The articles there are invariably of high quality and quite informative.

Their latest is Secret Link Uncovered Between Pure Math and Physics by Kevin Hartnett.

It features the work by number-theorist Minhyong Kim of Oxford University.

In it, Minhyong Kim comes out of the closet, revealing that many of his results on rational points of algebraic curves were inspired by analogies he sees between number theory and physics.

So far he refrained from mentioning this inspiration in papers because “Number theorists are a pretty tough-minded group of people,” he said.

Yesterday, Peter Woit had a post on this on his blog ‘Not Even Wrong’, stuffed with interesting links to recent talks and papers by Minhyong Kim.

Minhyong Kim’s ideas grew out the topic of arithmetic topology, that is, the analogy between number rings and $3$-dimensional compact manifolds and between their prime ideals and embedded knots.

In this analogy, which is based on the similarity between finite connected covers of manifolds on the one hand and connected etale extensions of rings on the other, the prime spectrum of $\mathbb{Z}$ should correspond (due to Minkowski’s result on discriminants and Perelman’s proof of the Poincare-conjecture) to the $3$-sphere $S^3$.

I’ve written more about this analogy here:

– What is the knot associated to a prime?

– Who dreamed up the knots=primes analogy?

– The birthday of the primes=knots analogy.

And probably I’ll mention it later this month when I give a couple of talks at the $\mathbb{F}_1$-seminar in Ghent.

]]>Prepping for this I try to read most of the popular math-books hitting the market.

The latest two explore how the internet changed the way we discuss, learn and do mathematics. Think Math-Blogs, MathOverflow and Polymath.

**‘Gina says’, Adventures in the Blogosphere String War**

The ‘string wars’ started with the publication of the books by Peter Woit:

Not even wrong: the failure of string theory and the search for unity in physical law

and Lee Smolin:

The trouble with physics: the rise of string theory, the fall of a science, and what comes next.

In the summer of 2006, Gil Kalai got himself an extra gmail acount, invented the fictitious ‘Gina’ and started commenting (some would argue trolling) on blogs such as Peter Woit’s own Not Even Wring, John Baez and Co.’s the n-Category Cafe and Clifford Johnson’s Asymptotia.

Gil then copy-pasted Gina’s comments, and the replies they provoked, into a leaflet and put it on his own blog in June 2009: “Gina says”, Adventures in the Blogosphere String War.

Back then, it was fun to waste an afternoon re-reading all of this, and I wrote about it here:

Now here’s an idea (June 2009)

Gina says, continued (August 2009)

With only minor editing, and including some drawings by Gil’s daughter, these leaflets have now resurfaced as a book…?!

After more than 10 years I had hoped that Gil would have taken this test-case to say some smart things about the math-blogging scene and its potential to attract more people to mathematics, or whatever.

In 2009 I wrote:

“Having read the first 20 odd pages in full and skimmed the rest, two remarks : (1) it shouldn’t be too difficult to borrow this idea and make a much better book out of it and (2) it raises the question about copyrights on blog-comments…”

**Closing the gap: the quest to understand prime numbers**

I can hear you sigh, but no, this is **not** yet another prime number book.

In May 2013, Yitang Zhang startled the mathematical world by proving that there are infinitely many prime pairs, each no more than 70.000.000 apart.

Perhaps a small step towards the twin prime conjecture but it was the first time someone put a bound on this prime gap.

Vicky Neal‘s book tells the story of closing this gap. In less than a year the bound of 70.000.000 was brought down to 246.

If you’ve read all popular prime books, there are a handful of places in the book where you might sigh: ‘oh no, not that story again’, but by far the larger part of the book explains exciting results on prime number progressions, not found anywhere else.

Want to know about sieve methods?

Which results made Tim Gowers or Terry Tao famous?

What is Szemeredi’s theorem or the Hardy-Littlewood circle method?

Ever heard about the Elliot-Halberstam or the Erdos-Turan conjecture? The work by Tao on Erdos discrepancy problem or that of James Maynard (and Tao) on closing the prime gap?

Closing the gap is the book to read about all of this.

But it is much more.

It tells about the origins and successes of the Polymath project, and details the progress made by Polymath8 on closing the gap, it gives an insight into how mathematics is done, what role conferences, talks and research institutes a la Oberwolfach play, and more.

