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Periods and the conjectures of Grothendieck and Kontsevich-Zagier


Ayoub, J (2014). Periods and the conjectures of Grothendieck and Kontsevich-Zagier. European Mathematical Society. Newsletter, (91):12-18.

Abstract

This paper concerns a class of complex numbers, called periods, that appear naturally when comparing two cohomology
theories for algebraic varieties (the first defined topologically and the second algebraically). Our goal is to explain the fundamental conjectures of Grothendieck and Kontsevich–Zagier that give very precise information about the transcendence properties of periods. The notion of motive (due to Grothendieck) plays an important conceptual role. Finally, we explain a geometric version of these conjectures. In contrast with the original conjectures whose solution seems to lie in a very distant future, if at all it exists, a solution for the geometric conjectures is within reach of the actual motivic technology.

Abstract

This paper concerns a class of complex numbers, called periods, that appear naturally when comparing two cohomology
theories for algebraic varieties (the first defined topologically and the second algebraically). Our goal is to explain the fundamental conjectures of Grothendieck and Kontsevich–Zagier that give very precise information about the transcendence properties of periods. The notion of motive (due to Grothendieck) plays an important conceptual role. Finally, we explain a geometric version of these conjectures. In contrast with the original conjectures whose solution seems to lie in a very distant future, if at all it exists, a solution for the geometric conjectures is within reach of the actual motivic technology.

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Additional indexing

Item Type:Journal Article, not refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2014
Deposited On:30 Sep 2014 10:40
Last Modified:21 Nov 2017 18:17
Publisher:European Mathematical Society Publishing House
ISSN:1027-488X
Related URLs:http://www.ams.org/mathscinet-getitem?mr=3202399 (Organisation)

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