It feels like a long time since I updated here, and I will surely forget some things I wished to mention. But here are a few items:
At vixra, Deep Jyoti Dutta, who I think was 19 when he wrote this paper, took a ratio of masses in deuterium-tritium fusion and found a factor of 𝜋^2. This struck me as something that might be explained in skyrmion theory, e.g. from integrating over a 4-dimensional solid angle.
Also at vixra but not under physics, a serious-but-joking-but-serious made-up religion, "Harmonology". Spoiler warning, the explanation is at the end.
A few months back I asked Math Overflow if there's any explanation as to why 𝜋 is specifically near 3, but the question was removed. This struck me as an odd failure of imagination, in the age of "the field with one element" and all the elaborate mappings of higher number theory. The question itself was inspired by the Church of Entropy's writings on the subject.
There was a recent flurry of events in mainstream and mainstream-alternative math and physics. The great John Conway died, from coronavirus. Stephen Wolfram and Eric Weinstein (see transcript, between 02:11:07 and 02:12:34) came out with their theories of everything. And it was announced that Mochizuki's disputed proof of the abc conjecture would be published in a Japanese journal.
German ex-wunderkind Peter Scholze has taken the lead in western skepticism about Mochizuki's proof. Mochizuki has a grid of copies of a ring, which is supposed to define an "arithmetic deformation theory" (Fesenko's term) based on separating out addition and multiplication. Scholze claims the grid is redundant, and can be replaced with a single copy of the ring, but then Mochizuki's conclusion doesn't follow.
For my part, rather than just believe that Mochizuki needs to answer Scholze, I am hoping to understand the overall argument whereby abc is reached. Mochizuki himself makes the intriguing assertion that an important step is to obtain the "equations"
Nh ≈ h and q^N ≈ q
where you start with an elliptic curve with certain "q-parameters", and construct a simulated elliptic curve with Faltings height h and q-parameters q^N. The step from q^N to q, sounds like knocking out powers greater than 1 (as when one defines the radical Rad(abc)), and the height might play a role in establishing an upper bound (as the abc inequality requires). So maybe the whole thing is a kind of symmetry or duality of Diophantine equations and their crystalline uplifts... I don't know, it's all very interesting but still way beyond me.
But another intriguing thing I found, in Mochizuki's recent expository article (section 4.4), is the idea that western mathematicians have had trouble understanding his work, because their thinking is guided by certain preconceptions about the nature of progress. He mentions two other paradigms, Grothendieck's pursuit of motives, and the Langlands program; whereas he places himself in a third tradition, anabelian geometry.
Since his antagonist Scholze is known as the discoverer of perfectoids, which have been central to the advance of the Langlands program, one might suppose that from a very high perspective, perhaps in a mathematics of the near future, one will think in terms of relations between three kinds of entities, motives, perfectoids, and - perhaps Mochizuki's frobenioids.