Introduction
The "infinite-dimensiona l groups " i n th e titl e o f thi s se t o f lectur e note s refe r
to a vas t collectio n o f concret e group s o f automorphism s o f variou s mathematica l
objects. Example s includ e unitar y group s o f Hilbert space s and , mor e generally, of
operator algebras , the infinite symmetri c group, groups of homeomorphisms, group s
of automorphism s o f measur e spaces , isometr y group s o f variou s metri c spaces ,
etc. Typicall y suc h group s ar e supportin g additiona l structures , suc h a s a natura l
topology, or sometimes the structure of an infinite-dimensional Li e group. I n fact, i t
appears that thos e additional structures are often encode d in the algebraic structur e
of the groups in question. Th e full exten t o f this phenomenon i s still unknown, but ,
for instance , a recen t astonishin g resul t o f Kechri s an d Rosenda l [123] state s tha t
there i s onl y on e non-trivia l separabl e grou p topolog y o n th e infinit e symmetri c
group Soo . I t coul d als o well be tha t ther e ar e relevan t structure s o n thos e group s
that hav e no t ye t bee n formulate d explicitely .
Those infinite-dimensiona l group s ar e bein g studie d fro m a numbe r o f differ -
ent perspective s i n variou s part s o f mathematics : mos t notably , representatio n
theory [170, 161] , logic an d descriptiv e se t theor y [14, 110], infinite-dimensiona l
Lie theor y [103, 171] , theor y o f topologica l group s o f transformation s [7 , 229 ]
and abstrac t harmoni c analysi s [21, 47] , as wel l a s fro m a purel y grou p theoreti c
viewpoint [11 , 22 ] an d fro m tha t o f set-theoreti c topolog y [206] . Together , thos e
studies mar k a ne w chapte r i n th e theor y o f group s o f transformations , a pos t
Gleason-Yamabe-Montgomery-Zippin developmen t [158].
The presen t se t o f lectur e note s outline s a n approac h t o th e stud y o f infinite -
dimensional group s base d o n som e ideas originatin g i n geometric functiona l analy -
sis, an d explorin g a n interpla y betwee n th e dynamica l propertie s o f thos e groups ,
combinatorial Ramsey-typ e theorems, an d geometry of high-dimensional structure s
(asymptotic geometri c analysis) .
The theor y i n questio n i s a result o f several developments , som e of which hav e
happened independentl y of , an d i n paralle l to , eac h other , s o i t i s a "partiall y
ordered set. " O n th e othe r hand , u p unti l no w those development s wer e invariabl y
absorbed int o th e theor y an d pu t int o connectio n t o eac h other , s o th e se t is ,
hopefully, "directed. "
It i s no t clea r ho w fa r bac k i n tim e on e need s t o g o toward s th e beginnings .
One o f th e mai n component s o f th e theor y th e phenomeno n o f concentratio n
of measure, th e subjec t o f study o f the modern-da y asymptoti c geometri c analysi s
wa s explicitel y describe d an d studie d i n a 1922 boo k b y Pau l Lev y [127] fo r
Euclidean spheres . However , anothe r majo r manifestatio n o f the phenomenon - th e
Law of Large Numbers - wa s stated by Emile Borel in the modern form alread y back
in 1904, see a discussion in [150]. An d accordin g to Gromov [93] , the concentratio n
l
http://dx.doi.org/10.1090/ulect/040/01
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