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30 D. LascarThere is a natural topology on Gal(T) (see [19] for details). It can be definedin two different ways.My favorite one is via the ultraproduct construction. Let (7, ; i £ I) be afamily of elements of Gal(T) and U an ultrafilter on J. Choose a saturated modelM and, for each i £ L an automorphism fi £ Aut(M) lifting 7$. Consider theultrapower M' = Yl i€U M. We can define the automorphism Y\ i€U fi on M'. Thisautomorphism acts on Bdd(M') = Bdd(T), so defines an element of Gal(T), say ß.This element ß should be considered as a limit of the family (7$ ; i £ I) along U.A subset X of Gal(T) is closed for the topology we are defining if it is closed forthis limit operation. You should be aware that the element ß may depend on thechoices of the /,'s, because the topology we are defining is not Hausdorff in general.The other way to define a topological structure on Gal(T) is to define a topologyon Bdd(T). If, as it is the case when T is stable, Gal(T) can be identified with agroup of permutation on the set of imaginary elements, then we just endow Gal(T)with the pointwise convergence topology (that is we consider the set of imaginaryelementswith the discrete topology). Otherwise, it is more complicated, and hereis what should be done in general:For each n ofM n /E is closed if and only if tp^(X) is the intersection of a family of subsetsdefinable with parameters. Gal(T) acts on M n /E and the topology on Gal(T) isdefined as the coarsest topology which makes all these actions (with various n andE) continuous.Now, we can prove:Theorem 5 1. Gal(T) is a topological compact group.2. It is Hausdorff if and only if it acts faithfully on the set of bounded hyperimaginaryelements, if and only if it acts faithfully on the set of finitary boundedhyperimaginary elements.3. It is profinite if and only if it acts faithfully on the set of bounded imaginaryelements.There is a Galois correspondence between the subgroups of Gal(T) and thebounded ultra-imaginary elements: every subgroup of Gal(T) is the stabilizer ofan ultra-imaginary element. The hyperimaginary elements correspond to the closedsubgroups and the imaginary elements correspond to the clopen subgroups of Gal (T).Let H 0 be the topological closure of the identity. Then H 0 is a normal subgroupof Gal(T). If we consider Gal(T) as a permutation group on Bdd(T), H 0is exactly the pointwise stabilizer of the set of bounded hyperimaginary elements.So, if we set GOIQ(T) = Gal(T)/H 0 , GOIQ(T) acts faithfully on the set of boundedhyperimaginary elements. As a quotient group, GOIQ(T) is canonically endowedwith a topology. This way, we get a compact Hausdorff group.Recently, L. Newelski ([22]) has proved that H 0 is either trivial or of cardinality2 N °.I would like to conclude this section by a conjecture. In all the known examplesof countable saturated structures where the small index property is false, there is a
Automorphism Groups of Saturated Structures 31non open subgroup of G al (T) of countable index (and, its preimage by the canonicalhomomorphism from Aut(M) onto Gal(T) is a non open subgroup of Aut(M) ofcountable index). If A has a cardinality strictly less than card(M), define AUì/A(M)as the subgroup of AUìA(M) generated by\J{Aut N (M) ; AÇN - K 0 . The smallindex property has a natural generalization. If we assume that A
Page 1 and 2: Beijing 2002August 20-28Proceedings
Page 3 and 4: ContentsSection 1. LogicE. Bouscare
Page 5 and 6: R. Pandharipande: Three Questions i
Page 7 and 8: Section 1. LogicE. Bouscaren: Group
Page 9 and 10: 4 E. Bouscaren2. Different notions
Page 11 and 12: 6 E. BouscarenWe now consider an al
Page 13 and 14: 8 E. Bouscarendimension would need
Page 15 and 16: 10 E. Bouscaren1. Either G is not o
Page 17 and 18: 12 E. Bouscaren[7] E. Bouscaren & F
Page 19 and 20: 14 J. Denef F. Loeserthis theory to
Page 21 and 22: 16 J. Denef F. Loeserwill require m
Page 23 and 24: 18 J. Denef F. Loeserseries P P (T)
Page 25 and 26: 20 J. Denef F. Loeserfor all m> n.
