Thank you, kind referee! :-)

Got a referee report back on a paper that took a long time to get right, but which I thought for a long time wasn’t worth even submitting for publication. My favourite remark from the report:

The proofs are clear and easy to follow, but the end results are non-trivial, and with clear applications.

There has been online discussion recently, which I have glimpsed but not followed in-depth, on who we as mathematicians write papers for. The stereotype of a genius mathematician whose papers are inscrutable to all but those on the inside circle who have access to the great man is not one that I find appealing. Yes, mathematics has prerequisites on knowledge, but it can also make sense when explained well. I am flattered that my proofs, which I struggled to write in an elementary way as possible, are seen by at least one person as clear, while also appreciating the scope of the theorem. There are of course things to fix and further clarify, and so this is refereeing at its best: seeing the worth of a paper (I am clearly no great judge of my own work), but finding ways to improve and suggest extra useful context. The scrutiny was very close, to the point of identifying an object that was written in a canonically isomorphic form, and in context, this was not correct. Had this paper been rejected from the journal, but I had gotten the same report, the blow would have been very much softened. So I repeat: thank you, kind referee!

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:

Screen Shot 2020-04-27 at 11.54.26 am

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:

M5-branes and Hypothesis H

I have been following Urs Schreiber and Hisham Sati’s work on what they call ‘Hypothesis H‘ with great interest (see there for detailed literature and history; I will be slack with references). This is a proposal that concerns the generalised cohomology theory in which M-theory’s C-field takes its values. On a grander scale, it’s about the definition of M-theory itself, which, famously, has been pictured as a blob with various tentacles, each of which is a known theory. Each tentacle is meant to be a different reduction of M-theory, under a limits. M-theory itself is the ‘body’ of the blob, but is completely unknown. Hypothesis H is an attempt to fill in this picture, and I want to explain a little about this, and a paper I have written that is a small contribution to this project.

Continue reading “M5-branes and Hypothesis H”

Small ideas on not-quite-covering maps

If you have done any algebraic geometry, then you will have come across descriptions of “covering families” of various types, usually described as a collection of maps with a) common codomain, b) some nice property, such that c) they are jointly surjective. For instance, an Zariski/étale/smooth/flat covering family of a scheme X is a collection of open embedding/étale/smooth/flat maps U_\alpha \to X such that \coprod_\alpha U_\alpha \to X is an epimorphism. (one might need more adjectives over mere flatness, but these can be safely added with no change of anything below.) One can play similar games in other categories; there is a notion of cover of manifolds that is a surjective submersion (or, in principle, a jointly surjective family of submersions), or a cover of a topological space that is an open surjection. That surjectivity is here is good and right: this makes the covers/covering families into subcanonical pretopologies, in that in the case of the single covering maps, these are regular epimorphisms, and really do capture the notion of “being a cover” (in the covering family version, an analogous property holds).

But what is common to the examples above (and many others) is that the other property also has nice features. Submersions, open maps, open embeddings, étale maps, smooth maps etc all contain the isomorphisms, are closed under composition, and are closed under pullback. As such these classes of maps, not assuming surjectivity, already form Grothendieck pretopologies, though they are far from being subcanonical. Further, the categories of schemes, manifolds and topological spaces exhibit a property called extensivity. As a result, coproducts behave as you expect disjoint unions to, rather than, say, direct sums of vector spaces (which are also coproducts). Moreover, the coproduct inclusions form a covering family in each of the cases given (so we have a superextensive pretopology). This implies that one can turn each of these pretopologies into a pretopology where every covering family consists of a single map (a singleton pretopology). This process preserves the property of being subcanonical, and so I will assume that we are working with a singleton subcanonical pretopology.

Another feature that these examples display is that given, for instance, a submersion X \to N, then given any surjective submersion M \to N, the resulting fold  map X \sqcup M \to N is a surjective submersion. The same is true for open maps, for open embeddings, for étale maps and so on. There is a recent PhD thesis by Giorgi Arabidze (not this Giorgi Arabidze!), that deals with internal groupoids in categories equipped with an abstraction of this type of structure, though I have a slightly different take. He starts from an arbitrary singleton pretopology (though calls it a “stronger pretopology”), whose maps are called partial covers, then singles out the subclass of partial covers that are also regular epimorphisms, and calls those covers. The covers then form a subcanonical singleton pretopology. I’m interested in starting from the latter, and seeing what sort of larger classes of maps I can take that have the fold map property above.

As an example, whenever a subcanonical superextensive pretopology is given by specifying that a covering family is a collection of jointly-surjective X-maps, for X a class of maps that form a pretopology on their own, then this is the sort of setup I’d like to abstract, but merely starting from a subcanonical singleton pretopology. So here is a first attempt at this. Nothing deep, just wanting to see if it seems like something that’s out there (and the name is totally provisional).

Definition: Let J be a subcanonical singleton pretopology on an extensive category S. Let K \supset J be another singleton pretopology such that

  1. Given any U \to X in K and V\to X in J, the fold map U+V\to X is in J
  2. K is local for the extensive pretopology.

We then call K a class of precovers for J.

By “local for the extensive topology” I mean that if I have a map in K with codomain a coproduct, if its corestriction to each summand is in K, then it is itself in K. This implies that the coproduct f+g\colon U+V\to X+Y of maps in K is again in K. What I am interested in is the largest possible class of precovers for a given pretopology as in the definition. As noted in the title, these maps are not quite covers, but can be ‘completed’ to a cover. Another way to think about this might be something like finding a class of maps in which J is a kind of ‘ideal’ for the operation of forming the fold map in a suitable slice category.

As an example that is somewhat contrasting to the above more geometric ones, consider an extensive category S with all finite limits (a ‘lextensive’ category; a topos is an example), a subcanonical singleton pretopology J, and consider the subcanonical singleton pretopology given by maps that admit J-local sections. Then the class of all maps in S form a class of precovers. For an more specific example of interest, take a pretopos and the singleton pretopology \mathrm{Reg} consisting of the regular epimorphisms (this, together with the extensive pretopology generate the coherent pretopology). What are the maps that admit \mathrm{Reg}-local sections? And then what is the largest class of precovers, is it all maps again?

 

AustMS 2019 talks

Just so I have these somewhere, these are the notes/slides for the two talks I gave at the annual Australian Mathematical Society talk last week. The first is a revision of this talk.

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.