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K3 surface containing the 1729 yellow cab

Fun: Whiskas has finally opened a Kitten College where diverse kitten are taught string theory and other things. The ad forgot to say that they plan to produce a Kitten Witten every year.
I wrote a rather technical and extensive blog post about similar topics in March 2015 and I don't plan to repeat it or exceed it right now.



But Ken Ono (not to be confused with John Lennon) and his graduate student Sarah Trebat-Leder from Emory University, Georgia (not in the former Soviet Union but in the former Confederate States of America) have just published a playful mathematical paper with some close enough ties to string theory:
The 1729 K3 Surface (arXiv, appeared in Research in Number Theory)
At least three news outlets, Science Daily, Futurity, and Livemint, have just written three nice and decent pieces of hype.




The story behind this piece of research is really cute. Ingenious Indian mathematician Srinivasa Ramanujan was a friend with every number but the friendship with some of them was more intimate. Once upon a time, he was ill with tuberculosis and his friend Hardy visited him in the hospital. "The yellow cab had a boring number on the plate, 1729. I hope it's not a bad sign."




Ramanujan replied: "On the contrary. It's a wonderful number. 1729 is the smallest number that may be written as the sum of two third powers in two inequivalent ways. Let us shall these numbers taxi-cab numbers." Indeed, you may verify that\[

1729 = 1^3 + 12^3 = 9^3+10^3

\] Every schoolkid with some knowledge in arithmetics may verify that his remark was right. In fact, 1729 is the smallest taxi-cab numbers. Most schoolkids as well as adults – including some very clever ones – would stop at this point. It looks like a curiosity, an inconsequential piece of recreational mathematics. You might think that it's silly for 1729 to appear all over the Futurama as well as in the Simpsons.

However, Ramanujan didn't stop there at all. Well, the next step may still look like recreational mathematics, a bit more contrived one, but it gets better after some time. Ramanujan generally studied the solutions to the related Diophantine equation\[

x^3 + y^3 = z^3 + w^3

\] where the D-adjective means that we only allow integer-valued solutions. He was able to find an infinite family of solutions parameterized by \(a,b\) because\[

(3a^2 + 5ab − 5b^2)^3 + (4a^2 − 4ab + 6b^2)^3 =\\
= −(5a^2 − 5ab − 3b^2)^3 + (6a^2 − 4ab + 4b^2)^3

\] There are various operations you can do to reinterpret the solutions as either integer solutions to some equations, or rational solutions to some equations. And similar cubic equations may be understood as equations for elliptic curves – which is, in the case of complex geometry, equivalent to the 2-real-dimensional surface with toroidal topology given by a nice complex equation.

Things are a bit technical and I don't plan to learn and teach all the details – the paper should be an okay place for that. But to make the story short, Ono et al. prove the following theorem:\[

y^2 = x^3 - 432 m(t)^2

\] understood as an equation over rational numbers \({\mathbb Q}\) is an elliptic K3-surface with rank 2 and Picard number 18. Note that \(432=1728/4\) and I must tell you that\[

m(t) = (7+t+t^2) (1+t+7t^2) (13 \!-\! 23t\! +\! 13t^2)

\] is a particular sixth-order polynomial whose structure may be obtained from Ramanujan's parameterization in a certain way. Picard number is some rank with a very abstract and contrived definition which is difficult to calculate and if I understand it well, the theorem is a cool result because they could find an answer to this esoteric question in a very simple way. If there is some more profound value behind the theorem, then I apologize but I haven't understood it yet.



The quartic surface, a simple subtype of a K3 surface

For someone like me – or (probably) you – who is inexperienced in this sort of algebraic geometry, it is difficult to even verify that the surface given by the equation above is a K3-surface at all ;-), let alone to calculate its Picard number. At least, you may verify the right dimension: there are three possibly complex coordinates \(x,y,t\) and you have one equation so what you're left with is complex-2-dimensional. Great! :-)

More seriously, what I feel and like are the interconnections between all the "next to trivial" structures in mathematics. You could say that the solutions to the Pythagorean theorem such as \(a^2+b^2=c^2+d^2\) are too simple and they're about as simple as some spheres etc. If you try to make just one small step for a man beyond this simple mathematics, you may end up studying cubic equations. And instead of spheres or tori, you study K3 surfaces. It's sort of inevitable because at least as a real-4-dimensional compactification manifold for string theory, K3 is the simplest choice after the torus.

It's fun to ask whether the K3-surface is too physical or too detached from physics; and whether it's too simple or too complex. I think that the naive critics of string theory would almost certainly answer that it's too complicated and irrelevant for physics. But string theorists know better. K3 is almost certainly relevant for physics and it is a simple complex surface. It's so simple that to get realistic compactifications, you need to use K3 as the fiber or do other things.

One could say that the particle spectrum is constructed out of some concepts, ideas, and characteristics of the compactification geometries – much like organisms are made of elements and compounds – and K3 is one of the simple enough objects or ideas that are surely needed in that game – the K3 is perhaps analogous to the helium or carbon atoms (while the hydrogen represents the sphere or tori).

At any rate, these mathematicians have surely made sense out of one page of Ramanujan's "lost notebook." The page is reprinted in their paper.

Also, I want to pick one more quote from the paper that made me laugh. At some point, they claim that the quotient of \(C\) by \(\omega\) is elliptic. In fact, it is given by\[

y^2 − 132003308704176245102247936y =\\
= x^3 − 755077852050168921425460\heartsuit\\
\heartsuit 2155146485718479954847046041.

\] I had to divide the "not so small" integer into two lines by the \(\heartsuit\) symbol so that the readers using the green blog template don't face excessively long lines of mathematics.

Would you guess that this the equation of the most prominent toroidal surface that emerges from a closer analysis of the "boring" number of Hardy's yellow cab? :-)

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