Proceedings of The International Congress of Mathematicians 2010 (ICM 2010): Vol. IV

Contents......Page 6
Section 1 Logic and Foundations......Page 10
1. Introduction......Page 12
2. Notation and Background......Page 15
3.1. The basis problem for uncountable linear orders: a case study.......Page 16
3.2. The Ramsey Theory of w1......Page 20
4. Combinatorial Principles......Page 21
5. Proper Forcings and How to Construct Them......Page 24
5.1. The OCA forcing......Page 25
5.2. The PID forcing......Page 26
6.2. Bases in quotients of Banach spaces......Page 27
6.3. Von Neumann’s problem on the existence of strictly positive measures......Page 28
6.4. The determinacy of Gale-Stewart games......Page 29
7. The Role of 2N0 = N2......Page 30
8. The Role of PFA in Proving Theorems in ZFC......Page 31
9. Open Problems......Page 33
References......Page 34
1. Introduction......Page 39
2. Some Background from Computability Theory......Page 40
3. Randomness......Page 41
3.1. Finite objects......Page 42
3.2. Measure, tests, and Martin-Löf randomness......Page 43
3.3. Randomness and differentiability......Page 44
3.4. A notion stronger than Martin-Löf randomness......Page 45
4. For △02, Close to Computable = Far From Random......Page 47
4.1. Background on the two properties......Page 48
4.2. Coincidence of the two properties......Page 49
5.1. Basics on cost functions......Page 55
5.2. Applications of cost functions......Page 57
6. The Turing-below-many Paradigm......Page 62
References......Page 64
1. Introduction......Page 67
2. K-holomorphic Functions......Page 69
3.1. The one-variable case......Page 72
3.2. Functions of several variables......Page 73
4.1. Basic definitions......Page 74
4.2. Compact complex manifolds......Page 75
4.3. K-algebraic and K-analytic sets......Page 76
4.5. The family of complex tori......Page 77
4.6. Mild manifolds......Page 78
5.1. Characterizing K-analytic sets......Page 79
6. Classical Holomorphic Functions in an o-minimal Setting......Page 81
6.2. The Weierstrass ℘-function and elliptic curves......Page 82
6.3. The theta functions and abelian varieties......Page 84
7.1. Definability of compact real analytic manifolds in Ran......Page 86
7.2. The proof of Theorem 5.7......Page 87
References......Page 88
Section 2 Algebra......Page 92
Introduction......Page 94
1.1. Basic definitions......Page 96
1.4. Examples from modular representation theory......Page 98
1.8. Further examples......Page 99
2.1. The spectrum......Page 100
2.2. Localization......Page 103
2.3. Support and decomposition......Page 105
2.4. Gluing and Picard groups......Page 106
2.6. Non-compact objects......Page 108
3.1. Classification of thick ⊗-ideals, after Hopkins......Page 111
3.2. Further computations......Page 113
3.3. Applications to algebraic geometry......Page 114
3.4. Applications to modular representation theory......Page 115
4.1. Computing the spectrum in more examples......Page 116
4.3. Residue fields......Page 117
References......Page 118
1. Introduction......Page 122
2. Jordan Type......Page 123
3. Modules of Constant Jordan Type......Page 124
4. Endotrivial Modules......Page 125
5. Vector Bundles on Projective Space......Page 126
6. Properties of Fi(M)......Page 128
7. Chern Classes......Page 129
8. Small Modules with Interesting Varieties......Page 130
9. Modules for (Z/p)2......Page 131
References......Page 132
Introduction......Page 134
Acknowledgments......Page 135
1. Total Positivity......Page 136
2. Cluster Algebras......Page 139
3. Triangulations and Laminations......Page 146
References......Page 151
0. Introduction......Page 155
1. Definitions of Canonical Dimension......Page 156
2. Incompressible Varieties......Page 158
3. Motives......Page 162
