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Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological ...
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. [1] Category theory is used in almost all areas of mathematics.
Andrew Hugh Wallace was born and raised in Edinburgh, Scotland. He received in 1946 an MA in mathematics from Edinburgh University and in 1949 a PhD from St. Andrews University with thesis Rational integral functions and associated linear transformations. [2] In the 1950s he was an assistant professor of mathematics at the University of Toronto ...
In algebraic topology, the fundamental group of a pointed topological space is defined as the group of homotopy classes of loops based at . This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an algebraic variety with the Zariski topology. In the classification of covering spaces, it is ...
Joseph J. Rotman (May 26, 1934 – October 16, 2016 [1]) was a Professor of Mathematics at the University of Illinois at Urbana–Champaign [2] and also a published author of 10 textbooks. Rotman was born in Chicago. He did his undergraduate and graduate work at the University of Chicago, where he received his doctorate in 1959 with a thesis in ...
Sylvia de Neymet. Solomon Lefschetz ForMemRS (Russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was a Russian-born American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations. [ 3 ][ 1 ][ 4 ][ 5 ]
Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through their homology and cohomology.
Algebraic cycle. In mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of the algebraic topology of V that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety.