Monday, May 21, 2012
As research into quantum gravity continues, it is wise to look at the mathematical issues at hand. Most would agree that ordinary algebraic geometry is not sufficient to tackle the problem in its ultimate form. The theory of motives, at least in its commutative form, is only recently finding applications in the study of scattering amplitudes and Calabi-Yau compactifications. Noncommutative algebraic geometry and its generalized motives, as an extension of Grothendieck's dream of building a gateway between algebraic geometry and the assortment of Weil cohomology theories (de Rham, Betti, l-adic, crystalline, etc.) seems to be a more appropriate tool in the study of quantum geometry. In the noncommutative framework, the role of algebraic varieties and classical Weil cohomologies is played by differential graded categories and numerous functorial invariants. The gateway category Mot, through which all invariants factor uniquely, is the category of noncommutative motives and the different invariants (the Grothendieck, higher K-theory, and cyclic homology groups, etc.) are simply different representations of the motivic category.