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Noncommutative motives and their applications

Marcolli, Matilde and Tabuada, Gonçalo (2015) Noncommutative motives and their applications. In: Commutative algebra and noncommutative algebraic geometry. Mathematical Sciences Research Institute publications. Vol.1. No.67. Cambridge University Press , Cambridge, pp. 191-214. ISBN 978-1-107-06562-8.

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This survey is based on lectures given by the authors during the program “Noncommutative algebraic geometry and representation theory” at the MSRI in the Spring 2013. It covers the recent work [44, 45, 46, 47, 48, 49, 50] on noncommutative motives and their applications, and is intended for a broad mathematical audience. In Section 1 we recall the main features of Grothendieck’s theory of motives. In Sections 2 and 3 we introduce several categories of noncommutative motives and describe their relation with the classical commutative counterparts. In Section 4 we formulate the noncommutative analogues of Grothendieck’s standard conjectures of type C and D, of Voevodsky’s smash-nilpotence conjecture, and of Kimura-O’Sullivan finite-dimensionality conjecture. Section 5 is devoted to recollections of the (super-)Tannakian formalism. In Section 6 we introduce the noncommutative motivic Galois (super-)groups and their unconditional versions. In Section 7 we explain how the classical theory of (intermediate) Jacobians can be extended to the noncommutative world. Finally, in Section 8 we present some applications to motivic decompositions and to Dubrovin’s conjecture.

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Additional Information:© 2015 Mathematical Sciences Research Institute. Marcolli was partially supported by the grants DMS-0901221, DMS-1007207, DMS-1201512, and PHY-1205440. Tabuada was partially supported by the National Science Foundation CAREER Award #1350472 and by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matemática e Aplicações). The authors are very grateful to the organizers Michael Artin, Victor Ginzburg, Catharina Stroppel, Toby Stafford, Michel Van den Bergh, and Efim Zelmanov for kindly giving us the opportunity to present our recent work. They would like also to thank the anonymous referee for comments and corrections.
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Funda��o para a Ci�ncia e a Tecnologia (FCT)UID/MAT/00297/2013
Series Name:Mathematical Sciences Research Institute publications
Issue or Number:67
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:79024
Deposited By: Ruth Sustaita
Deposited On:13 Jul 2017 02:25
Last Modified:03 Oct 2019 18:15

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