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Invariants of Legendrian Knots 391ators in all monomials. Thus, if K and K are Legendrian isotopie then the gradedhomology ring H (A, 9) is anti-isomorphic to itself, and there exist £+,£!_ G H (A, 9)such that deg(£|_) = 1, deg(£L) = —1, £-£+ = 1. But one can check that suchclasses do not exist (see [14, 15] for details) and hence K and K are not Legendrianisotopie. Note that 'first order invariants' such as Poincaré polynomials are uselessin distinguishing Legendrian mirror knots.3. Admissible decompositions of fronts3.1. DefinitionsIn this section, we present the invariants of Legendrian knots constructed in [2].These invariants are defined in terms of the front projection.Given a a-generic oriented Legendrian knot L, denote by C(L) the set ofits points corresponding to cusps of a(L). The Maslov index p:L\ C(L) —¥ Y =Z/ro(L)Z is a locally constant function, uniquely defined up to an additive constantby the following rule: the value of p jumps at points of C(L) by ±1 as shown inFigure 4. We call a crossing of S = a(L) Maslov if p takes the same value on bothits branches.ß = i+l~^^ / \ ^~-li = i+l[1 = i-^ \ / ^-// = iFigure 4: Jumps of the Maslov index near cuspsAssume that S = a(L) is a union of closed curves Xi,..., X n that have finitelymanyself-intersections and meet each other at finitely many points. Then we callthe unordered collection {Xi,... ,X n } a decomposition of S. A decomposition{Xi,...,X n } is called admissible if it satisfies certain conditions, which we aregoing to define. The first two are as follows:(1) Each curve Xi bounds a topologically embedded disk: X t = dB t .(2) For each i £ {l,...,n}, q £ R, the set Bi(q) = {« G R | (q,u) £ Bi} is eithera segment, or consists of a single point « such that (q,u) is a cusp of S, or isempty.Conditions (1) and (2) imply that each curve X t has exactly two cusps (and hencethe number of curves is half the number of cusps). Each X t is divided by cuspsinto two pieces, on which the coordinate q is a monotone function. Near a crossingx £ XiCi Xj, the decomposition of S may look in one of the three ways representedin Figure 5. Conditions (1) and (2), in particular, rule out the decomposition shownin Figure 5a. We call the crossing point x switching if X t and Xj are not smoothnear x (Figure 5b), and non-switching otherwise (Figure 5c).(3) If (qo, u) £ Xi fl Xj (i ^ j) is switching then for each q ^ q 0 sufficiently closeto qo the set Bi(q) fl Bj(q) either coincides with Bi(q) or Bj(q), or is empty.
392 Yu. V. ChekanovFigure 5: Local decompositions(4) Every switching crossing is Maslov.We call a decomposition admissible if it satisfies Conditions (l)-(3), and gradedadmissible if it also satisfies Condition (4). Denote by Adm(S) (resp. Adm + (S))the set of admissible (resp. graded admissible) decompositions of S. Given D £Adm(S), denote by Sw(£>) the set of its switching points. Define 0(D) = #(-D)) —#(Sw(D)).Theorem 3.1. If a-generic Legendrian knots L,L' c R 3 are Legendrian isotopiethen there exists a one-to-one mapping g: Adm(
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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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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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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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Differential Geometry via Harmonic
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then eitherorDifferential Geometry
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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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Page 357 and 358: Section 5. TopologyMladen Bestvina:
Page 359 and 360: 374 Mladen Bestvinaof Out(F n ) coc
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Page 369 and 370: 384 Mladen Bestvina63 John R. Stall
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Page 373 and 374: 388 Yu. V. Chekanovwithout loss of
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Knots, von Neumann Signatures, and
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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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