# Thuses

(This is a guest post by David Benjamin Lim)

I would like to make an announcement concerning a new math blog, Thuses.com, started by Stanford math graduate students Slava Naprienko, Bogdan Zavyalov and myself. You may already be aware of Mathoverflow and how it has become extremely valuable to the mathematical community. Unfortunately on Mathoverflow you cannot share little things in math – often not enough for a paper – that you find cool with others.

We feel that the current format in which mathematical ideas are disseminated throughout the community is suboptimal: The arxiv does not support comments and hence is not conducive to discussing mathematical ideas. On the other hand, if you want to publish a “folklore” result, there is not really a place to do this apart from a personal blog or website. It would be nice if there would be a one-stop place to share math, which is what motivated us to start Thuses.

The things you post on Thuses don’t even need to be new or folklore results. They can be expository in nature, e.g. a new take on a paper or a hairy calculation that’s “known” but no mathematician has ever bothered to write down. Or even a list of challenge problems, such as Peter Scholze’s post on Kevin Buzzard’s blog here. Note however that posts on introductory grad level math, e.g. “Introduction to Scheme Theory.” are not welcome on Thuses.

For an example of a post that is really awesome, take a look at this post of Sean Cotner and Bogdan. It is exactly the kind of thing that we want to see on Thuses, and answers one of Richard Taylor’s questions during the Stanford Number Theory Learning Seminar.

Come to Thuses and spread the word of math! For more ways to support us, please look at our support page here.

# Journal of Combinatorial Theory Series A editorial board quits; forms Diamond Open Access replacement

Good news for Open Access fans and non-fans of publishing giant Elsevier: another editorial board of an Elsevier-published journal (well, “most” of the editors) have quit and set up a replacement journal. The new journal is called Combinatorial Theory, and will be published starting the second quarter 2021 on the University of California’s eScholarship platform.

Like a number of other good mathematics journals, it charges no publication fees and is free to read; the journal will publish articles under a Creative Commons Attribution (CC By 4.0) license. The Editorial Board of Comb. Theory is top-notch (including, I note, recent George Szekeres Medal winner Ole Warnaar).

Added, since Andrés Caicedo pointed out Ardila’s tweet thread on the matter:

# Congratulations Roger Penrose!

If Witten, a physicist, can get a Fields Medal, then it is only fair that a mathematician get a Physics Nobel Prize.

# I feel accepted

Now, after multiple rejections and two and a half years of doing the rounds of journals, my joint paper Extending Whitney’s extension theorem: nonlinear function spaces with Alexander Schmeding (arXiv:1801.04126) has been accepted to appear in the Annales de l’Institut Fourier. I wrote a little bit about something from it here. This paper provides the proof that is needed for another joint paper of mine (this one with Raymond Vozzo), on mapping stacks of differentiable stacks (the title just mentions orbifolds, but it works for any differentiable stack as codomain).

# Two recent research theses from my department

One is in complex analytic geometry, and the other is in number theory, with a dash of differential geometry. I had the pleasure of seeing both Haripriya and Ben develop from their first forays into research to producing two excellent theses, both of which won a commendation.

Abstract: Let $M$ be an open Riemann surface. A recent result due to Forstnerič and Lárusson [8] says that, for a closed conical subvariety $A \subset \mathbb{C}^n$ such that $A \setminus \{0\}$ is an Oka manifold, the weak homotopy type of the space of non-degenerate holomorphic $A$-immersions of $M$ into $\mathbb{C}^n$ is the same as that of the space of holomorphic (or equivalently, continuous) maps from $M$ into $A\setminus \{0\}$. In their paper, the authors sketch the proof of this theorem through claiming analogy with a related result, and invoking advanced results from complex and differential geometry, including seminal theorems from Oka theory. The work contained in this thesis was motivated by the absence of a self-contained proof for the special case where $A = \mathbb{C}$ – as, perhaps, the first geometrically interesting case that one would consider. We remedy the absence by providing a fully detailed, self-contained proof of this case; namely, the parametric h-principle for holomorphic immersions of open Riemann surfaces into $\mathbb{C}$. We outline this more precisely as follows. Take a holomorphic 1-form $\theta$ on $M$ which vanishes nowhere. We denote by $\mathcal{I}(M, \mathbb{C})$ the space of holomorphic immersions of $M$ into $\mathbb{C}$, and denote by $\mathcal{O}(M, \mathbb{C}^*)$ the space of nonvanishing holomorphic functions on $M$. We prove, in all detail, that the continuous map

$\mathcal{I}(M, \mathbb{C}) \to \mathcal{O}(M, \mathbb{C}^*), f \mapsto df/\theta,$

is a weak homotopy equivalence. This gives a full description of the weak homotopy type of $\mathcal{I}(M, \mathbb{C})$, as the target space $\mathcal{O}(M, \mathbb{C}^*)$ is known by algebraic topology (Remark 5.2.3).

