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July 14, 2019

What Happened At The Magnitude Workshop

Posted by Tom Leinster

A week ago we had a short workshop on magnitude at the University of Edinburgh, organized by Heiko Gimperlein, Magnus Goffeng and me. If that sounds familiar to you, it might be because I advertised it here before. The slides from the talks are now on the website. You can also see a list of open problems.

Anyway, it was a great meeting, focused on the magnitude of metric spaces (as opposed to enriched categories more generally), and roughly evenly split between the analytic and homological aspects of magnitude. It included talks from our own Simon Willerton and Mike Shulman, as well as other experts in a wide variety of different fields (as the official name of the workshop suggests: “Magnitude 2019: Analysis, Category Theory, Applications”). And Emily Roff, who’s doing a PhD with me, spoke about our work on the maximum diversity of a compact metric space.

Heiko and Magnus also invited some experts in the theory of capacity to help us out, knowing that this is something highly relevant to magnitude — even though it now seems that the kinds of questions about capacity that we’re asking do not yet have answers. I was particularly happy to see people from the algebraic side taking part in discussions on primarily analytic questions, and vice versa.

The talks were arranged so that each day started with some introductory stuff (schedule here), before going into more depth. So if you’re curious to find out more and read some of the talk slides, that’s where you might want to start.

Posted at July 14, 2019 10:24 AM UTC

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Re: What Happened At The Magnitude Workshop

Thank you for posting this! I am happy to learn about this although I was unable to attend the conference.

One question I am interested in is whether two graphs differ by a connected Whitney twist have isomorphic magnitude homology (posed in Hepworth-Willerton ‘15). I do not see this problem in the open problem list. Is it solved?

Posted by: Yuzhou Gu on July 14, 2019 6:28 PM | Permalink | Reply to this

Re: What Happened At The Magnitude Workshop

That’s a very good point. As far as I know that problem isn’t solved, but I guess no one thought to put it on the list.

Posted by: Tom Leinster on July 14, 2019 7:37 PM | Permalink | Reply to this

Re: What Happened At The Magnitude Workshop

It appears on the list now. I hope that is not an abuse of the list! I still need to add my conjecture about the derivative of the magnitude of odd balls.

Posted by: Simon Willerton on July 15, 2019 10:58 AM | Permalink | Reply to this

Re: What Happened At The Magnitude Workshop

I believe that this remains unsolved, and I am pleased that Simon has abused the list in this way.

I wonder what the truth of the matter is? It seems impossible that Whitney-twisted graphs have isomorphic magnitude homology, because what on earth would the map be?

But then, if the magnitude homologies are not isomorphic, then what kind of homological relationship could explain the invariance of magnitude under this sort of Whitney twist? It cannot be a Mayer-Vietoris sequence because the magnitude of the Whitney sum (if that’s the right terminology) is not given by the inclusion-exclusion formula. One possibility would be that there is a spectral sequence computing the magnitude homology of a Whitney sum, whose E 2E_2 term depends only on the pieces.

Posted by: Richard Hepworth on July 15, 2019 1:38 PM | Permalink | Reply to this

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