4. General Generalized Severi-Brauer Varieties......Page 165
5. Dimension of Upper Motive......Page 166
References......Page 168
1. Definition of Essential Dimension......Page 171
2. First Examples......Page 173
3. The Fixed Point Method......Page 177
4. Central Extensions......Page 180
5. Essential Dimension at p and Two Types of Problems......Page 183
6. Finite Groups of Low Essential Dimension......Page 186
7.1. Strongly incompressible elements......Page 189
7.3. Cyclic groups......Page 190
7.6. Essential dimension of PGLn......Page 191
7.8. Exceptional groups......Page 192
7.9. Groups whose connected component is a torus......Page 193
References......Page 194
Quadratic Forms, Galois Cohomology and Function Fields of p-adic Curves......Page 198
1. Quadratic Forms and Galois Cohomology Groups......Page 199
2. Galois Cohomology Groups of Function Fields of Surfaces......Page 202
3. The u-invariant......Page 203
4. The Chow Group of 0-cycles......Page 204
References......Page 206
Section 3 Number Theory......Page 210
1. Introduction......Page 212
2.1. The modulo p local correspondence......Page 215
2.2. Over E: first properties......Page 217
2.3. (φ, Γ)-modules and the theorems of Colmez and of Paškūnas......Page 219
2.4. Local-global compatibility......Page 223
3.1. Why the GL2(Qp) theory cannot extend directly......Page 225
3.2. So many representations of GL2(F)......Page 226
3.3. Questions on local-global compatibility......Page 229
4.1. Invariant lattices and admissible filtrations......Page 231
4.2. Supersingular modules and irreducible Galois representations......Page 233
4.3. Serre weights and Galois representations......Page 234
References......Page 236
1. Selmer Groups......Page 240
2. Behavior Under Congruences......Page 244
3. Artin Twists......Page 249
4. Parity......Page 252
References......Page 256
Artin’s Conjecture on Zeros of p-adic Forms......Page 258
References......Page 265
Introduction......Page 267
1. Nonarchimedean Analytic Spaces......Page 269
2. Witt Vectors......Page 271
3. Filtered Isocrystals and Weak Admissibility......Page 273
4. Admissibility at Rigid Analytic Points......Page 275
5. Admissibility at General Analytic Points......Page 280
6. The Universal Crystalline Local System......Page 282
7. Further Remarks......Page 284
References......Page 286
1. Introduction......Page 289
2.1. Case ℓ = p. Serre’s weight [32]......Page 294
2.3. Statement......Page 295
3. Existence of Compatible Systems......Page 296
4. The Strategy of the Proof......Page 297
References......Page 299
Introduction......Page 303
1.1. Potentially semi-stable representations......Page 306
1.2. Deformation rings......Page 308
2.1. Local Langlands and IK-representations......Page 310
2.2. Formulation of the conjecture......Page 311
3.1. Statements......Page 313
3.2. A sketch of the proofs......Page 315
References......Page 319
1.1. The complex points......Page 321
1.3. Canonical models......Page 323
2.1. Intersection homology......Page 324
2.3. L2 cohomology of Shimura varieties and discrete automorphic representations......Page 326
3.1. The sheaf-theoretic point of view on intersection homology......Page 328
3.2. Perverse sheaves......Page 329
3.3. Intermediate extensions and the intersection complex......Page 330
3.5. Application to Shimura varieties......Page 332
4. Counting Points on Shimura Varieties......Page 333
5.1. The topological case......Page 335
5.2. Algebraic construction of weighted cohomology......Page 336
5.3. Application to the cohomology of Shimura varieties......Page 339
References......Page 340
Introduction......Page 344
1.1. Euler numbers......Page 346
1.2. Characteristic classes......Page 348
1.3. Conductor formula......Page 350
