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 .
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