Answer by sdr for What is the status of the theory of motives?
I think one recent conceptual advance is Scholze's proposal of a universal cohomology theory valued in "shtukas over $\operatorname{Spec}(\mathbb{Z})$" -- see his ICM address, Section 9. The proposal...
View ArticleAnswer by Donu Arapura for What is the status of the theory of motives?
Of course Grothendieck gave a construction of a category of what we would now call pure motives, but he couldn't prove that it was abelian etc. without assuming the standard conjectures. These...
View ArticleWhat is the status of the theory of motives?
It has been almost 60 years since Grothendieck conceived the conjectural theory of motives in order to grasp the common behavior of the most important (Weil) cohomology theories.But what is the status...
View ArticleAnswer by Raphael Albrecht for What is the status of the theory of motives?
Here is an excellent recent overview article by Clement Dupont from Grothendieck Centre in Montpellier, covering the origins and most of the recent developments for the non-expert mathematician. This...
View ArticleAnswer by Alexey Do for What is the status of the theory of motives?
This is not very recent but it is worth mentioning. One approach to the existence of category of mixed motives $MM(k)$ ($k$ a field of characteristic zero) is via the existence of the motivic Galois...
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