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Grothendieck's Contributions In The 1950s
He generalized differences to schemes and extending the Riemann-Roch theorem. -
His Contribution In 1955
He started to work on sheaf theory and homological algebra. Sheaf theory is a way to organize information over certain pieces of a shape, so everything is put into place smoothly when joined together. Similar to sheaf theory, homological algebra is a part of math that uses algebra to get information about shapes by breaking them down into simpler pieces. -
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Grothendieck's Contributions In The 1960s
He introduced the theory of schemes, which allowed certain of Weil's number theory concepts to be solved. -
His Contribution In 1961
He consolidated themes in geometry, number theory, topology and complex analysis. Topology is a study of geometric properties and dimensional correlation unaffected by the continuous change of shape or size of figures. Complex analysis is the study of complex numbers together with their byproduct, manipulation, and other properties. -
His Contributions In 1962
He introduced the idea of K-theory which is the study of a ring that classifies vector bundles over a space. Vector bundles allow us inspect geometric properties from a more algebraic point of view. He also revolutionized homological algebra.