He assured us that he would *not* be apologetic when mentioning ideas from category theory, but *proud*.

His talk is quite nice for other reasons too:

• Tom Leinster, Magnitude homology.

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Abstract.Magnitude homology is a homology theory of enriched categories, proposed by Michael Shulman late last year. It therefore specializes to (among other things) a homology theory of ordinary categories and a homology theory of metric spaces. This metric homology theory is something new, and is different from persistent homology. As a sample result, the first homology of a subset X of Euclidean space detects whether X is convex.Like all homology theories, magnitude homology has an Euler characteristic, defined as the alternating sum of the ranks of the homology groups. Often this sum diverges, so we have to use some formal trickery to evaluate it. In this way, we end up with an Euler characteristic that is often not an integer. This number is called the magnitude of the enriched category. In the setting of compact metric spaces, magnitude is closely related to volume, surface area, curvature, and other classical invariants of geometry. In the special setting of finite metric spaces, magnitude appears to provide information of interest about point-sets, such as the apparent dimension and number of clusters at different length scales.

I will give an overview of these developments, assuming little categorical knowledge.

One of my main efforts is to break down the ‘pure math wall’ separating category theory and applied mathematics. Because engineers and chemists don’t learn category theory from mathematicians, they wind up reinventing it for themselves! A fair amount of work is required simply to explain this in detail, and I have a bunch of papers doing this.

I’m also running a special session on applied category theory at the AMS meeting at U.C. Riverside on 4-5 November 2017, and I’ll be speaking at a big applied category theory workshop at the Lorentz Center in Leiden in April 2018—more details about that later.

In theoretical computer science, on the other hand, the utility of category theory is well-understood.

]]>Way back in my graduate school days I was surprised to hear something as abstract as algebraic topology had applications to robotic vision and motion.

I’ll be interested to read how far that application has progressed since then.

Bob Clark

]]>I guess you were reading an archaic version of my notes! I’ve revised them many times since then, so please grab the latest version… if you want one where everything is true.

I discovered a nasty technical mistake on the train to Sapporo. It’s fixed now, and I’ve given my talk.

]]>Huh. I swear there was another page there, involving, if I recall, three different -Rips complexes, essentially establishing an order of sorts. Your indices of choice for that were , and , but an unused could be found in the introductory paragraph just before that, instead of a . This seems to now be gone entirely. Did you update the slides or am I going crazy?

Though as I see it right now, page 28, to me, does contain one equation:

.

You might, however, find this on page 27 instead: For my browser pdf viewer at least, there is, for some reason, one duplicate page 18/19. If that doesn’t happen on your end, you’ll probably have one page fewer. (For me there are 32 pages total)

I don’t see an or anywhere in the talk, and especially not on page 28, which has no equations. What am I missing?

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