Looking for a gift for that niece of yours interested in maths? Look no further. Closing the gap is a great book!

]]>It seems to be that time of the year again.

The twitter-account of the ever optimistic @math_jin is probably the best source for (positive) news about IUT/ABC. He now announces the latest version of Yamashita’s ‘summary’ of Mochizuki’s proof:

山下剛さんのIUTサーベイが更新されました。

Go Yamashita

A proof of the abc conjecture after Mochizuki.

preprint. last updated on 18/Nov/2017.https://t.co/XtnMEO3zoQ#IUTABC— math_jin (@math_jin) November 18, 2017

Another informed source is Ed Frenkel. He sometimes uses his twitter-account @edfrenkel to broadcast Ivan Fesenko‘s enthusiasm.

Big news on Mochizuki's groundbreaking IUT: Over 1000 comments on his 4 papers have been addressed & the final versions sent back to the journal for approval. Hopefully, will be published soon.

Here's Ivan Fesenko's interview about IUT on the AMS website.https://t.co/6GLk3Xh0lm— Edward Frenkel (@edfrenkel) November 17, 2017

Googling further, I stumbled upon an older (newspaper) article on the subject: das grosse ABC by Marlene Weiss, for which she got silver at the 2017 science journalism awards.

In case you prefer an English translation: The big ABC.

Here’s her opening paragraph:

“In a children’s story written by the Swiss author Peter Bichsel, a lonely man decides to invent his own language. He calls the table “carpet”, the chair “alarm clock”, the bed “picture”. At first he is enthusiastic about his idea and always thinks of new words, his sentences sound original and funny. But after a while, he begins to forget the old words.”

The article is less optimistic than other recent popular accounts of Mochizuki’s story, including:

Monumental proof to torment mathematicians for years to come in Nature by Davide Castelvecchi.

Hope Rekindled for Perplexing Proof in Quanta-magazine by Kevin Hartnett.

Baffling ABC maths proof now has impenetrable 300-page ‘summary’ in the New Scientist by Timothy Revell.

Marlene Weiss fears a sad ending:

“Table is called “carpet”, chair is called “alarm clock”, bed is called “picture”. In the story by Peter Bichsel, the lonely man ends up having so much trouble communicating with other people that he speaks only to himself. It is a very sad story.”

Perhaps things will turn out for the better, and we’ll hear about it sometime.

In six months, I’d say…

]]>“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!

]]>Last month, John Duncan, Michael Mertens and Ken Ono managed to do just that.

Inevitably, they had to suffer through a photoshoot and give their university’s PR-people some soundbites.

**CAPTION**

In the simplest terms, an elliptic curve is a doughnut shape with carefully placed points, explain Emory University mathematicians Ken Ono, left, and John Duncan, right. “The whole game in the math of elliptic curves is determining whether the doughnut has sprinkles and, if so, where exactly the sprinkles are placed,” Duncan says.

**CAPTION**

“Imagine you are holding a doughnut in the dark,” Emory University mathematician Ken Ono says. “You wouldn’t even be able to decide whether it has any sprinkles. But the information in our O’Nan moonshine allows us to ‘see’ our mathematical doughnuts clearly by giving us a wealth of information about the points on elliptic curves.”

(Photos by Stephen Nowland, Emory University. See here and here.)

Some may find this kind of sad, or a bad example of over-popularisation.

I think they do a pretty good job of getting the notion of rational points on elliptic curves across.

That’s what the arithmetic of elliptic curves is all about, finding structure in patterns of sprinkles on special doughnuts. And hey, you can get rich and famous if you’re good at it.

Their Nature-paper Pariah moonshine is a must-read for anyone aspiring to write a math-book aiming at a larger audience.

It is an introduction to and a summary of the results they arXived last February O’Nan moonshine and arithmetic.

**Update (October 21st)**

John Duncan send me this comment via email:

“Strictly speaking the article was published in Nature Communications (https://www.nature.com/ncomms/). We were also rejected by Nature. But Nature forwarded our submission to Nature Communications, and we had a great experience. Specifically, the review period was very fast (compared to most math journals), and the editors offered very good advice.

My understanding is that Nature Communications is interested in publishing more pure mathematics. If someone reading this has a great mathematical story to tell, I (humbly) recommend to them this option. Perhaps the work of Mumford–Tate would be more agreeably received here.

