By: Jimmy Ortega
MM103 College Mathematics
Alexander Grothendiek, is one of the most important mathematicians of the twentieth century. Grothendieck shot to the top of the mathematical world in the 1950s, when he was only in his twenties. Leaving aside an initial interest in analysis, he had started to work on problems in algebraic geometry, an area of math that, as the name suggests, uses algebra to study geometric objects. The branch of category theory is now called Grothendieck topologies, and conducted research in algebraic geometry, including the introduction of étale cohomolgy and proof of the Weil conjectures. His work unified complex analysis, geometry, number theory, and topology, and spanned a breathtaking spectrum of mathematics, including the cohomological interpretation of L-functions, crystalline cohomology, derived categories, formalisms for local and global duality (the 'six operations'), functional analysis, geometric objects via represent able functions, sheaf cohomology as derived functions, and the 'yoga of weights'. Cohomology is an invariant of a topological space, formally "dual" to homology, and so it detects "holes" in a space. Cohomology has more algebraic structure than homology, making it into a graded ring (with multiplication given by the so-called "cup product"), whereas homology is just a graded Abelian group invariant of a space (Weisstein, 2012).The overlaps of geometry and algebra are familiar even to school students. A circle of radius 1, for example, can be described, as a result of Pythagoras's theorem, by the algebraic equation x2 + y2 = 1, and the equation y = x2 describes a parabola. More complicated algebraic expressions describe more complex and higher-dimensional geometric objects. In its most general form, this idea is encapsulated in mathematical objects called algebraic varieties. It is work on just these varieties that propelled Grothendieck to mathematical stardom. In...