Page 27 and 28: 22 J. Denef F. Loeseri > 1. One ver
Page 29 and 30: ICM 2002 • Vol. II • 25-33Autom
Page 31 and 32: Automorphism Groups of Saturated St
Page 33: Automorphism Groups of Saturated St
Page 37 and 38: Automorphism Groups of Saturated St
Page 39 and 40: ICM 2002 • Vol. II • 37-45Repre
Page 41 and 42: 3. The Blanchfield pairingRepresent
Page 43 and 44: Representations of Braid Groups 41n
Page 45 and 46: Representations of Braid Groups 43A
Page 47 and 48: Representations of Braid Groups 45T
Page 49 and 50: 48 A.Bondal D.Orlovis believed to b
Page 51 and 52: 50 A.Bondal D.OrlovDefinition 4 [BK
Page 53 and 54: 52 A.Bondal D.OrlovLet Y be a smoot
Page 55 and 56: 54 A.Bondal D.OrlovIn particular, w
Page 57 and 58: 56 A.Bondal D.Orlov[BVdB] Bondal A.
Page 59 and 60: 58 M. Levine(PB) Let £ be a rank r
Page 61 and 62: 60 M. Levine2. Let 1Z d%m (X) be th
Page 63 and 64: 62 M. LevineNow, if P = P(ci,. •
Page 65 and 66: 64 M. LevineProof. For CH*, this us
Page 67 and 68: 66 M. Levineis an isomorphism. Sinc
Page 69 and 70: 68 Cheryl E. Praegeranalysis. Analo
Page 71 and 72: 70 Cheryl E. Praegers-arc-transitiv
Page 73 and 74: 72 Cheryl E. Praegercase a good kno
Page 75 and 76: 74 Cheryl E. Praeger5. Simple group
Page 77 and 78: 76 Cheryl E. Praeger[15] C. H. Li,
Page 79 and 80: 78 Markus Rost1. Norm varietiesAll
Page 81 and 82: 80 Markus RostWe apply the degree f
Page 83 and 84: 82 Markus Rost4. Hubert's 90 for sy
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84 Markus RostFor n = 2 one can tak
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ICM 2002 • Vol. II • 87^92Dioph
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Diophantine Geometry over Groups an
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Diophantine Geometry over Groups an
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ICM 2002 • Vol. II • 93^103None
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Noncommutative Projective Geometry
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106 Dimitri Tamarkinalgebra and def
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108 Dimitri TamarkinThe product on
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110 Dimitri TamarkinProof. Let F =
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112 Dimitri Tamarkin2.5.9.Similarly
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114 Dimitri TamarkinComputeJ2,jl(a
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116 Dimitri Tamarkin3.2.1.To descri
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ICM 2002 • Vol. II • 119-128Con
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Converse Theorems, Functoriality, a
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ICM 2002 • Vol. II • 129^138Con
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Constructing and Counting Number Fi
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ICM 2002 • Vol. II • 139-148Ana
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Analyse p-adique et Représentation
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ICM 2002 • Vol. II • 149-162Equ
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Equiv. Bloch-Kato Conjecture and No
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[17;[is;[19;[20;[21[22[23;[24;[25;[
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ICM 2002 • Vol. II • 163-171Tam
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Tamagawa Number Conjecture for zeta
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174 S. S. Kudlaseries with represen
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t.176 S. S. Kudlais the exponential
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178 S. S. Kudlawhere Z = J2ì n ìP
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180 S. S. Kudla(ii) T £ Sym 2 (Z)>
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182 S. S. KudlaTheorem 6. ([13], [9
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Elliptic Curves and Class Field The
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198 E. UllmoSoit X une variété al
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200 E. UllmoThéorème 2.4 Soit X u
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202 E. UllmoQuestion 4.1 Soit x n =
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204 E. UllmoNotons que cet énoncé
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206 E. Ullmo[17] H. Oh. Uniform Poi
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208 T. D. WooleyWaring's problem as
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210 T. D. Wooleycircumstances one h
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212 T. D. Wooleywith 1 < z,w < F, M
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214 T. D. Wooleyand also byl f \K(a
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216 T. D. Wooleytions involving nor
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Section 4. Differential GeometryB.
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222 B. Andrewspotentially applicabl
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224 B. Andrewssurface can look metr
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226 B. Andrewsboundary of the set {
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228 B. Andrewseach t > 0, to either
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230 B. AndrewsUSSR Izv. 20 (1983).[
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232 Robert Bartnikprovides an impor
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234 Robert BartnikTheorem 2 If (M,
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236 Robert BartnikThe horizon condi
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238 Robert BartnikTheorem 8 Suppose
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240 Robert BartnikPhysics, LNP 212.