By the way, our Nature Communications article is open access, available at https://www.nature.com/articles/s41467-017-00660-y.”

]]>There will be one talk a month, on a tuesday evening from 18hr-20hr. Among the lecturers are the ‘usual suspects’:

Pierre Cartier (October 24th) will discuss the state of functional analysis before Grothendieck entered the scene in 1948 and effectively ‘killed the subject’ (said Dieudonné).

Alain Connes (November 7th) will talk on the origins of Grothendieck’s introduction of toposes.

In fact, toposes will likely be a recurrent topic of the seminar.

Laurant Laforgue‘s title will be ‘La notion de vérité selon Grothendieck'(January 9th) and on March 6th there will be a lecture by Olivia Caramello.

Also, Colin McLarty will speak about them on May 3rd: “Nonetheless one should learn the language of topos: Grothendieck on building houses”.

The closing lecture will be delivered by Georges Maltsiniotis on June 5th 2018.

Further Grothendieck news, there’s the exhibition of a sculpture by Nina Douglas, the wife of Michael Douglas, at the Simons Center for Geometry and Physics (h/t Jason Starr).

It depicts Grothendieck as shepherd. The lambs in front of him have Riemann surfaces inserted into them and on the staff is Grothendieck’s ‘Hexenkuche’ (his proof of the Riemann-Roch theorem).

In fact, I’m also re-reading Alexander Masters’ biography of Simon Norton, The genius in my basement – the biography of a happy man.

If you’re in for a suggestion, try to read these two books at about the same time. I believe it is beneficial to both stories.

Whatever. Sooner rather than later the topic of Conway’s game of life pops up.

Conway’s present pose is to yell whenever possible ‘I hate life!’. Problem seems to be that in book-indices in which his name is mentioned (and he makes a habit of checking them all) it is for his invention of the game of Life, and not for his greatest achievement (ihoo), the discovery of the surreal numbers.

If you have an hour to spare (btw. enjoyable), here are Siobhan Roberts and John Conway, giving a talk at Google: “On His LOVE/HATE Relationship with LIFE”

By synchronicity I encounter the game of life now wherever I look.

Today it materialised in following up on an old post by Richard Green on G+ on Gaussian primes.

As you know the Gaussian integers $\mathbb{Z}[i]$ have unique factorization and its irreducible elements are called Gaussian primes.

The units of $\mathbb{Z}[i]$ are $\{ \pm 1,\pm i \}$, so Gaussian primes appear in $4$- or $8$-tuples having the same distance from the origin, depending on whether a prime number $p$ remains prime in $\mathbb{Z}[i]$ or splits.

Here’s a nice picture of Gaussian primes, taken from Oliver Knill’s paper Some experiments in number theory

Note that the natural order of prime numbers is changed in the process (look at the orbits of $3$ and $5$ (or $13$ and $17$).

Because the lattice of Gaussian integers is rectangular we can look at the locations of all Gaussian primes as the living cell in the starting position on which to apply the rules of Life.

Here’s what happens after one move (left) and after three moves (right):

Knill has a page where you can watch life on Gaussian primes in action.

Even though the first generations drastically reduce the number of life spots, you will see that there remains enough action, at least close enough to the origin.

Knill has this conjecture:

**When applying the game of life cellular automaton to the Gaussian primes, there is motion arbitrary far away from the origin.**

What’s the point?

Well, this conjecture is equivalent to the twin prime conjecture for the Gaussian integers $\mathbb{Z}[i]$, which is formulated as

“there are infinitely pairs of Gaussian primes whose Euclidian distance is $\sqrt{2}$.”

]]>The story is well-known.

End of June 1990, Grothendieck phoned Jean Malgoire warning him to come asap if he wanted to safeguard the better part of G’s mathematical archive, for he was making a bonfire…

A second handover apparently took place on July 28th 1995.

Malgoire kept these notes (in huge Pampers boxes!) until 2010 when he got cold feet as a result of Grothendieck’s letter. He then donated the boxes to the University of Montpellier in 2012.

After Grothendieck’s death in 2014, Montpellier started a project to scan each and every page and put them online, with the backing of Grothendieck’s children (and generous financial support from the local authorities).

So here we are now, and… nobody seems to care.

I’m aware only of this post on MathOverflow by someone who wants to LaTex some of the material on motives.