2. Geometric Ramification Theory......Page 352
2.1. Ramification groups of a local field......Page 353
2.2. Ramification along a divisor......Page 357
2.3. Characteristic cycles......Page 360
References......Page 363
1. Introduction......Page 366
2. Spectral Expansions, and Expansions into Incomplete Eisenstein and Poincare Series......Page 370
3. Relation to L-functions and the Subconvexity Problem......Page 371
4. Inner Products with Poincare Series and the Shifted Convolution Problem......Page 375
5. Mean Values of Multiplicative Functions and Weak Subconvexity......Page 377
6. Sieve Methods and Holowinsky’s Work......Page 383
7. Proof of Mass Equidistribution......Page 386
8. The Escape of Mass Argument......Page 387
References......Page 388
1. Arithmetic Counting Problems......Page 392
1.1. Context......Page 393
2. Number Fields......Page 394
2.2. The asymptotic constant......Page 395
2.3. The asymptotic constant: Bhargava’s heuristic......Page 396
2.4. General groups; the role of the Schur multiplier......Page 397
2.5. Lifting invariants over global fields......Page 399
3. Function Fields and Hurwitz Spaces......Page 402
3.2. Stable homology......Page 403
3.3. Hurwitz schemes......Page 405
3.4. Stable components and the Schur correction......Page 406
4.1. Number fields......Page 408
References......Page 409
Section 4 Algebraic and Complex Geometry......Page 412
1. Introduction......Page 414
2. The Context for Many of the Problems......Page 416
3. Intersection Theory in an Ordinary Grassmannian......Page 418
3.2. Numerical relations between intersection numbers and invariant theory......Page 419
3.3. The Geometric Horn Property......Page 420
4. Quantum Cohomology......Page 421
4.1. Fulton’s conjecture......Page 423
5. Intersection Theory in an Arbitrary Homogeneous Space......Page 424
5.1. Intersection of Schubert varieties in a G/P......Page 425
5.2. A deformation of cohomology......Page 426
5.3. The deformed product and invariant theory......Page 427
6.1. Moduli spaces and theta functions......Page 428
6.2. Strange duality......Page 429
6.3. Other perspectives......Page 430
References......Page 432
1. Introduction......Page 436
2. Varieties of General Type......Page 442
2.1. Reid’s 3-fold exact plurigenera formula......Page 447
3. Varieties of Log General Type......Page 448
3.1. Automorphism groups of varieties of general type......Page 449
3.2. Varieties of intermediate Kodaira dimension......Page 450
3.3. Moduli spaces of varieties of general type......Page 452
3.5. Fano varieties......Page 453
References......Page 454
Hyperkähler Manifolds and Sheaves......Page 459
1. Introduction......Page 460
2. Non-separation for Hyperkähler Manifolds......Page 461
3. Twistor Spaces......Page 463
4. Non-separation for Sheaves and Complexes......Page 465
5. Open Problems......Page 467
References......Page 468
1. Generalities on Mixed Motives......Page 470
2. Non-commutative Setting......Page 472
3. Hodge-to-de Rham Spectral Sequence......Page 476
4. Review of Filtered Dieudonné Modules......Page 478
5. FDM in the Non-commutative Case......Page 481
6.1. Stable homotopy category and homology......Page 484
6.2. Equivariant categories......Page 486
6.3. Mackey functors......Page 487
7. Cyclotomic Traces......Page 490
7.1. Topological cyclic homology......Page 491
7.2. Cyclotomic complexes......Page 493
7.3. Comparison theorem......Page 494
7.4. Back to ring spectra......Page 496
8. Hodge Structures......Page 497
References......Page 502
1.1. Moduli spaces of stable maps......Page 506
1.2. Perfect obstruction theory and the virtual fundamental class......Page 507
1.4. GW invariants of noncompact Calabi-Yau 3-folds......Page 508
2.1. Genus g = 0......Page 510
