CHAPTER 1
Ouverture
Noncommutative geometry , a s developed by Conne s starting i n th e
early '80 s ([17], [19], [22]) , extends th e tool s o f ordinary geometr y t o
treat space s that ar e quotients, for which the usual "rin g of functions" ,
defined a s functions invarian t wit h respec t t o the equivalenc e relation ,
is to o smal l t o captur e th e informatio n o n th e "inne r structure " o f
points in the quotient space . Typically , for such spaces functions o n the
quotients ar e just constants , while a nontrivial ring of functions, whic h
remembers th e structur e o f th e equivalenc e relation , ca n b e define d
using a noncommutative algebr a o f coordinates, analogou s to the non -
commuting variable s o f quantum mechanics . Thes e "quantu m spaces "
are defined b y extending th e GePfand-Naimar k correspondenc e
X loc.comp . Hausdorf f spac e ^ Co(X) abelia n C*-algebr a
by dropping the commutativit y hypothesi s i n the right han d side . Th e
correspondence the n become s a definitio n o f wha t i s on th e lef t han d
side: a noncommutative space .
Such quotient s ar e abundan t i n nature . The y arise , fo r instance ,
from foliations . Severa l recen t result s als o sho w tha t noncomniuta -
tive space s aris e naturally i n number theor y an d arithmeti c geometry .
The first instanc e o f such connections between noncommutativ e geom -
etry an d number theor y emerge d i n the work of Bost an d Conne s [10],
which exhibit s a very interestin g noncommutativ e spac e wit h remark -
able arithmeti c propertie s relate d t o clas s fiel d theory . Thi s reveal s a
very useful dictionar y tha t relate s the phenomena of spontaneous sym -
metry breaking in quantum statistical mechanics to the mathematics of
Galois theory. Thi s spac e can be viewed a s the spac e of 1-dimensional
Q-lattices u p t o scale , modul o th e equivalenc e relatio n o f commensu -
rability (c/ . [33]) . Thi s spac e is closely related t o the noncommutativ e
space used by Connes to obtain a spectral realization of the zeros of the
Riemann zet a function , [24] . I n fact , thi s i s agai n th e spac e o f com -
mensurability classe s o f 1-dimensional Q-lattices , bu t wit h th e scal e
factor als o taken int o account .
More recently, other results that poin t t o deep connections betwee n
noncommutative geometr y an d numbe r theor y appeare d i n th e wor k
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http://dx.doi.org/10.1090/ulect/036/01
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