Handbook of Homotopy Theory

It seems there’s a new book in preparation dedicated to collating survey articles on various topics in contemporary homotopy theory. The chapters are slowly turning up on the arXiv. There are also odd chapters only on people’s webpages (eg this one by Paul Balmer). I haven’t seen a webpage or announcement of the final product yet, so I don’t know how long it’s going to turn out to be. But it all looks very cool!

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Call to All Mathematicians to Make Publications Electronically Available

This was the title of a recommendation endorsed by the International Mathematical Union Executive Committee in 2001. Pretty much the whole text is as follows:

Open access to the mathematical literature is an important goal. Each of us can contribute to that goal by making available electronically as much of our own work as feasible.
Our recent work is likely already in computer readable form and should be made available variously in TeX source, dvi, pdf (Adobe Acrobat), or PostScript form. Publications from the preTeX era can be scanned and/or digitally photographed. Retyping in TeX is not as unthinkable as first appears.
Our action will have greatly enlarged the reservoir of freely available primary mathematical material, particularly helping scientists working without adequate library access.

“Call to All Mathematicians to Make Publications Electronically Available”, IMU Committee on Electronic Information and Communication

Needless to say I heartily endorse this position.

The Google Translate is not enough

I track the incoming links to my post about Mochizuki’s Report with interest, and the volume coming from Japanese-language websites is not insignificant (surprise!). Every now and then I run comments mentioning me through Google Translate, and what comes out is slightly mysterious. For instance, the comment

自分はその時点では懸念と言えるようなものじゃないと思うんですけどね。雑感という感じで 
タオの懸念ていうのがはっきり見えるのは、やっぱり去年末の応用可能性とかについて述べたコメかなと 
Robertsの発言については個人的にはちょっと違うかなと思っている 
山下さんの言う通り、もはや単なるgeometric morphismだけの議論ではないだろうという点がね 
確かに最先端の圏論には望月さんは熟知していないだろうけど、しかしLurieとかの理論もGMの設定なんでね

https://rio2016.5ch.net/test/read.cgi/math/1541001291/689

is translated as

I think that I am not a concern at that point. What 
makes Tao’s fears clear as a feeling of touch is that as I mentioned by Mr. Yamashita who 
thinks that Roberts’ remarks are slightly different from rice that talked about the possibility of application at the end of last year and so on 
, It is no longer a matter of just geometric morphism It is 
true that Mr. Mochizuki is not familiar with the state-of-the-art category theory, but the theory like Lurie is also the setting of GM

I am not under the illusion that generally these comments may be nothing more than Reddit-level spitballing by mathematics students, but if someone is saying Go Yamashita has read my notes, then that is cause for hope that a fruitful dialogue could be opened up. I’m not sure about “state-of-the-art category theory”, but I don’t think that Lurie-level categories are necessary. Also, the statements being discussed by myself and Terry Tao are quite old, before the (exciting?) events of this year.  However, I don’t want to read too much into what seems like rampant rumour-mongering in the source thread over at 5channel on the basis of bad software translation. If anyone with a decent level of Japanese could give me an idiomatic translation of the above quoted passage, I would be grateful. You will get some free internet points in return, or a coffee of your choice if we ever meet in person.

In praise of Replacement

This is the title of a quite long paper from the set theorist Akihiro Kanamori:

Abstract: This article serves to present a large mathematical perspective and historical basis for the Axiom of Replacement as well as to affirm its importance as a central axiom of modern set theory

The Bulletin of Symbolic Logic
Volume 18, Number 1, March 2012 (author-hosted pdf)

The Axiom of Replacement is what separates our modern concept of ZFC from Zermelo’s original 1908 set theory. Without it there are naive constructions one might do, like the countably-infinite iterated powerset construction (or the relatively small cardinal \beth_\omega), that may not exist. On the other hand, most uses arise purely in set theory, their results can often be arrived at by combinations of other axioms or weaker axioms. One celebrated result near the border between set theory and non-set theory, Borel determinacy, is in fact equivalent to the axiom of Replacement. One reason this axiom is sometimes held in suspicion by category theorists of a certain bent is that the set theory ETCS defined by properties of the category of sets is agnostic about Replacement—it doesn’t hold (and nor does it fail to hold) in the equiconsistent system BZC (similar to Zermelo’s original system). ETCS as a system can be phrased entirely using concepts used every day by all mathematicians (eg composition of functions is associative, subsets have indicator functions, etc). One view (perhaps one built of straw) is that even if an inconsistency were to be found in ZFC, with its demand that an arbitrary function from a set to the class of all sets has a range that is a set, since ETCS is concerned with such ordinary objects and operations, the culprit would surely be Replacement. But Kanamori spends a good 38 pages justifying Replacement, so we cannot immediately dismiss it. And in any case, axioms equivalent to Replacement can be found to strengthen ETCS (or any other structural set theory in a similar style).

Moufang loops

I’ve been working through a partially completed project I started with Ben Nagy, an undergrad student at Adelaide. Ben wrote code to calculate the twisted cocycle (sometimes called a ‘factor set’ in the finite group literature) for the Parker loop, following an algorithm of Griess. We found, experimentally, a nice basis of the extended Golay code that makes certain subloops of the Parker loop split canonically as a direct product, as well as a cute way to encode the full twisted cocycle Golay \times Golay \to \mathbb{Z}/2 in a much smaller amount of data:

Enough data to reconstruct the multiplication of the Parker loop

Here black = 1 and white = 0, and this should be read as being made of four blocks, each labelled by a pair of complementary 6-dimensional subcodes of Golay. That South-East block has been making me curious, so I decided to have a go at figuring out what it gave. It turns out that that subloop can be further simplified, to give a code loop extending a 3-dimensional subcode. Given the classification of 16-element Moufang loops, this code loop should be M_{16}(C_2\times C_4), but I’m trying to show this directly, rather than ruling out the competition.

Trusty old Nokia 3310 for that nostalgic 00s digital photo look

I feel like I’m back in Algebra II finding subgroups from a multiplication table. (Later: aha, found it!)

In memoriam

My great-grandfather, W. W. Woodhouse, was enlisted during the First World War; I do not know if or where he saw active combat. My grandfather, Raymond ‘Curly’ Woodhouse, served in Borneo in the Second World War, despite only having one eye, as a signaller. He lived, but his experience had a profound impact on my family that I still feel today. Unfortunately I was too young to understand while he was still with us.