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Course on categorical actions February 7, 2014

Posted by Ben Webster in Shamelss Self Promotion.
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I have the excellent luck to be sending this semester in Paris, thanks to the Fondation Sciences Mathématiques de Paris.  Part of the deal is that I’m giving a weekly course at the “graduate level” (though I think I have more professors than graduate students in the course) on higher representation theory.  Also thanks to FSMP, the course is being videotaped and posted online;  the first installment is up here.  I’m also posting the videos and additional commentary on a WordPress site; if you have any questions, you can always ask them there (or here, but maybe it’s more germane there).   

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Comments

1. Mahler - February 8, 2014

Nice lecture, thank you. A small comment: The notion of homology is not due to Poincare, but to Noether and Hopf.

2. Noah Snyder - February 9, 2014

Do you have a source for that? Wikipedia says it was Poincare in “Analysis Situs” from 1895. Since 1895 is certainly before Noether or Hopf, I guess your claim is that homology doesn’t actually appear in Poincare’s work? It certainly looks to me like Section 5 “Homologies” is giving a definition of Homology.

3. Mahler - February 9, 2014

I think the difference is that Ben is talking about ”homology groups”, while Poincare (as far as I recall) is talking about ”homology classes” in his (nevertheless groundbreaking) ”Analysis Situs”. I think I read about it in

Hilton, Peter (1988), “A Brief, Subjective History of Homology and Homotopy Theory in This Century”, Mathematics Magazine (Mathematical Association of America) 60 (5): 282–291

See also

http://en.wikipedia.org/wiki/Homology_%28mathematics%29#History

4. Noah Snyder - February 9, 2014

Ah ok, I see what you’re saying. Though Poincare does talk in the Betti number section about linear independence of homology classes, and I’m not sure what that means if he wasn’t thinking of them in some sense as a group. Though perhaps the point is that the modern algebraic point of view (like the one Noether had) was not established in Poincare’s time, so Poincare wouldn’t really have been thinking of it as a “group” but just as “things you can add.”

5. Mahler - February 10, 2014

Yes, that is my point. Poincare introduced ”all” these notions familiar to every topologist nowadays. Of course I do not have a proof, but it seems that Poincare was thinking more in terms of ”numbers” than ”structures”.

It took a long time to realize that they are more than ”mere numbers” (or whatever), but ”interesting structures” on their own. And if one thinks about ”groups” and not ”numbers”, then it is clear that one has the notion of homomorphism between them – one has ”categorified” the ”numbers”.

Anyway, I think, if you want to use ”Homology” as an example of categorification – a good idea by the way, since ”everyone” is familiar with it – then, in my opinion, one should mention the ”algebraic point of view” from ~1925-1930.

But the lecture/video is still great!

6. Erka - March 12, 2014

Haha, since you re in Paris, and related to your 2007 questionning regarding the pronunciation of the name “Koszul”, you could start promoting the French way, wich sounds like /kosyl/… Indeed the name is of polish ascent but modified several times by the French civil registration along the 19 century. It is therefore not polish at all despite it meaning “jersey”, since the 30 or so persons bearing this name in the world all hold it from this attempt to “frenchisized” it (with great success, obviously :-)
My grand father Jean-Louis Koszul is doing very well, actually.


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