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Cohomology theories

Grothendieck's construction of new cohomology theories, which use algebraic techniques to study topological objects, has influenced the development of algebraic number theory, algebraic topology, and representation theory. As part of this project, his creation of topos theory, a category-theoretic generalization of point-set topology, has influenced the fields of set theory and mathematical logic.

The Weil conjectures were formulated in the later 1940s as a set of mathematical problems in arithmetic geometry. They describe properties of analytic invariants, called local zeta functions, of the number of points on an algebraic curve or variety of higher dimension. Grothendieck's discovery of the ℓ-adic étale cohomology, the first example of a Weil cohomology theory, opened the way for a proof of the Weil conjectures, ultimately completed in the 1970s by his student Pierre Deligne. Grothendieck's large-scale approach has been called a "visionary program". The ℓ-adic cohomology then became a fundamental tool for number theorists, with applications to the Langlands program.

Grothendieck's conjectural theory of motives was intended to be the "ℓ-adic" theory but without the choice of "ℓ", a prime number. It did not provide the intended route to the Weil conjectures, but has been behind modern developments in algebraic K-theory, motivic homotopy theory, and motivic integration. This theory, Daniel Quillen 's work, and Grothendieck's theory of Chern classes, are considered the background to the theory of algebraic cobordism, another algebraic analogue of topological ideas.