Perhaps this is due to the less than optimal presentation of the material, or more likely, Grothendieck’s terrible handwriting. Perhaps the University of Montpellier should partner up with the (older generation of) French pharmacists.

But then, there’s this artistic gem in the archive: cote 154 systemes the pseudo-droites written in 1983-84.

It is written on ancient matrix-plotter page. Here’s a typical example

Which mathematical department wouldn’t want to acquire a framed version of one of these original pages?

That’s the point I wanted to make early may in this G+-post, hoping to raise money to safeguard the Lasserre part of Grothendieck’s gribouillis.

When in need for a header picture, I’ll use a page of Grothendieck’s gallery No 154 from now on.

Here’s evidence that Grothendieck was working on GUTS! (literally).

]]>The idea appears to be that ABC involves both the additive and multiplicative nature of integers, making rings into ‘2-dimensional objects’ (and clearly we use both ‘dimensions’ in the theory of schemes).

So, perhaps we should try to ‘dismantle’ scheme theory, and replace it with something like geometry over the field with one element $\mathbb{F}_1$.

The usual $\mathbb{F}_1$ mantra being: ‘forget all about the additive structure and only retain the multiplicative monoid’.

So perhaps there is yet another geometry out there, forgetting about the multiplicative structure, and retaining just the addition…

This made me wonder.

In the forgetting can’t be that hard, can it?-post we have seen that the forgetful functor

\[

F_{+,\times}~:~\mathbf{rings} \rightarrow \mathbf{sets} \]

(that is, forgetting both multiplicative and additive information of the ring) is representable by the polynomial ring $\mathbb{Z}[x]$.

So, what about our ‘dismantling functors’ in which we selectively forget just one of these structures:

\[

F_+~:~\mathbf{rings} \rightarrow \mathbf{monoids} \quad \text{and} \quad F_{\times}~:~\mathbf{rings} \rightarrow \mathbf{abelian~groups} \]

Are these functors representable too?

Clearly, ring maps from $\mathbb{Z}[x]$ to our ring $R$ give us again the elements of $R$. But now, we want to encode the way two of these elements add (or multiply).

This can be done by adding extra structure to the ring $\mathbb{Z}[x]$, namely a comultiplication $\Delta$ and a counit $\epsilon$

\[

\Delta~:~\mathbb{Z}[x] \rightarrow \mathbb{Z}[x] \otimes \mathbb{Z}[x] \quad \text{and} \quad \epsilon~:~\mathbb{Z}[x] \rightarrow \mathbb{Z} \]

The idea of the comultiplication being that if we have two elements $r,s \in R$ with corresponding ring maps $f_r~:~\mathbb{Z}[x] \rightarrow R \quad x \mapsto r$ and $f_s~:~\mathbb{Z}[x] \rightarrow R \quad x \mapsto s$, composing their tensorproduct with the comultiplication

\[

f_v~:~\mathbb{Z}[x] \rightarrow^{\Delta} \mathbb{Z}[x] \otimes \mathbb{Z}[x] \rightarrow^{f_r \otimes f_s} R

\]

determines another element $v \in R$ which we can take either the product $v=r.s$ or sum $v=r+s$, depending on the comultiplication map $\Delta$.

The role of the counit is merely sending $x$ to the identity element of the operation.

Thus, if we want to represent the functor forgetting the addition, and retaining the multiplication we have to put on $\mathbb{Z}[x]$ the structure of a **biring**

\[

\Delta(x) = x \otimes x \quad \text{and} \quad \epsilon(x) = 1 \]

(making $x$ into a ‘group-like’ element for Hopf-ists).

The functor $F_{\times}$ forgetting the multiplication but retaining the addition is represented by the **Hopf-ring** $\mathbb{Z}[x]$, this time with

\[

\Delta(x) = x \otimes 1 + 1 \otimes x \quad \text{and} \quad \epsilon(x) = 0 \]

(that is, this time $x$ becomes a ‘primitive’ element).

Perhaps this adds another feather of weight to the proposal in which one defines algebras over the field with one element $\mathbb{F}_1$ to be birings over $\mathbb{Z}$, with the co-ring structure playing the role of descent data from $\mathbb{Z}$ to $\mathbb{F}_1$.

