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nLab > Latest Changes: noncommutative motive

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2013

    added to noncommutative motive a brief version of the Definition due to Blumberg-Gepner-Tabuada.

    Also added (with brief comments) their references and the dg-category theoretic precursors by Denis-Charles Cisinski and Tabuada.

    (Deserves to be expanded further, certainly, just a quick note so far.)

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJun 9th 2013
    • (edited Jun 9th 2013)

    http://www.ihes.fr/~maxim/TEXTS/ncmotives%20(Skoda%20notes).pdf

    This is a partial writeup of a Kontsevich’s talk on nc motives, LaTeX typed by me on the basis of his transparencies (I was not present at the talk). The last transparency is missing, and also there are some possible errors due my misunderstanding at that time and possible introduced by some readability problems on the original transparencies. Now I could probably made it better as many complementary material is available (and I have later seen the last transparency). Some of the definitions from the beginning are, in more detail, in the huge article

    • L.Katzarkov, M.Kontsevich, T.Pantev, Hodge theoretic aspects of mirror symmetry, Proceedings of Symposia in Pure Mathematics vol. 78 (2008), “From Hodge theory to integrability and TQFT: tt*-geometry”, eds. Ron Y. Donagi and Katrin Wendland, 87-174, arxiv/0806.0107, pdf

    Stefan Schwede has given some useful clarifications/motivation of various conditions Kontsevich is putting on good A ∞A_\infty-categories in one of his expositional talks in a series on stable categories in Bonn. I have notes somewhere, but not easy to find.

    The video of Maxim Kontsevich of his talk on non-commutative motives at IAS Princeton can be found here.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 9th 2013

    Thanks, Zoran! Very useful.

    I have added all those pointers to the References here. Please further edit as need be.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJun 9th 2013
    • (edited Jun 9th 2013)

    I made some corrections in the entry. Katzarkov et al does not have the definition of motives, but the Kontsevich’s framework of saturated, proper, smooth etc. A ∞A_\infty-categories representing noncommutative varieties, which is used and sketched in Kontsevich’s approach to the noncommutative motives in the talks above.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 13th 2013
    • (edited Jun 13th 2013)
    • CommentRowNumber6.
    • CommentAuthoradeelkh
    • CommentTimeJul 21st 2013
    • (edited Jul 21st 2013)

    I added some references to the approach of Toen-Vaquie-Vezzosi.

    By the way, do the videos here work for anyone?

    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeJul 21st 2013

    I tested one or two. They look fine.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeAug 23rd 2013

    added a brief remark Relation to correspondences equipped with cocycles pointing out theorem 9.36 in Blumberg-Gepner-Tabuada 10.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 11th 2015
    • (edited Nov 11th 2015)

    I have filled in various previously missing definitions and facts at the beginning of ncg motives – As the universal additive invariant

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