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Topology: Algebraic Topology
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Topology is a qualitative study of shapes and other mathematical objects. It has evolved from features of geometry where specific measurments are not relevant. Modern topology uses tools from algebra and heavily interacts with other branches of mathematics such as dynamical systems, analysis, or algebraic geometry.
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Topology is a large branch of mathematics that includes many subfields. The most basic division within topology is point-set topology, which investigates such concepts as compactness, connectedness, and countability; algebraic topology, which investigates such concepts as homotopy, homology; and geometric topology, which studies manifolds and their embeddings, including knot theory.
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There is an excellent, if somewhat dated, collection of "Reviews in Topology" by Norman Steenrod, a sorted collection of the relevant reviews from Math Reviews (1940-1967). Many now-classical results date from that period. Most of the reviews in that collection are of algebraic and differential topology....
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Topology at Columbia University has enjoyed a long tradition. Illustrious professors from the past include Samuel Eilenberg, who is responsible for the foundations of algebraic topology, and Lipman Bers, whose ideas in complex variables played an influential role in Thurston's program for three-dimensional manifolds. The senior faculty in the topology group at Columbia are Joan Birman (emerita), Troels Jorgenson, Mikhail Khovanov, John Morgan, Walter Neumann, and Peter Ozsváth; junior faculty include Ciprian Manolescu and Dylan Thurston; and there is ... a number of post-doctoral researchers and visitors. The closest connections with the research interests other mathematicians not strictly in the topology group, include David Bayer, Robert Friedman, Brian Greene, Richard Hamilton, Melissa Liu, and Michael Thaddeus. The topology group has a number of informal student seminars, a regular Topology seminar (which often meets twice on Fridays), and also a gauge theory seminar which meets on Fridays.
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Algebraic Topology