” for quantum computers.

A lot of money is being thrown at the quantum computing hype, causing people to leave academia for quantum computing firms. A recent example (hitting the press even in Belgium) being the move of Bob Coecke from Oxford University to Cambridge Quantum Computing.

Sure, quantum computing is an enticing idea, and we have fantastic quantum algorithms such as Shor’s factorisation algorithm and Grover’s search algorithm.

The (engineering) problem is building quantum computers with a large enough number of qubits, which is very difficult due to quantum decoherence. To an outsider it may appear that the number of qubits in a working quantum computer is growing at best linearly, if not logarithmic, in sharp contrast to Moore’s law for classical computers, stating that the number of transistors in an integrated circuit doubles every two years.

Quantum computing evangelists assure us that this is nonsense, and that we should replace Moore’s law by Neven’s law claiming that the computing power of quantum computers will grow not just exponentially, but doubly exponentially!

What is behind these exaggerated claims?

In 2015 the NSA released a policy statement on the need for post-quantum cryptography. In the paper “A riddle wrapped in an enigma”, Neil Koblitz and Alfred Menezes carefully examined NSA’s possible strategies behind this assertion.

Can the NSA break PQC? Can the NSA break RSA? Does the NSA believes that RSA-3072 is much more quantum-resistant than ECC-256 and even ECC-384?, and so on.

Perhaps the most plausible of all explanations is this one : the NSA is using a diversion strategy aimed at Russia and China.

Suppose that the NSA believes that, although a large-scale quantum computer might eventually be built, it will be hugely expensive. From a cost standpoint it will be less analogous to Alan Turing’s bombe than to the Manhattan Project or the Apollo program, and it will be within the capabilities of only a small number of nation-states and huge corporations.

Suppose also that, in thinking about the somewhat adversarial relationship that still exists between the U.S. and both China and Russia, especially in the area of cybersecurity, the NSA asked itself “How did we win the Cold War? The main strategy was to goad the Soviet Union into an arms race that it could not afford, essentially bankrupting it. Their GNP was so much less than ours, what was a minor set-back for our economy was a major disaster for theirs. It was a great strategy. Let’s try it again.”

This brings us to the claim of quantum supremacy, that is, demonstrating that a programmable quantum device can solve a problem that no classical computer can solve in any feasible amount of time.

In 2019, Google claimed “to have reached quantum supremacy with an array of 54 qubits out of which 53 were functional, which were used to perform a series of operations in 200 seconds that would take a supercomputer about 10,000 years to complete”. In December 2020, a group based in USTC reached quantum supremacy by implementing a type of Boson sampling on 76 photons with their photonic quantum computer. They stated that to generate the number of samples the quantum computer generates in 20 seconds, a classical supercomputer would require 600 million years of computation.

Richard Borcherds rants against the type of problems used to claim quantum ‘supremacy’. He proposes the ‘teapot problem’ which a teapot can solve instantaneously, but will be impossibly hard for classical (and even quantum) computers. That is, any teapot achieves ‘teapot supremacy’ over classical and quantum computers!

Another point of contention are the ‘real-life applications’ quantum computers are said to be used for. Probably he is referring to Volkswagen’s plan for traffic optimization with a D-Wave quantum computer in Lisbon.

“You could give these guys a time machine and all they’d use it for was going back to watch some episodes of some soap opera they missed”

Enjoy!

I’m on vacation, and re-reading two ‘metabiographies’:

and

Siobhan Roberts : Genius At Play: The Curious Mind of John Horton Conway

.

Siobhan Roberts’ book is absolutely brilliant! I’m reading it for the n-th time, first on Kindle, then hardcopy, and now I’m just flicking through its pages, whenever I want to put a smile on my face.

So, here’s today’s gem of a Conway quote (on page 150):

Pure mathematicians usually don’t found companies and deal with the world in an aggressive way. We sit in our ivory towers and think.

(Conway complains his words were taken out of context, in an article
featuring Stephen Wolfram.)

If only university administrations worldwide would accept the ‘sitting in an ivory tower and think’-bit as the job description, and evaluation criterium, for their pure mathematicians.

Sadly… they prefer managers to thinkers.

This reminds me of another brilliant text, perhaps not receiving the attention it deserves:

Daniel J. Woodhouse : An open letter to the mathematical community.

Woodhouse offers a reaction to the ‘neoliberal upper management and bloated administration’ of universities:

Within the sphere of pure mathematics — the oldest and most successful of humanity’s intellectual endeavors — I believe our best chance at preserving the integrity and dignity of our tradition is to return to our Pythagorean roots. We should become a cult.

How?

Let us seclude ourselves in mountain caves and daub mysterious equations in blood across rock-faces to ward off outsiders. Let us embrace our most impenetrable mathematical texts as sacred and requiring divinely distributed revelation.

Why?

I am convinced that the current system has dulled our understanding of the value we offer through our instruction. Modern mathematical techniques are the foundation of modern science, medicine, and technology, and we should be the literal, rather than metaphorical, high priests of this temple. Only by withholding our insights will we be able to reassert the intrinsic worth of our knowledge.

