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Sunday, November 24, 2013

Galois Groups and Harmonic Analysis

Filed under: General,Geometry — Tags: — m759 @ 9:29 AM

"In 1967, he [Langlands] came up with revolutionary
insights tying together the theory of Galois groups 
and another area of mathematics called harmonic
analysis. These two areas, which seem light years
apart
, turned out to be closely related."

— Edward Frenkel, Love and Math, 2013

"Class field theory expresses Galois groups of
abelian extensions of a number field F
in terms of harmonic analysis on the
multiplicative group of [a] locally compact
topological ring, the adèle ring, attached to F."

— Michael Harris in a description of a Princeton
    mathematics department talk of October 2012

Related material: a Saturday evening post.

See also Wikipedia on the history of class field theory.
For greater depth, see Tate's [1950] thesis and the book
Fourier Analysis on Number Fields .

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