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242 P. Biran2. Various intersection
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244 P. BiranTheorem E'. Let (M,oj)
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246 P. Biransymplectic packings in
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248 P. Birangroup of Hamiltonian di
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250 P. Birannumber of steps it take
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252 P. Biranbe used to obtain many
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254 P. Biran[6] P. Biran, A stabili
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ICM 2002 • Vol. II • 257-271Bla
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Black Holes and the Penrose Inequal
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[23;[24;[25;[26;[27[28;[29'[30^Blac
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274 Xiuxiong ChenFor simplicity, in
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276 Xiuxiong ChenTheorem 0.1. [14]
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278 Xiuxiong ChenQuestion 0.6. (Don
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280 Xiuxiong Chencurvature in S 2 c
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282 Xiuxiong Chen[12] X. X. Chen an
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284 Weiyue DingSehrödinger flows a
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286 Weiyue DingIn order to prove Th
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288 Weiyue DingFor the first term i
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290 Weiyue Dingwhich satisfies the
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ICM 2002 • Vol. II • 293^302Dif
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Differential Geometry via Harmonic
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then eitherorDifferential Geometry
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ICM 2002 • Vol. II • 303-313Ind
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Index Iteration Theory for Symplect
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316 Anton PetruninTheorem A. Given
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318 Anton PetruninRiemannian manifo
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320 Anton Petruninspace. In the cas
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ICM 2002 • Vol. II • 323-338Col
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Collapsed Riemannian Manifolds with
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ICM 2002 • Vol. II • 339-349Com
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Complex Hyperbolic Triangle Groups
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352 Paul Seideldeformation spaces.
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354 Paul Seidelforgetting all the p
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356 Paul SeidelTheorem 3. Si,...,S
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358 Paul Seidelwhich must be of ord
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360 Paul Seidel[10] E. Getzler, Bat
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362 Weiping Zhang(iii) The heat ker
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364 Weiping Zhang3. The index theor
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366 Weiping Zhangwhere ch(g) is the
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- T p ( P d M , > 0 , 9 P d M , > 0
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Section 5. TopologyMladen Bestvina:
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374 Mladen Bestvinaof Out(F n ) coc
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376 Mladen Bestvina3. Culler-Vogtma
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378 Mladen BestvinaDefinition 8. A
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380 Mladen Bestvinaand then gluing
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382 Mladen BestvinaAnal. 10 (2000),
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384 Mladen Bestvina63 John R. Stall
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386 Yu. V. Chekanovplu I,Figure 1:
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388 Yu. V. Chekanovwithout loss of
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390 Yu. V. ChekanovFigure 2: Lagran
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392 Yu. V. ChekanovFigure 5: Local
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394 Yu. V. Chekanovderived from the
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396 M. Furatadimensional vector spa
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398 M. FurataHowever in the Seiberg
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400 M. Furata6. Seiberg-Witten-Floe
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402 M. Furata[14] K. Fukaya & K. On
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ICM 2002 • Vol. II • 405-114Gé
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Géométrie de Contact 407- la sph
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Géométrie de Contact 4093) chaque
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412 E. Giroux- en tout point p où
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414 E. Giroux[EU] Y. ELIASHBERG, Cl
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416 L. Hesselholt1. Algebraic üC-t
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418 L. Hesselholt(i) a pro-log diff
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420 L. HesselholtThe canonical map
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422 L. Hesselholttogether with a mu
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424 L. Hesselholtwhere \'- GK —^
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ICM 2002 • Vol. II • 427-436Sym
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Symplectic Sums and Gromov-Witten I
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ICM 2002 • Vol. II • 437-146Kno
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Knots, von Neumann Signatures, and
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ICM 2002 • Vol. II • 447-156Str
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Strings and the Stable Cohomology o
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ICM 2002 • Vol. II • 457-168Non
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Non-zero Degree Maps between 3-Mani
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Section 6. Algebraic and Complex Ge
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472 Hélène Esnaultto ^klX)/k carr
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474 Hélène Esnaultf*c 2 (E, V) is
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476 Hélène EsnaultTheorem 3.1 ([5
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478 Hélène Esnaultno longer the c
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480 Hélène EsnaultQuestion 5.4. W
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ICM 2002 • Vol. II • 483-494Hil
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Hilbert Schemes of Points on Surfac
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and use this to define operatorsHil
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ICM 2002 • Vol. II • 495-502Vec
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Vector Bundles on a K3 Surface 497s
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Vector Bundles on a K3 Surface 499A
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Vector Bundles on a K3 Surface 501a
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ICM 2002 • Vol. II • 503-512Thr
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Three Questions in Gromov-Witten Th
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ICM 2002 • Vol. II • 513^524Upd
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Update on 3-folds 515in the hyperbo
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Update on 3-folds 517In many contex
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Update on 3-folds 5193.3. Explicit