I hope these few paragraphs have wetted your appetite to read the manifesto in full, and then take action!

I’m back in Antwerp for over a week now, and finally got hold of our copy of Shimura’s “Introduction to the arithmetic theory of automorphic functions”.

The sad story of disappearing libraries at our university, and possibly elsewhere (everywhere?).

Over 20 years ago our maths department shared a building with the language departments, as well as a library.

The ground floor was taken up by languages, science books were in the cellar. There were years I spend more time on the ground floor than in the maths section.

I must have read most of the Dutch novels published between 1980 and 2000. For some time I could even pass as a Joyce-scholar, at least to those interested in a tiny part of Finnegans Wake.

All that changed when they united the three different branches of Antwerp university and we had to move to another campus.

We were separated from the language departments (they moved to the center of town) and, sadly, also from their library.

On the positive side, we moved to a nice building with a gorgeous library. And, an added bonus, it was on the same floor as my office. To kill an hour it was fun to stroll over to the library and spend some time between books and journals.

Then, some years ago, they closed down the maths-library and moved a tiny fraction of it to the science-library (located at a different campus).

Administration argued that too few people visited the library to keep it open.

But more important, they needed the space to create what they call a ‘study landscape’: a lounge where students can hang out, having enough power outlets for all their computers and smartphones.

So, the maths-library had to go for a place where, during term, students can recharge their phones, and during examination periods like now, students can sit together to study.

It seems that millennials need to have visual confirmation that their fellow students are also offline.

Today even the science-library is transformed into such a study-landscape, and only a handful of math-books remain on the shelves (well-hidden behind another door).

For the few odd ones, like me, who still want to browse through a book occasionally, you have to request for it online.

A few days later you get an email saying that your request is granted (they make it sound as if this is a huge favour), and then they need some more days to get the book from the storehouse and deliver it (sometimes randomly) to one of the few remaining university libraries, sorry, study landscapes…

Smullyan found himself in a very strange country indeed!

All the inhabitants of this country are completely truthful – they always tell you honestly what they believe, but the trouble is that about half of the population are totally mad, and all their beliefs are wrong!

The other half are totally sane and accurate in their judgments, all their beliefs are correct.

Smullyan felt honoured to be invited by the Presidential couple.

The President was the only one who said anything, and what he said was:

“My wife once said that I believe that she believes I am mad.”

What can be deduced about the President’s sanity, and that of his wife?

Reference: This is a minor adaptation of Problem 4.5 in Raymond Smullyan‘s book Logical Labyrinths.

My one and only resolution for 2018: ban vampires from my life!

Here’s the story.

In the 1920’s, Montparnasse was at the heart of the intellectual and artistic life in Paris because studios and cafés were inexpensive.

Artists including Picasso, Matisse, Zadkine, Modigliani, Dali, Chagall, Miro, and the Romanian sculptor Constantin Brancusi all lived there.

You’ll find many photographs of Picasso in the company of others (here center, with Modigliani and Salmon), but … not with Brancusi.

From A Life of Picasso: The Triumphant Years, 1917-1932 (Vol 3) by John Richardson:

“Brancusi disapproved of one of of Picasso’s fundamental characteristics—one that was all too familiar to the latter’s fellow artists and friends—his habit of making off not so much with their ideas as with their energy. “Picasso is a cannibal,” Brancusi said. He had a point. After a pleasurable day in Picasso’s company, those present were apt to end up suffering from collective nervous exhaustion. Picasso had made off with their energy and would go off to his studio and spend all night living off it. Brancusi hailed from vampire country and knew about such things, and he was not going to have his energy or the fruits of his energy appropriated by Picasso.”

I learned this story via Austin Kleon who made this video about it:

Earlier today, John Duncan (of moonshine fame) emailed he was unable to post a comment to the previous post:
“I went to post a comment but somehow couldn’t convince the website to cooperate.”

There’s little point in maintaining a self-hosted blog if people cannot comment on it. If you tried, you got this scary message:

Catchable fatal error: Object of class WP_Error could not be converted to string in /wp-includes/formatting.php on line 1031

The days I meddled with wordpress core php-files are long gone, and a quick Google search didn’t come up with anything helpful.

In despair, there’s always the database to consider.

Here’s a screenshot of this blog’s database in phpMyAdmin:

No surprise you cannot comment here, there isn’t even a wp_comments table in the database! (though surprisingly, there’s a table wp_commentmeta…)

Two weeks ago I moved this blog to a new iMac. Perhaps the database got corrupted in the process, or the quick export option of phpMyAdmin doesn’t include comments (unlikely), or whatever.

Here’s what I did to get things working again. It may solve your problem if you don’t have a backup of another wordpress-blog with a functional wp_comments table.

1. Set up a new WordPress blog in the usual way, including a new database, let’s call it ‘newblog’.

2. In phpMyAdmin drop all tables in newblog except for wp_comments.

3. Export your blog’s database, say ‘oldblog’, via the ‘quick export’ option in phpMyAdmin to get a file oldblog.sql.

4. If this file is small you can use phpMyAdmin to import it into newblog. If not you need to do it with this terminal-command

mysql -h localhost -u root – p newblog < oldblog.sql

and have the patience for this to finish.

