theory, but neither he nor any other expert saw how to accomplish this until such a theory was expressed by Grothendieck. This program culminated in the proofs of the Weil conjectures, the last of which was settled by Grothendieck's student Pierre Deligne in the early 1970s after Grothendieck had largely withdrawn from mathematics. Major mathematical contributions In Grothendieck's retrospective Récoltes et Semailles, he identified twelve of his contributions that he believed qualified as "great ideas". In chronological order, they are: Topological tensor products and nuclear spaces "Continuous" and "discrete" duality (derived categories, "six operations") Yoga of the Grothendieck–Riemann–Roch theorem K-theory relation with intersection theory Schemes Topoi Étale cohomology and l-adic cohomology Motives and the motivic Galois group (Grothendieck ⊗-categories) Crystals and crystalline cohomology, yoga of "de Rham coefficients", "Hodge coefficients"... "Topological algebra": ∞-stacks, derivators; cohomological formalism of topoi as inspiration for a new homotopical algebra Tame topology Yoga of anabelian algebraic geometry, Galois–Teichmüller theory "Schematic" or "arithmetic" point of view for regular polyhedra and regular configurations of all kinds Here the term yoga denotes a kind of "meta-theory" that may be used heuristically; Michel Raynaud writes the other terms "Ariadne's thread" and "philosophy" as effective equivalents. Grothendieck wrote that, of these themes, the largest in scope was topoi, as they synthesized algebraic geometry, topology, and arithmetic. The theme that had been most extensively developed was schemes, which were the framework "par excellence" for eight of the other themes (all but 1, 5, and 12). Grothendieck wrote that the first and last themes, topological tensor products and regular configurations, were of more modest size than the others. Topological tensor products had played the role of a tool rather than of a source of inspiration for further developments; but he expected that regular configurations could not be exhausted within the lifetime of a mathematician who devoted oneself to it. He believed that the deepest themes were motives, anabelian geometry, and Galois–Teichmüller theory. Influence Grothendieck is considered by many to be the greatest mathematician of the twentieth century. In an obituary David Mumford and John Tate wrote: Although mathematics became more and more abstract and general throughout the 20th century, it was Alexander Grothendieck who was the greatest master of this trend. His unique skill was to eliminate all unnecessary hypotheses and burrow into an area so deeply that its inner patterns on the most abstract level revealed themselves–and then, like a magician,