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foundation for this major field, because of its great expressive power as well as technical depth. In that setting one can use birational geometry, techniques from number theory, Galois theory and commutative algebra, and close analogues of the methods of algebraic topology, all in an integrated way.

Its influence spilled over into many other branches of mathematics, for example the contemporary theory of D-modules. (It also provoked adverse reactions, with many mathematicians seeking out more concrete areas and problems. Grothendieck is one of the few mathematicians who matches the French concept of maître à penser; some go further and call him maître-penseur.)

The bulk of Grothendieck's published work is collected in the monumental, and yet incomplete, Éléments de géométrie algébrique (EGA) and Séminaire de géométrie algébrique (SGA). Perhaps Grothendieck's deepest single accomplishment is the invention of the étale and l-adic cohomology theories, which explain an observation of André Weil's, that there is a deep connection between the topological characteristics of a variety and its diophantine (number theoretic) properties. For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers. Weil realized that to prove such a connection one needed a new cohomology theory, but neither he nor any other expert saw how to do this until such a theory was found by Grothendieck. This program culminated in the proofs of the Weil conjectures by Grothendieck's student Pierre Deligne in the early 1970s after Grothendieck had largely withdrawn from mathematics.

Major mathematical topics (from Récoltes et Semailles)
He wrote a retrospective assessment of his mathematical work (see the external link La Vision below). As his main mathematical achievements ("maître-thèmes"), he chose this collection of 12 topics (his chronological order):

Topological tensor products and nuclear spaces
"Continual" and "discrete" duality (derived categories and "six operations").
Yoga of the Grothendieck-Riemann-Roch theorem (K-theory, relation with intersection theory).
Schemes.
Toposes.
Étale cohomology including l-adic cohomology.
Motives and the motivic Galois group (and Grothendieck categories)
Crystals and crystalline cohomology, yoga of De Rham and Hodge coefficients.
Topological algebra,

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