# How much algebra is there in Noncommutative Geometry?

My Professor of Homological Algebra got me into some Hochschild (co)homology and then suggested to continue with formally smooth algebras, noncommutative differential forms and so forth.

Now, my personal interest has always been algebra, especially noncommutative (ring and module theory, category theory, some homological algebra). Moreover, my studies have been concerned almost exclusively with such subjects. Hence, I am very bad at calculus, differential geometry etc. I do have the basics, though. I know it's recommendable to have a decent knowledge in most of the subjects, but I got far enough in algebra not needing very much knowledge of calculus, say.

What I want to ask you is the following: Assuming that I got on the way to noncommutative geometry (so it seems to me...), how much algebra is there in? As I said, I really want a career in noncommutative algebra and it would be pretty unpleasant to get involved intensively in a subject which is not my cup of tea. So far, so good, I am enjoying the subject and it interests me in a personal way (not just for school), but I am only at the beginning, I have met only the most basics.

I read that Connes somehow started the subject of NG wanting to extend differential geometry for arbitrary (noncommutative) rings, but would you say the current research work in NG, cyclic homology and the like is algebra at its finest or does it have deep and links with something else (what?)?

OR am I getting this wrong and there are some other paths ahead, starting from noncommutative differential forms, formally smooth algebras, Hochschild (co)homology and the like?

Thank you.

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I'm a PhD student in noncommutative algebra - I work on Hopf algebras - and it is definitely, definitely hard to avoid noncommutative geometry. That said, from my vantage point the geometric side of things looks much more algebraic than differential, and the differential sort is less relevant and more easily ignored.

It's also... I don't think you need to be an expert in geometry in order to do noncommutative algebra, but I think that lacking some degree of foundational knowledge and/or not being comfortable with the language of geometry will hurt you (at least, it's been pretty frustrating for me - I don't have much of a geometric background). You don't have to do noncommutative geometry, there's a lot of work being done on the algebraic side of things, but as a noncommutative algebraist you'll be working alongside geometry and it will be a handicap if you try to ignore that side of things.

The massive caveats here are that I am a) a PhD student, so hardly have a good overview even of my little part of the field, b) really only speaking about said little part of the field and have no idea about other varieties of NCA/NCG. Especially the relevance of noncommutative differential geometry vs nc algebraic geometry is something which I am guessing varies widely.

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Thank you! In the meantime, I went a bit deeper in my studies and I find your answer extremely appropriate. By the way, I'll be studying Hopf algebras starting this summer and seriously consider a PhD in this direction. – AdrianM May 23 '12 at 9:07

It seems to me that nobody has been able to do very much with noncommutative geometry who did not have a strong background in several traditional commutative subjects. It also looks from the outside as though the subject has been very dependent on guidance from a small number of leaders with extraordinary intuition. This could make it a difficult area in which to specialize.

Those are only my superficial thoughts as an outsider to NCG who sometimes reads papers in that field. I would love to discover that these remarks are completely wrong. There are people here or on Mathoverflow who do research in NCG and could give you a more informative answer.

Also, as you probably know, there are several different meanings of NCG, such as Connes operator-algebraic NCDG, Kontsevich, Rosenberg, string-theoretic geometry, and others.

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Yes, I did note the different trends. Also, it's obvious the commutative necessity. It's algebra, nevertheless. My concern is about differential geometry and the like relations (complex analysis, diff topology...). This is my Achiles' heel. – AdrianM Apr 29 '12 at 10:56