Branches of Mathematics
Algebraic K-Theory
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Algebraic K-theory is a branch of mathematics connecting geometry, topology, ring theory and number theory, assigning to a geometric, algebraic or arithmetic object a sequence of abelian groups called its K-groups.
Facts
Central QuestionWhat the K-groups of a given ring or space actually are, a question the field's own methods make notoriously difficult to answer even for objects as basic as the ring of integers, whose higher K-groups remain an outstanding computational problem. 1 Key DebateHow far Alexander Grothendieck's single Grothendieck group, K0, defined in his study of intersection theory on algebraic varieties in the late 1950s, could be extended to a whole sequence of higher K-groups capturing deeper information, an extension that took over a decade of further work after Grothendieck's own original definition to complete. 1 Cross-Tradition Connections
Sources
1. Algebraic K-Theory (Wikipedia)
WikipediaOpening paragraph, first sentenceQuote, Opening paragraph, first sentence
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups.
1. Algebraic K-Theory (Wikipedia)
WikipediaOpening paragraph, on K-group difficultyQuote, Opening paragraph, on K-group difficulty
They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.
1. Algebraic K-Theory (Wikipedia)
WikipediaHistory / The Grothendieck group K0Quote, History / The Grothendieck group K0
K-theory was discovered in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only K0, the zeroth K-group, but even this single group has plenty of applications.
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