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Clifford Algebras and the Duflo Isomorphism 641Theorem. [1] The quantization mapQ := (sym®q) o i(Si) : Wg -• Wgintertwines the contraction operators, Lie derivatives, and differentials on Wg andon Wg.The fact that Q intertwines the two differentials d 14 , d w relies on a numberof special properties of the function S e , including the CDYBE.Put differently, the quantization map Q defines a new, graded non-commutativering structure on the Weil algebra Wg, in such a way that the derivations i)fi, Lf , dare still derivations for the new ring structure, and in fact become inner derivations.Notice that Q restricts to the quantization map for Clifford algebras q : Ag —¥ Cl(g)on the second factor and to the Duflo map on the first factor, but is not just theproduct of these two maps.5. Equivariant cohomologyH. Cartan in [3] introduced the Weil algebra Wg as an algebraic model for thealgebra of differential forms on the classifying bundle EG, at least in the case Gcompact.In particular, it can be used to compute the equivariant cohomology HQ(M)(with real coefficients) for any G-manifold M. Yet if h ,Lf h ,d Rh denote the contractionoperators, Lie derivatives, and differential on the de Rham complex 0(Af )of differential forms. LetH B (M) = H((Wg ® tt(M)) basic , d w + d m )where (Wg ® 0(Af ))t> as ic is the subspace annihilated by all Lie derivatives Lf' +L Rh and all contraction operators if' + i Rh . Cartan's result says that H S (M) =HQ(M,R) provided G is compact.Alore generally, we can define H B (A) for any g-differential algebra A. YetR S (A) be defined by replacing Wg with Wg. The quantization map Q : Wg —¥ Wginduces a map Q : H S (A) —¥ R S (A).Theorem. [1] For any g-differential algebra A, the vector space isomorphism Q :H s (Â) —¥ R s (A) is in fact an algebra isomorphism.Our proof is by construction of an explicit chain homotopy between the twomaps Wg ® Wg —¥ Wg given by "quantization followed by multiplication" and"multiplication followed by quantization", respectively. Taking A to be the trivial g-differential algebra (i.e. A = Q(point)), the statement specializes to Duflo's theoremfor quadratic g.References[1] A. Alekseev & E. Aleinrenken, The non-commutative Weil algebra, Invent.Math. 139 (2000), 135^172.
642 E. Aleinrenken[2] A. Alekseev & E. Aleinrenken, Clifford algebras and the classical dynamicalYang-Baxter equation (in preparation).[3] H. Cartan, Notions d'algèbre différentielle; application aux groupes de Lie etaux variétés où opère un groupe de Lie., Colloque de topologie (espaces fibres),Bruxelles, (1950).[4] Al. Duflo, Opérateurs différentiels bi-invariants sur un groupe de Lie, Ann. Sci.École Norm. Sup. 10 (1977), 265^288.[5] P. Etingof & A. Varchenko, Geometry and classification of solutions of theclassical dynamical Yang-Baxter equation, Comm. Math. Phys., 192 (1988),77-120.[6] P. Etingof & O. Schiffmann, On the moduli space of classical dynamical r-matrices, Math. Res. Lett. 8 (2001), 157^170.[7] B. Kostant, A cubic Dirac operator and the emergence of Euler number multipletsof representations for equal rank subgroups, Duke Math. J. 100 (1999),no. 3, 447-501.[8] B. Kostant & S. Sternberg, Symplectic reduction, BRS cohomology, andinfinite-dimensional Clifford algebras, Ann. Physics 176 (1987), no. 1, 49-113.
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Beijing 2002August 20-28Proceedings
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ContentsSection 1. LogicE. Bouscare
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R. Pandharipande: Three Questions i
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Section 1. LogicE. Bouscaren: Group
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4 E. Bouscaren2. Different notions
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6 E. BouscarenWe now consider an al
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8 E. Bouscarendimension would need
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10 E. Bouscaren1. Either G is not o
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12 E. Bouscaren[7] E. Bouscaren & F
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14 J. Denef F. Loeserthis theory to
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16 J. Denef F. Loeserwill require m
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18 J. Denef F. Loeserseries P P (T)
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20 J. Denef F. Loeserfor all m> n.
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22 J. Denef F. Loeseri > 1. One ver
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ICM 2002 • Vol. II • 25-33Autom
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Automorphism Groups of Saturated St
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ICM 2002 • Vol. II • 37-45Repre
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3. The Blanchfield pairingRepresent
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Representations of Braid Groups 41n
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Representations of Braid Groups 43A
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Representations of Braid Groups 45T
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48 A.Bondal D.Orlovis believed to b
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50 A.Bondal D.OrlovDefinition 4 [BK
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52 A.Bondal D.OrlovLet Y be a smoot
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54 A.Bondal D.OrlovIn particular, w
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56 A.Bondal D.Orlov[BVdB] Bondal A.
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58 M. Levine(PB) Let £ be a rank r
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60 M. Levine2. Let 1Z d%m (X) be th
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62 M. LevineNow, if P = P(ci,. •
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64 M. LevineProof. For CH*, this us
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66 M. Levineis an isomorphism. Sinc
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68 Cheryl E. Praegeranalysis. Analo
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70 Cheryl E. Praegers-arc-transitiv
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72 Cheryl E. Praegercase a good kno
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74 Cheryl E. Praeger5. Simple group
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76 Cheryl E. Praeger[15] C. H. Li,
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78 Markus Rost1. Norm varietiesAll
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80 Markus RostWe apply the degree f
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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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Converse Theorems, Functoriality, a
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Constructing and Counting Number Fi
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Analyse p-adique et Représentation
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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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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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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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Page 606 and 607: 630 Vikram Bhagvandas AlehtaDefinit
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Page 646 and 647: Modular Representations 671e) Any p
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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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