Motive (algebraic geometry)

In mathematics, a motive (or motif) is an abstract object in category theory, useful in algebraic geometry for a unified approach to cohomology theories. For every adequate equivalence relation there is a category of pure motives with respect to that relation.

Quotes

• Even though the Weil's Conjectures have been proved by Deligne without appealing to the theory of motives, an enlarged and in part still conjectural theory of mixed motives has in the meanwhile proved its usefulness in explaining conceptually, some intriguing phenomena arising in several areas of pure mathematics, such as Hodge theory, K-theory, algebraic cycles, polylogarithms, L-functions, Galois representations etc.
  • Caterina Consani: (3 November 2007)"Noncommutative geometry and motives (a quoi servent les endomotifs?)". arXiv:0711.0477 [math.QA]. (p. 2)

• Grothendieck's dream was to produce, for any system of polynomial equations, the essential nugget that would remain after everything apart from the shared topological flavour of the system was washed away. Perhaps borrowing the French musical term for a recurring theme, Grothendieck dubbed this the motif of the system.
  • Daniel R. Grayson: (6 November 2017)"Vladimir Voevodsky (1966–2017)". Nature 551 (7679): 169. DOI:10.1038/d41586-017-05477-9.

• The unlikely interplay between motives and quantum field theory has recently become an area of growing interest at the interface of algebraic geometry, number theory, and theoretical physics. The first substantial indications of a relation between these two subjects came from extensive computations of Feynman diagrams carried out by Broadhurst and Kreimer ..., which showed the presence of multiple zeta values as results of Feynman integral calculations. From the number theoretic viewpoint, multiple zeta values are a prototype case of those very interesting classes of numbers which, although not themselves algebraic, can be realized by integrating algebraic differential forms on algebraic cycles in arithmetic varieties. Such numbers are called periods, ... and there are precise conjectures on the kind of operations (changes of variables, Stokes formula) one can perform at the level of the algebraic data that will correspond to relations in the algebra of periods. As one can consider periods of algebraic varieties, one can also consider periods of motives. In fact, the nature of the numbers one obtains is very much related to the motivic complexity of the part of the cohomology of the variety that is involved in the evaluation of the period.
  • Matilde Marcolli: (2 Jul 2009)"Feynman integrals and motives". arXiv:0907.0321 [math-ph]. (p. 2)