2.2. Genus g = 1......Page 512
2.3. Genus g ≥ 2......Page 513
3.1. The topological vertex......Page 514
4.1. GW/DT Correspondence......Page 516
References......Page 517
1.1. Curves and Surfaces......Page 522
1.2. Strong and weak factorisation......Page 525
1.3. Flips and Flops......Page 526
2. Minimal Model Program......Page 529
3. Local Approach to Termination......Page 534
4. Global Approach to Termination......Page 539
5. Local-global Approach to Termination......Page 541
References......Page 546
1. L2 Extension Results......Page 549
2. Effective Pseudoeffectivity of Relative Pluricanonical Bundles......Page 552
3. Minimal Singularities Metrics and their Restriction Properties......Page 555
4. Non-vanishing......Page 560
References......Page 563
Introduction......Page 567
1. Motivic Cohomology......Page 570
2. Finiteness Conjecture on Motivic Cohomology......Page 573
3. Cohomological Hasse Principle......Page 575
4. Results on Cohomological Hasse Principle......Page 578
5. Application: Finiteness of Motivic Cohomology......Page 582
6. Application: Special Values of Zeta Functions......Page 583
7. Application: Higher Class Field Theory......Page 586
8. Application: Resolution of Quotient Singularities......Page 588
References......Page 591
1. Introduction......Page 595
2. Betti Tables......Page 597
3. Facets of the Cone and Cohomology Tables......Page 600
4. Existence......Page 607
5. Applications, Extensions of the Basic Theory and Open Questions......Page 608
References......Page 610
1. Introduction......Page 612
2. The Singular Case: First Steps......Page 616
3.1. A Chow ring......Page 618
3.3. Codimension of support......Page 619
3.4. Hodge conjecture......Page 622
3.5. Fine structure......Page 625
4.1. Some new tools from K-theory......Page 628
References......Page 629
Heuristics......Page 633
2.1. Parallel transport......Page 634
2.2. The ordinary double point......Page 635
2.3. Families of quadrics......Page 637
2.5. The Manolescu isomorphism......Page 639
2.7. ALE spaces as affine blow ups......Page 642
2.8. The Seidel-Smith construction......Page 643
3.1. Surface ordinary double point......Page 647
3.2. Adjoint quotient and the Flag variety......Page 649
4. Homological Mirror Symmetry......Page 650
4.1. Surfaces......Page 651
5. Hilbert Schemes of ALE Spaces and Khovanov Cohomology......Page 653
5.2. The construction......Page 654
5.3. Maps between ALE spaces......Page 655
5.5. Bigrading and Khovanov cohomology......Page 658
References......Page 659
Introduction......Page 661
1.1. Définition des invariants......Page 663
1.2. Bornes inférieures et optimalité......Page 665
1.4. Calculs......Page 667
2.1. Invariants relatifs réels......Page 669
2.2. Invariants relatifs imaginaires......Page 672
3.1. Définition des invariants dans les variétés algébriques réelles convexes......Page 675
3.2. Extension aux variétés symplectiques réelles fortement semi-positives......Page 677
3.3. Optimalité, congruences et calculs dans le cas de l\'ellipsoïde de dimension trois......Page 679
4.1. Absence de membranes J-holomorphes......Page 680
4.2. Présence de membranes J-holomorphes......Page 682
Références......Page 685
Section 5 Geometry......Page 688
1. Boundaries of Random Walks on Groups......Page 690
2. Applications of Entropy Criterion......Page 695
3. Choquet-Deny Theorems. Groups of Intermediate Growth. Applications of Random Walks to Growth of groups......Page 698
3.1. Application to growth......Page 700
4. Complete Description of Poisson-Furstenberg Boundaries......Page 702
5. Different Scales of Amenability. Asymptotic Invariants Related to Boundaries......Page 705
References......Page 708
1. Introduction......Page 714
