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. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
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 Classification
Pure or Applied Connections
Associated With
Includes
Source Birch-Tate conjecture (Wikipedia)
Source MacTutor biography of Hyman Bass, St Andrews
Source Wikipedia: John Milnor
Source Weibel's Conjecture (Wikipedia)
Sources
1. Algebraic K-Theory (Wikipedia)
WikipediaOpening 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.
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.
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.
View the Source Birch-Tate conjecture (Wikipedia)
Includes: Birch-Tate Conjecture, LeadQuote, Includes: Birch-Tate Conjecture, Lead
specifically in algebraic K-theory, the Birch-Tate conjecture is a conjecture proposed by Bryan John Birch and John Tate
Weibel's Conjecture (Wikipedia)
Includes: Weibel's Conjecture, LeadQuote, Includes: Weibel's Conjecture, Lead
Weibel's conjecture gives a criterion for vanishing of negative algebraic K-theory groups
Wikipedia: John Milnor
Includes: John Milnor, LeadQuote, Includes: John Milnor, Lead
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems.
MacTutor biography of Hyman Bass, St Andrews
Includes: Hyman Bass, BiographyQuote, Includes: Hyman Bass, Biography
but let us note that he himself gives his research interests as algebraic K-theory; number theory; group theory ( geometric methods ) ; and algebraic geometry.
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