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.