5. Change in your wp-config file the oldblog database to newblog.

Happy commenting!

Please allow for a couple of end-of-semester bluesy ramblings. I just finished grading the final test of the last of five courses I lectured this semester.

Most of them went, I believe, rather well.

As always, it was fun to teach an introductory group theory course to second year physics students.

Personally, I did enjoy our Lie theory course the most, given for a mixed public of both mathematics and physics students. We did the spin-group $SU(2)$ and its connection with $SO_3(\mathbb{R})$ in gruesome detail, introduced the other classical groups, and proved complete reducibility of representations. The funnier part was applying this to the $U(1) \times SU(2) \times SU(3)$-representation of the standard model and its extension to the $SU(5)$ GUT.

Ok, but with a sad undertone, was the second year course on representations of finite groups. Sad, because it was the last time I’m allowed to teach it. My younger colleagues decided there’s no place for RT on the new curriculum.

Soit.

The final lecture is often an eye-opener, or at least, I hope it is/was.

Here’s the idea: someone whispers in your ear that there might be a simple group of order $60$. Armed with only the Sylow-theorems and what we did in this course we will determine all its conjugacy classes, its full character table, and finish proving that this mysterious group is none other than $A_5$.

Right now I’m just a tad disappointed only a handful of students came close to solving the same problem for order $168$ this afternoon.

Clearly, I gave them ample extra information: the group only has elements of order $1,2,3,4$ and $7$ and the centralizer of one order $2$ element is the dihedral group of order $8$. They had to determine the number of distinct irreducible representations, that is, the number of conjugacy classes. Try it yourself (Solution at the end of this post).

For months I felt completely deflated on Tuesday nights, for I had to teach the remaining two courses on that day.

There’s this first year Linear Algebra course. After teaching for over 30 years it was a first timer for me, and probably for the better. I guess 15 years ago I would have been arrogant enough to insist that the only way to teach linear algebra properly was to do representations of quivers…

Now, I realise that linear algebra is perhaps the only algebra course the majority of math-students will need in their further career, so it is best to tune its contents to the desires of the other colleagues: inproducts, determinants as volumes, Markov-processes and the like.

There are thousands of linear algebra textbooks, the one feature they all seem to lack is conciseness. What kept me going throughout this course was trying to come up with the shortest proofs ever for standard results. No doubt, next year the course will grow on me.

Then, there was a master course on algebraic geometry (which was supposed to be on scheme theory, moduli problems such as the classification of fat points (as in the car crash post, etale topology and the like) which had a bumpy start because class was less prepared on varieties and morphisms than I had hoped for.

Still, judging on the quality of the papers students are beginning to hand in (today I received one doing serious stuff with stacks) we managed to cover a lot of material in the end.

I’m determined to teach that first course on algebraic geometry myself next year.

Which brought me wondering about the ideal content of such a course.

Half a decade ago I wrote a couple of posts such as Mumford’s treasure map, Grothendieck’s functor of points, Manin’s geometric axis and the like, which are still quite readable.

In the functor of points-post I referred to a comment thread Algebraic geometry without prime ideals at the Secret Blogging Seminar.

As I had to oversee a test this afternoon, I printed out all comments (a full 29 pages!) and had a good time reading them. At the time I favoured the POV advocated by David Ben-Zvi and Jim Borger (functor of points instead of locally ringed schemes).

Clearly they are right, but then so was I when I thought the ‘right’ way to teach linear algebra was via quiver-representations…

We’ll see what I’ll try out next year.

You may have wondered about the title of this post. It’s derived from a paper Raf Bocklandt (of the Korteweg-de Vries Institute in Amsterdam) arXived some days ago: Reflections in a cup of coffee, which is an extended version of a Brouwer-lecture he gave. Raf has this to say about the Brouwer fixed-point theorem.

“The theorem is usually explained in worldly terms by looking at a cup of coffee. In this setting it states that no matter how you stir your cup, there will always be a point in the liquid that did not change position and if you try to move that part by further stirring you will inevitably move some other part back into its original position. Legend even has it that Brouwer came up with the idea while stirring in a real cup, but whether this is true we’ll never know. What is true however is that Brouwers refections on the topic had a profound impact on mathematics and would lead to lots of new developments in geometry.”

I wish you all a pleasant end of 2016 and a much better 2017.

As to the 168-solution: Sylow says there are 8 7-Sylows giving 48 elements of order 7. The centralizer of each of them must be $C_7$ (given the restriction on the order of elements) so two conjugacy classes of them. Similarly each conjugacy class of an order 3 element must contain 56 elements. There is one conjugacy class of an order 2 element having 21 elements (because the centralizer is $D_4$) giving also a conjugacy class of an order 4 element consisting of 42 elements. Together with the identity these add up to 168 so there are 6 irreducible representations.

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.