2.1. The definition and examples......Page 715
2.2. Some existence results of balanced metrics......Page 716
2.3. The construction of balanced metrics on #k(S3×S3)......Page 717
3. The Strominger System......Page 718
3.1. Non-Kähler solutions on some Kähler Calabi-Yau threefolds......Page 719
3.2. Solutions on some non-Kähler Calabi-Yau threefolds......Page 720
3.4. The explicit solution on the torus bundle over the Eguchi-Hanson space......Page 721
4. Form-type Calabi-Yau Equations......Page 722
References......Page 723
Locally Homogeneous Geometric Manifolds......Page 726
2. The Classification Question......Page 727
3. Ehresmann Structures and Development......Page 728
5. The Hierarchy of Geometries......Page 730
6. Deforming Ehresmann Structures......Page 731
7. Representations of the Fundamental Group......Page 732
8. Thurston’s Geometrization of 3-manifolds......Page 733
9. Complete Affine 3-manifolds......Page 734
10. Affine Structures on Closed Manifolds......Page 736
11. Hyperbolic Geometry on 2-manifolds......Page 737
12. Complex Projective 1-manifolds, Flat Conformal Structures and Spherical CR Structures......Page 739
13. Surface Groups: Symplectic Geometry and Mapping Class Group......Page 742
References......Page 744
1. Introduction......Page 754
2. Examples......Page 755
3. Why Is the Systolic Inequality Hard?......Page 756
5. Minimal Surface Theory......Page 758
6. Topological Dimension Theory......Page 761
7. Scalar Curvature......Page 763
8. The Federer-Fleming Averaging Argument......Page 767
9. Notions of Size in Riemannian Geometry......Page 771
References......Page 775
1.1. A toy question......Page 778
1.2. Boundary rigidity......Page 779
1.3. Filling volumes and minimal fillings......Page 781
2. Some Implications......Page 782
2.3. E. Hopf ’s theorem......Page 783
3. Minimality in a Banach Space......Page 784
3.1. Isometric representations......Page 785
3.2. Defining the surface area in L∞......Page 786
3.3. Sketch-proof of theorems 3.1 and 3.2.......Page 787
4. Finslerian Case......Page 790
References......Page 791
0. Introduction......Page 794
1.1. Dirac operators and quantization......Page 796
1.2. Bergman kernel on Cn......Page 798
2.1. Asymptotic expansion of Bergman kernel......Page 800
2.2. Asymptotic expansion of Toeplitz operators......Page 803
2.3. The Kähler case......Page 804
3.1. Quantization commutes with reduction......Page 806
3.2. Berezin-Toeplitz quantization and reduction......Page 807
3.3. The Kähler case......Page 810
4.1. Quantization formula on Cn......Page 812
4.2. Vergne’s conjecture......Page 813
References......Page 815
1.1. The Yamabe problem......Page 820
1.2. The set of solutions and some conjectures......Page 821
1.3. A compactness theorem......Page 823
1.4. Noncompactness results......Page 826
2. The Connectedness Problem......Page 830
2.1. Some applications to General Relativity......Page 833
References......Page 835
1. Introduction......Page 839
2.1. Homogeneous spaces with 4-dimensional isometry group......Page 842
2.2. Homogeneous spaces with 3-dimensional isometry group......Page 844
3.1. Integrability equations in E3(k,T)......Page 845
3.2. Stability and index of CMC surfaces......Page 847
4.1. Rotational compact CMC surfaces......Page 848
4.2. The Alexandrov problem in E3(k,T)......Page 849
4.3. The Hopf problem in E3(k,T)......Page 850
4.4. The isoperimetric problem in E3(k,T)......Page 851
5. CMC Spheres in Sol3......Page 852
5.1. Proof of Theorem 5.2: uniqueness......Page 853
6. Surfaces of Critical CMC......Page 855
6.1. Harmonic Gauss maps......Page 856
6.2. Half-space theorems......Page 859
6.3. The classification of entire graphs......Page 860
7.1. The Collin-Rosenberg theorem......Page 863