Abstract: We present some results related to the areas of theta functions, modular forms, Gauss sums and reciprocity. After a review of background material, we recount the elementary theory of modular forms on congruence subgroups and provide a proof of the transformation law for Jacobi’s theta function using special values of zeta functions. We present a new proof, obtained during work with Michael Eastwood, of Jacobi’s theorem that every integer is a sum of four squares. Our proof is based on theta functions but emphasises the geometry of the thrice-punctured sphere.

Next, we detail some investigations into quadratic Gauss sums. We include a new proof of the Landsberg–Schaar relation by elementary methods, together with a second based on evaluations of Gauss sums. We give elementary proofs of generalised and twisted Landsberg–Schaar relations, and use these results to answer a research problem posed by Berndt, Evans and Williams. We conclude by proving some sextic and octic local analogues of the Landsberg–Schaar relation.

Finally, we give yet another proof of the Landsberg–Schaar relation based on the relationship between Mellin transforms and asymptotic expansions. This proof makes clear the relationship between the Landsberg–Schaar relation and the existence of a metaplectic Eisenstein series with certain properties. We note that one may promote this correspondence to the setting of number fields, and furthermore, that the higher theta functions constructed by Banks, Bump and Lieman are ideal candidates for future investigations of such correspondences.

# Dupuy and Hilado’s work on unravelling Mochizuki

It’s taken a while, but now there are some papers starting to be released, here:

Before now these were circulating privately, or just linked in some tweets of Taylor. Just to clarify, Taylor has said publicly that if he had been a referee for the IUT papers, he would not have recommended publication, from the point of view of exposition and readability. But he and Anton Hilado are engaging seriously with the material at hand, and in particular are treating Corollary 3.12 as basically a conjecture, and figuring out how one can work with it, much like any other conjecture in number theory.

EDIT: The papers are now available on the arXiv:

# Terry Tao’s first paper: “Perfect numbers”

Terry Tao has just been awarded the inaugural Riemann prize, and as a result I discovered he had his first mathematics paper published at age 8, in a (now defunct) journal for school mathematics in my home state of South Australia. Since this rare item only appears available reproduced as an appendix in a scanned pdf of a 1984 article in an education journal, I thought I’d re-typeset it. So here it is:

Terence Tao, Perfect numbers, Trigon (School Mathematics Journal of the Mathematical Association of South Australia) 21 (3), Nov. 1983, p. 7–8. (pdf)

Note that this appeared 13 years before his earliest listed paper in Math Reviews/MathSciNet.

Edit: I made a GitHub repository for the paper, the LaTeX source, and the code (working, after minor edits) from it. I passed it on to Tao already.

# Third solution to writing 3 as a sum of three third powers!

Andrew Booker and Andrew Sutherland have found, using the the Charity Engine distributed computing platform, a third solution in integers to the equation $x^3 + y^3 + z^3 = 3$, so we now know each of

• 13 + 13 + 13

• 43 + 43 + (-5)3

• 5699368212219623807203 + (-569936821113563493509)3 + (-472715493453327032)3 (verify!)

is equal to 3. It is conjectured that there are infinitely many integer solutions, but we seem to be nowhere near that.

# Dupuy: computations conditional on IUT3 Corollary 3.12

Just a quick note to advertise some slides by Taylor Dupuy recently presented at Rice University

Explicit Computations in IUT, slides for talk in AGNT Seminar, Rice University April 8, 2019 (pdf)

This is joint work with Anton Hilado modelled on his series of YouTube videos presented earlier this year (and there are links to them in the slides, for technical details). Note that all of this is explicitly stated by Dupuy as being conditional on Corollary 3.12 in Mochizuki’s third IUT paper, the written proof of which is not accepted by almost the entire number theory community.

I see the benefit here as at least simplifying what seems like the sound part of Mochizuki’s work (even if dependent on something still regarded as conjectural) to ordinary mathematics; no dismantling alien ring structures or odd metaphors about how school students get confused by logarithms. Not only that, but the statement of Corollary 3.12 is rendered in ordinary mathematics, rather than in the language of Frobenioids and their ilk.