The latest on Mochizuki

Once in every six months there’s a flurry of online excitement about Mochizuki’s alleged proof of the abc-conjecture.

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:

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

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…

Life on Gaussian primes

At the moment I’m re-reading Siobhan Roberts’ biography of John Horton Conway, Genius at play – the curious mind of John Horton Conway.

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 $\sqrt{2}$.”

How to dismantle scheme theory?

In several of his talks on #IUTeich, Mochizuki argues that usual scheme theory over $\mathbb{Z}$ is not suited to tackle problems such as the ABC-conjecture.

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 $\mathbf{Spec}(\mathbb{Z})/\mathbb{F}_1$.

Moonshine for everyone

Today, Samuel Dehority, Xavier Gonzalez, Neekon Vafa and Roger Van Peski arXived their paper Moonshine for all finite groups.

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’!