7.2. Minimal surfaces of finite total curvature in ℍ2 × ℝ......Page 865
References......Page 867
1. Introduction......Page 871
2. “Thick” Knots......Page 872
3. Methods I: Algorithmic Unsolvability of the Diffeomorphism Problem and its Applications......Page 875
4. Methods II: Kolmogorov Complexity and Time-bounded Kolmogorov Complexity......Page 878
5. Disconnectedness of Sublevel Sets of Riemannian Functionals......Page 879
6. Methods III: Simplicial Norm, Homology Surgery, Arithmetic Groups......Page 882
7. Disconnectedness of Sublevel Sets of Riemannian Functionals: Current Work and Some Open Questions......Page 883
8. Higher-dimensional Cycles in Sublevel Sets......Page 885
9. Morse Landscapes of the Length Functional......Page 886
References......Page 889
1. Introduction......Page 891
2. Statement of the Result......Page 892
2.1. The non-obstructed case......Page 895
2.3. Extremal versus constant scalar curvature metrics......Page 896
2.4. The case of projective spaces......Page 897
2.5. The blow up of P2 at two points......Page 898
2.6. The case of toric varieties......Page 899
3. Overview of the construction......Page 900
3.1. Perturbation of extremal metrics......Page 901
3.2. The origin of the constraints......Page 903
References......Page 905
1. Introduction......Page 908
2. Gromov-Hausdorff Convergence......Page 909
3. Basic Results......Page 910
4. Reconstruction of Low-dimensional Collapsed Manifolds......Page 912
5. Essential Coverings......Page 914
6. Reconstruction by Spectral Data......Page 916
References......Page 920
Section 6 Topology......Page 924
1. Introduction......Page 926
Acknowledgements......Page 929
2.1. Lagrangian correspondences......Page 930
2.2. Symmetric products......Page 931
2.3. Heegaard-Floer homology......Page 932
3.1. Positive perturbations and partial wrapping......Page 933
3.2. The algebra of a decorated surface......Page 935
3.3. Generating the partially wrapped Fukaya category......Page 938
4. Yoneda Embedding and Invariants of Bordered 3-manifolds......Page 943
5. Relation to Bordered Heegaard-Floer Homology......Page 946
References......Page 949
1. Introduction......Page 951
2. Hochschild Homology and the Todd Class......Page 952
3. Factorization Algebras......Page 954
4. Descent and Factorization Homology......Page 957
5. Main Theorem......Page 959
6. Factorization Algebras from Quantum Field Theory......Page 960
7. Deformation Quantization in Quantum Field Theory......Page 961
8. Holomorphic Chern-Simons Theory......Page 964
References......Page 967
1. Introduction......Page 969
3. Generalized Smale Conjecture for Hyperbolic 3-manifolds and the Log(3)/2 Theorem......Page 970
4. Marden’s Tameness Conjecture, Bers - Sullivan - Thurston Density Conjecture and Thurston’s Ending Lamination Conjecture......Page 972
5. Volumes of Hyperbolic 3-manifolds......Page 973
6. Ending Lamination Space......Page 976
References......Page 978
The Classification of p–compact Groups and Homotopical Group Theory......Page 982
1. Root Data over the p–adic Integers......Page 984
2. p–compact Groups and their Classification......Page 987
2.1. Construction of p–compact groups......Page 991
2.2. Uniqueness of p–compact groups......Page 993
2.3. Lie theory for p–compact groups......Page 996
3. Finite Loop Spaces......Page 998
4. Steenrod’s Problem of Realizing Polynomial Rings......Page 1001
5. Homotopical Finite Groups, Group Actions......Page 1003
References......Page 1005
1. Introduction......Page 1011
2. The Action of the Mapping Class Group on the Curve Graph......Page 1013
3. The Action of the Mapping Class Group on Teichmüller Space......Page 1017
4. A Geometric Model for the Mapping Class Group......Page 1018