As, for example, in my note The coordinate biring of $\mathbf{Spec}(\mathbb{Z})/\mathbb{F}_1$.

]]>Originally, Moonshine was thought to be connected to the Monster group. McKay and Thompson observed that the first coefficients of the normalized elliptic modular invariant

\[

J(\tau) = q^{-1} + 196884 q + 21493760 q^2 + 864229970 q^3 + \ldots

\]

could be written as sums of dimensions of the first few irreducible representations of the monster group:

\[

1=1,~\quad 196884=196883+1,~\quad 21493760=1+196883+21296876,~\quad … \]

Soon it transpired that there ought to be an infinite dimensional graded vectorspace, the moonshine module

\[

V^{\sharp} = \bigoplus_{n=-1}^{\infty}~V^{\sharp}_n \]

with every component $V^{\sharp}_n$ being a representation of the monster group $\mathbb{M}$ of which the dimension coincides with the coefficient of $q^n$ in $J(\tau)$.

It only got better, for any conjugacy class $[ g ]$ of the monster, if you took the character series

\[

T_g(\tau) = \sum_{n=-1}^{\infty} Tr(g | V^{\sharp}_n) q^n \]

you get a function invariant under the action of the subgroup

\[

\Gamma_0(n) = \{ \begin{bmatrix} a & b \\ c & d \end{bmatrix}~:~c = 0~mod~n \} \]

acting via transformations $\tau \mapsto \frac{a \tau + b}{c \tau + d}$ on the upper half plane where $n$ is the order of $g$ (or, for the experts, almost).

Soon, further instances of ‘moonshine’ were discovered for other simple groups, the unifying feature being that one associates to a group $G$ a graded representation $V$ such that the character series of this representation for an element $g \in G$ is an invariant modular function with respect to the subgroup $\Gamma_0(n)$ of the modular group, with $n$ being the order of $g$.

Today, this group of people proved that there is ‘moonshine’ for any finite group whatsoever.

They changed the definition of moonshine slightly to introduce the notion of moonshine of depth $d$ which meant that they want the dimension sequence of their graded module to be equal to $J(\tau)$ under the action of the normalized $d$-th Hecke operator, which means equal to

\[

\sum_{ac=d,0 \leq b < c} J(\frac{a \tau + b}{c}) \]

as they are interested in the asymptotic behaviour of the components $V_n$ with respect to the regular representation of $G$.

What baffled me was their much weaker observation (remark 2) saying that you get ‘moonshine’ in the form described above, that is, a graded representation $V$ such that for every $g \in G$ you get a character series which is invariant under $\Gamma_0(n)$ with $n=ord(g)$ (and no smaller divisor of $n$), for every finite group $G$.

And, more importantly, you can explain this to any student taking a first course in group theory as all you need is Cayley’s theorem stating that any finite group is a subgroup of some symmetric group $S_n$.

Here’s the idea: take the original monster-moonshine module $V^{\sharp}$ but forget all about the action of $\mathbb{M}$ (that is, consider it as a plain vectorspace) and consider the graded representation

\[

V = (V^{\sharp})^{\otimes n} \]

with the natural action of $S_n$ on the tensor product.

Now, embed a la Cayley $G$ into $S_n$ then you know that the order of $g \in G$ is the least common multiple of the cycle lengths of the permutation it it send to. Now, it is fairly trivial to see that the character series of $V$ with respect to $g$ (having cycle lengths $(k_1,k_2,\dots,k_l)$, including cycles of length one) is equal to the product

\[

J(k_1 \tau) J(k_2 \tau) \dots J(k_l \tau) \]

which is invariant under $\Gamma_0(n)$ with $n = lcm(k_i)$ (but no $\Gamma_0(m)$ with $m$ a proper divisor of $n$).

For example, for $G=S_4$ we have as character series of $(V^{\sharp})^{\otimes 4}$

\[

(1)(2)(3)(4) \mapsto J(\tau)^4 \]

\[

(12)(3)(4) \mapsto J(2 \tau) J(\tau)^2 \]

\[

(12)(34) \mapsto J(2 \tau)^2 \]

\[

(123)(4) \mapsto J(3 \tau) J(\tau) \]

\[

(1234) \mapsto J(4 \tau) \]

Clearly, the main results of the paper are much more subtle, but I’m already happy with this version of ‘moonshine for everyone’!

]]>