FOREWORD x i
in gravitational vacuum) , whereas the Galoi s avata r definitel y belong s
to number theory .
In th e Marcolli lectures , stressin g numbe r theory , physic s theme s
pop u p a t th e en d o f Chapte r 2 (Chaoti c Cosmolog y i n genera l rela -
tivity), th e beginnin g o f Chapte r 3 (formalis m o f quantu m statistica l
mechanics), and finally, sec. 5 of Chapter 4 where some models of black
holes in general relativity tur n ou t t o have the sam e mathematical de -
scription a s oo-adic fibers of curves in Arakelov geometry. Th e reemer -
gence o f Gauss ' Galoi s grou p Gal
a6
(Q/Q) i n Bost-Conne s symmetr y
breaking, an d o f Gauss' statistics of continued fraction s i n the Chaoti c
Cosmology models, shows that connection s with classical mathematic s
are a s strong a s ever .
Hopefully, thi s livel y expositio n wil l attrac t youn g researchers an d
incite the m t o engag e themselve s i n exploratio n o f the ric h ne w terri -
tory.
Yuri I. Manin.
Bonn, March 17, 2005.
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