5. Geometry and Rigidity of MCG(S)......Page 1022
6. Resemblance with Lattices......Page 1024
References......Page 1028
1.1. Floer homology of 3-manifolds......Page 1031
1.2. Contact geometry preliminaries......Page 1032
1.3. The ECH chain complex......Page 1033
1.4. The ECH index......Page 1035
1.5. Example: the ECH of an ellipsoid......Page 1039
1.6. Some additional structures on ECH......Page 1041
1.7. Analogues of ECH in other contexts......Page 1043
2.1. Generalizations of the Weinstein conjecture......Page 1044
2.2. The Arnold chord conjecture......Page 1045
2.3. Obstructions to symplectic embeddings......Page 1046
References......Page 1048
1. Introduction......Page 1051
2. The State of Play......Page 1053
3. Congruence Covers......Page 1056
4. Abelian Covers......Page 1058
5. Counting Finite Covers......Page 1061
6. The Behaviour of Algebraic Invariants in Finite Covers......Page 1062
7. The Behaviour of Geometric and Topological Invariants in Finite Covers......Page 1064
8. Two Approaches to the Virtually Haken Conjecture......Page 1068
9. Covering Spaces of Hyperbolic 3-orbifolds and Arithmetic 3-manifolds......Page 1070
10. Group-theoretic Generalisations......Page 1073
11. Subgroup Separability, Special Cube Complexes and Virtual Fibering......Page 1075
References......Page 1076
0. Introduction......Page 1080
1.1. Borel Conjecture......Page 1081
1.2. Fundamental groups of closed manifolds......Page 1082
1.3. Novikov Conjecture......Page 1083
1.6. Vanishing of the reduced projective class group......Page 1084
1.7. Vanishing of the Whitehead group......Page 1085
1.8. The Bass Conjecture......Page 1086
2.1. The K-theoretic Farrell-Jones Conjecture for torsion-free groups and regular coefficient rings......Page 1087
2.2. The L-theoretic Farrell-Jones Conjecture for torsion-free groups......Page 1089
3.1. Classifying spaces for families......Page 1090
3.4. The Farrell-Jones Conjecture......Page 1091
4.1. The work of Farrell-Jones and the status in 2004......Page 1093
4.3. Inheritance properties......Page 1094
5. Computational Aspects......Page 1096
6. Methods of Proof......Page 1097
7.2. Solvable groups......Page 1100
7.7. Classification of (non-aspherical) manifolds......Page 1101
References......Page 1102
Introduction......Page 1108
1. Moduli Problems for Commutative Rings......Page 1110
2. Higher Category Theory......Page 1111
3. Higher Algebra......Page 1116
4. Formal Moduli Problems......Page 1120
5. Tangent Complexes......Page 1123
6. Noncommutative Geometry......Page 1127
References......Page 1133
1. Introduction......Page 1135
2. Weil-Petersson Measure on Mg,n......Page 1138
3. Asymptotic Behavior of Weil-Petersson Volumes and Tautological Intersection Pairings......Page 1144
4. Random Riemann Surfaces of High Genus......Page 1149
References......Page 1151
1. Introduction......Page 1155
2. Preliminaries......Page 1156
3. Simply Connected Surfaces of General Type with pg = 0......Page 1158
3.1. An example with pg = 0 and K2 = 2......Page 1160
3.2. An example with pg = 0 and K2 = 3......Page 1162
3.3. An example with pg = 0 and K2 = 4......Page 1164
4. Other Surfaces Via ℚ-Gorenstein Smoothings......Page 1165
References......Page 1166
1. Contact 3-manifolds......Page 1168
Knots in contact topology......Page 1169
Overtwisted versus tight dichotomy......Page 1171
Contact surgery......Page 1172
Open book decompositions and Giroux’s theorem......Page 1173
Ozsváth–Szabó homologies of 3-manifolds......Page 1174
Contact Ozsváth–Szabó invariants......Page 1176
Legendrian and transverse invariants......Page 1178
Surgery along knots in S3......Page 1179
Legendrian and transverse knots......Page 1184
References......Page 1185
Author Index......Page 1188