show how the solution of old problems fell out in straightforward ways now that their real nature had been revealed. By the 1970s, Grothendieck's work was seen as influential, not only in algebraic geometry and the allied fields of sheaf theory and homological algebra, but influenced logic, in the field of categorical logic. Geometry Grothendieck approached algebraic geometry by clarifying the foundations of the field, and by developing mathematical tools intended to prove a number of notable conjectures. Algebraic geometry has traditionally meant the understanding of geometric objects, such as algebraic curves and surfaces, through the study of the algebraic equations for those objects. Properties of algebraic equations are in turn studied using the techniques of ring theory. In this approach, the properties of a geometric object are related to the properties of an associated ring. The space (e.g., real, complex, or projective) in which the object is defined, is extrinsic to the object, while the ring is intrinsic. Grothendieck laid a new foundation for algebraic geometry by making intrinsic spaces ("spectra") and associated rings the primary objects of study. To that end, he developed the theory of schemes that informally can be thought of as topological spaces on which a commutative ring is associated to every open subset of the space. Schemes have become the basic objects of study for practitioners of modern algebraic geometry. Their use as a foundation allowed geometry to absorb technical advances from other fields. His generalization of the classical Riemann–Roch theorem related topological properties of complex algebraic curves to their algebraic structure and now bears his name, being called "the Grothendieck–Hirzebruch–Riemann–Roch theorem". The tools he developed to prove this theorem started the study of algebraic and topological K-theory, which explores the topological properties of objects by associating them with rings. After direct contact with Grothendieck's ideas at the Bonn Arbeitstagung, topological K-theory was founded by Michael Atiyah and Friedrich Hirzebruch. 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