Abstract:

Algebraic varieties have many cohomology groups attached to them: Betti cohomology, de Rham cohomology, étale cohomology, and crystalline cohomology. But they are not too different. In fact, under reasonable assumptions, there are comparison isomorphisms relating them. This led Grothendieck to the idea of having a "universal cohomology theory" which any other (good) cohomology theory is a particular case of. This is (conjecturally) realised with his construction of the category of pure motives, whose objects can be thought of as the "essence" of algebraic varieties. In this talk, I will try to explain and motivate this construction and show that when we restrict our attention to the simplest varieties one can think of, namely points, one recovers the familiar notion of Galois representation that is well-known to number theorists.

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