Lots of things have happened and not been noted, sorry. I will try to catch up a bit over the next few weeks.

In the past few weeks I have spoken at two student-organised conferences. First, in July, was the Postgraduate Group Theory Conference. It was held over three days, at three different venues in London (City, Kings and Imperial), the first time this has been done, I think. I was unable to attend the first two days, since I was recovering from a mild brush with covid, but on the final day I went along to Imperial for the event.

Then this week I was in Edinburgh for GEARS, the Glasgow and Edinburgh Algebra Research Seminar, held in the Bayes Centre.

No prizes for guessing what I talked about in these meetings. And I am not going to give you a blow-by-blow account. I will just mention one talk from the Edinburgh meeting, by William Bevington. He was talking about a result, I think by Andreas Blass, asserting that there is a cohomology theory which “detects” the failure of the Axiom of Choice.

Maybe this is not so surprising. The Axiom of Choice states that every partition has a section, while vanishing cohomology means that every cocycle is a coboundary, not such a different idea. So all you have to do is to build a cohomology theory without topology, by using discrete spaces.

But it did lead us to wonder whether various properties of non-AC universes would be reflected in the cohomology arising in this case.

Two final remarks. First you need the Axiom of Choice to prove that there is a group structure defined on any set. (Proof: Exercise!) Second, as William remarked, of equivalents of the Axiom of Choice, the Axiom itself is obviously true, the Well-Ordering Principle obviously false, and Zorn’s Lemma, well, who knows?

I very much enjoy student-organised meetings. They have a real buzz of excitement to them that just doesn’t happen at most meetings.

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About Peter Cameron

I count all the things that need to be counted.

Cohomology “detects” the failure of the fundamental theorem of calculus (a La Newton).

Doing some grazing online (“digging” seems inappropriate when one does hardly any work) I see that Blass’s result got mentioned on the n-category cafe back in 2013.

Oddly that blog post doesn’t feel the need to link to Blass’s paper, which is:

A. Blass, Cohomology detects failures of the axiom of choice. Trans. Amer. Math. Soc. 279 (1983), 257-269.

I haven’t done more than skim over it, but to me the more interesting parts are those where Blass investigates what can be said if one puts restrictions on the coefficient group G that one is using for cohomology H^1(X;G). In particular, it appears that abelian G are not enough to get the characterization of AC.