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ICM 2002 • Vol. II • 221-230Positively Curved Surfacesin the Three-sphereB. Andrews*AbstractIn this talk I will discuss an example of the use of fully nonlinear parabolicflows to prove geometric results. I will emphasise the fact that there is a widevariety of geometric parabolic equations to choose from, and to get the bestresults it can be very important to choose the best flow. I will illustrate thisin the setting of surfaces in a three-dimensional sphere.There are quite a few relevant results for surfaces in the sphere satisfyingvarious kinds of curvature equations, including totally umbillic surfaces,minimal surfaces and constant mean curvature surfaces, and intrinsically flatsurfaces. Parabolic flows can strengthen such results by allowing classes ofsurfaces satisfying curvature inequalities rather than equalities: This was firstdone by Huisken, who used mean curvature flow to deform certain classes ofsurfaces to totally umbillic surfaces. This motivates the question "What is theoptimal result of this kind?" — that is, what is the weakest pointwise curvaturecondition which defines a class of surfaces which retracts to the space ofgreat spheres?The answer to this question can be guessed in view of the examples. Toprove it requires a surprising choice of evolution equation, forced by the requirementthat the pointwise curvature condition be preserved.I will conclude by mentioning some other geometric situations in whichstrong results can be proved by choosing the best possible evolution equation.2000 Mathematics Subject Classification: 53C44, 53C40.Keywords and Phrases: Surfaces, Curvature, Parabolic equations.1. IntroductionMy aim in this talk is to demonstrate the use of fully nonlinear parabolic evolutionequations as tools for proving results in differential geometry. I will emphasisethe fact that there is a wide variety of flows which are geometrically defined and* Centre for Mathematics and its Applications, Australian National University, ACT 0200,Australia. E-mail: andrews@maths.anu.edu.au
222 B. Andrewspotentially applicable to geometric problems, and that there is great benefit to behad by choosing the flow carefully. I will focus on a particular application, relatingto surfaces in the 3-sphere, but the method has much wider applicability.There are some well-known examples of geometric evolution equations of thekind I want to consider: Eells and Sampson [8] used a heat flow to prove existenceof harmonic maps into non-positively curved targets; Hamilton considered the flowof Riemannian metrics in the direction of their Ricci tensor, and proved that itdeforms metrics of positive Ricci curvature on three-manifolds [12] and metrics ofpositive curvature operator on four-manifolds [13] to constant curvature metrics.The Ricci flow also gives results in higher dimensions, proved by Huisken [14],Nishikawa [24] and Margerin [19]—[21], if the curvature tensor is suitably pinched.The mean curvature flow of submanifolds of Euclidean space is also well-known asthe gradient descent flow of the area functional, and because it arises in modelsof interfaces such as in annealing metals. The examples I will concentrate on areclosest to the last example, as they are evolution equations describing submanifoldsmoving with curvature-dependent velocity. There are many parabolic flows of thiskind, particularly for the codimension one (hypersurface) case: William Firey [11]introduced the motion by Gauss curvature as a model for pebbles wearing away asthey tumble, and other flows which have been considered include motion by powersof Gauss curvature [28], [6], the square root of the scalar curvature [7], the harmonicmean of the principal curvatures [2]-[3], and the reciprocal of the mean curvature[17]. More generally, one can take the velocity to be a function of the principalcurvatures which is monotone increasing in each argument.This gives a huge variety of flows to choose from, so it makes sense to choosethe flow carefully to suit the problem. I will illustrate a strategy for choosing theflow by asking that some desired curvature inequality be preserved under the flow.I will begin, in the next two sections, by discussing some old results concerningsurfaces in the three-sphere. This motivates the results of the later sections.2. Constant mean curvature surfacesThere is a well-known result of Simons [27] which says that a minimal hypersurfacein a S n+1 with the squared norm of the second fundamental form \A\ 2less than n is in fact totally geodesic (hence a great n-sphere). This result comesfrom an application of Simons' identity which relates the second derivatives of meancurvature to the Laplacian of the second fundamental form:VjVjF = Ahij + \A\ 2 hij - Hh\hpj + F^y - n%.From this we can deduce if the hypersurface is minimal (so F = 0)0 = A|,4| 2 - 2|V-4| 2 + 2|,4| 2 (|,4| 2 - n).If |.4| 2 < n at a maximum, then the maximum principle implies |.4| 2 is identicallyzero,and the result follows. Also, if the maximum of |.4| 2 is equal to n, then Mmust be a product S k (a) x S n^k(b) in R k+1 x R n+1^k, with radii a and 6 determinedby the fact that M lies in S n+1 C R n+2 and is minimal.
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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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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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Page 175 and 176: 178 S. S. Kudlawhere Z = J2ì n ìP
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Page 179 and 180: 182 S. S. KudlaTheorem 6. ([13], [9
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Page 193 and 194: 198 E. UllmoSoit X une variété al
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Page 203 and 204: 208 T. D. WooleyWaring's problem as
Page 205 and 206: 210 T. D. Wooleycircumstances one h
Page 207 and 208: 212 T. D. Wooleywith 1 < z,w < F, M
Page 209 and 210: 214 T. D. Wooleyand also byl f \K(a
Page 211 and 212: 216 T. D. Wooleytions involving nor
Page 213: Section 4. Differential GeometryB.
Page 217 and 218: 224 B. Andrewssurface can look metr
Page 219 and 220: 226 B. Andrewsboundary of the set {
Page 221 and 222: 228 B. Andrewseach t > 0, to either
Page 223 and 224: 230 B. AndrewsUSSR Izv. 20 (1983).[
Page 225 and 226: 232 Robert Bartnikprovides an impor
Page 227 and 228: 234 Robert BartnikTheorem 2 If (M,
Page 229 and 230: 236 Robert BartnikThe horizon condi
Page 231 and 232: 238 Robert BartnikTheorem 8 Suppose
Page 233 and 234: 240 Robert BartnikPhysics, LNP 212.
Page 235 and 236: 242 P. Biran2. Various intersection
Page 237 and 238: 244 P. BiranTheorem E'. Let (M,oj)
Page 239 and 240: 246 P. Biransymplectic packings in
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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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