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Update on 3-folds 521The modem view
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Update on 3-folds 523is entertainin
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ICM 2002 • Vol. II • 525-532Sur
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Sur les Algèbres Vertex Attachées
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ICM 2002 • Vol. II • 533-541Top
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Topology of Singular Algebraic Vari
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ICM 2002 • Vol. II • 545-554Har
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Harmonie Analysis on Real Reductive
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[BS3][BS4][Be]Harmonie Analysis on
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ICM 2002 • Vol. II • 555-570On
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On the Dynamical Yang-Baxter Equati
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ICM 2002 • Vol. II • 571-582Geo
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Geometrie Langlands Correspondence
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ICM 2002 • Vol. II • 583-597On
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On the Local Langlands Corresponden
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600 A. KlyachkoAmong commonly known
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602 A. KlyachkoBy Theorem 2.3 this
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604 A. Klyachkois stable iff for ev
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606 A. KlyachkoUnitary spectral pro
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608 A. Klyachkospectra. By Metha-Se
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610 A. KlyachkoAgain our experience
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612 A. Klyachko[9] G. Faltings, Mum
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ICAl 2002 • Vol. II • 615-627Br
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Branching Problems of Unitary Repre
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630 Vikram Bhagvandas AlehtaDefinit
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632 Vikram Bhagvandas Alehtais in t
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634 Vikram Bhagvandas AlehtaTheorem
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ICAl 2002 • Vol. II • 637^642Cl
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Clifford Algebras and the Duflo Iso
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Clifford Algebras and the Duflo Iso
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ICAl 2002 • Vol. II • 643^654Re
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Representations of Yangians 6451.2.
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Representations of Yangians 6478 9
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Representations of Yangians 6492. T
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Representations of Yangians 651irre
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Representations of Yangians 6532.4.
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ICAl 2002 • Vol. II • 655-666Au
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Automorphic L-functions and Functor
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ICAl 2002 • Vol. II • 667^677Mo
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Modular Representations 669(l.b.2)
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Modular Representations 671e) Any p
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Modular Representations 673conjugac
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Modular Representations 675A genera
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Modular Representations 677^-repres
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ICAl 2002 • Vol. II • 681-690Va
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Value Distribution and Potential Th
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ICAl 2002 • Vol. II • 691-700Th
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The Branch Set of a Quasiregular Al
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ICAl 2002 • Vol. II • 701-709Ha
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Harmonie Aleasure and "Locally Flat
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712 N. Lerner1. From Hans Lewy to N
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714 N. Lernergood cases in (1.5) (w
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716 N. Lerner2. Counting the loss o
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718 N. LernerSolvability with loss
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720 N. Lerner6] L.Hörmander, On th
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722 C. ThieleA basic point of singu
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724 C. ThieleFigure 1: "Cone"< en I
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726 C. Thieleis negative. If all of
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728 C. Thielealso in these degenera
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730 C. ThieleThus he needed good co
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732 C. Thiele[6] Gilbert J., Nahmod
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I p : — \ J ( Z i , . . . , Z m )
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736 S. ZelditchTheorem 1 [2] The pa
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738 S. Zelditchfor U C
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740 S. Zelditch[Pi,Pj] = 0 and whos
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742 S. Zelditch[11] V. F. Lazutkin,
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744 Xiangyu Zhou+00 at the boundary
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746 Xiangyu Zhouconnected component
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748 Xiangyu ZhouWightman [32], Jost
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750 Xiangyu ZhouBrief proof is as f
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752 Xiangyu Zhou[11] Al. Jarnicki,
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Section 9. Operator Algebras andFun
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758 Semyon Alesker0.1.1 Definition,
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760 Semyon Alesker2. Translation in
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762 Semyon Alesker2.3. Valuations i
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764 Semyon AleskerReferences[1] Ale
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766 P. Bianewith classical cumulant
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768 P. BianeObserve thatr(ai ... a
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770 P. Biane4. Noncrossing cumulant
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772 P. Bianethis gives the order of
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774 P. Biane[20] R. Speicher, Combi
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776 D. Bischsubfactors has develope
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778 D. Bisch(for all k > 0), where
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780 D. BischObserve that by constru
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782 D. BischIt should be evident th
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784 D. Bisch[4] D. Bisch, Bimodules
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ICAl 2002 • Vol. II • 787^794Fr
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Free Probability, Free Entropy and
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ICAl 2002 • Vol. II • 795^812Ba
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Banach KK-theory and the Baum-Conne
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ICAl 2002 • Vol. II • 813-822On
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On Some Inequalities for Gaussian A
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Author IndexAlesker, Semyon 757Andr
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