Foreword

Noncommutative geometr y nowaday s look s a s a vast buildin g site .

On the one hand, practitioners of noncommutative geometry (o r ge-

ometries) alread y built u p a large and swiftly growin g body o f exciting

mathematics, challengin g traditiona l boundarie s an d subdivisions .

On the other hand , noncommutativ e geometr y lacks common foun -

dations: fo r many interesting constructions of "noncommutative spaces"

we cannot eve n sa y fo r sur e whic h o f them lea d t o isomorphi c spaces ,

because they ar e not object s o f an all-embracin g categor y (lik e that o f

locally ringed topologica l space s in commutativ e geometry) .

Matilde Marcolli' s lectures reflect thi s spiri t o f creative growt h an d

interdisciplinary research .

She start s Chapte r 1 with a sketc h o f philosoph y o f noncommuta -

tive geometr y a la Alai n Connes . Briefly , Conne s suggest s imaginin g

C*-algebras a s coordinat e rings . H e the n supplie s severa l bridge s t o

commutative geometr y b y his construction o f "ba d quotients " o f com-

mutative space s via crossed products an d hi s treatment o f noncommu -

tative Riemannian geometry. Finally , algebraic tools like X-theory an d

cyclic cohomology serv e to furthe r enhanc e geometri c intuition .

Marcolli the n proceed s t o explainin g som e recen t development s

drawing upo n he r recen t wor k wit h severa l collaborators . A commo n

thread i n all of them i s the stud y o f various aspect s o f uniformization :

classical modula r group , Schottk y groups . Th e modula r grou p act s

upon th e complex half plane , partiall y compactifie d b y cusps: rationa l

points o f the boundar y projectiv e line . Th e actio n become s "bad " a t

irrational points , an d her e i s wher e noncommutativ e geometr y enter s

the game . A wealt h o f classica l numbe r theor y i s encode d i n th e co -

efficients o f modula r forms , thei r Melli n transforms , Heck e operator s

and modula r symbols . Thei r counterpart s livin g a t th e noncommuta -

tive boundar y hav e onl y recentl y starte d t o unrave l themselves , an d

Marcolli give s a beautifu l overvie w o f wha t i s alread y understoo d i n

Chapters 2 and 3 .

IX