for questions about motives in algebraic geometry, including constructions of categories of motives and motivic sheaves, and aspects of the standard conjectures.
The theory of motives in algebraic geometry is a cohomology theory for algebraic varieties. It is distinguished by the property that it unifies the "Weil cohomology" theories, namely Betti cohomology, de Rham cohomology, $\ell$-adic étale cohomology, and $p$-adic crystalline cohomology. In this theory, varieties can be decomposed into combinations of more fundamental components, called motives.
Grothendieck defined a pseudo-abelian category of pure motives, unifying Weil cohomology theories of smooth projective varieties, using correspondences. Much later, there were independent constructions of triangulated categories of mixed motives by Hanamura, Levine, and Voevodsky. However, many of the desired properties of this theory rest on open conjectures concerning the existence of suitable cycles and correspondences. Most notably, we have the Hodge conjecture in de Rham cohomology, the Tate conjecture in étale cohomology, and Grothendieck's standard conjectures for Weil cohomology theories on algebraic varieties.
