ICM-2018, Rio de Janeiro: Proceedings, Vol.2

Number Theory......Page 6
Algebraic and Complex Geometry......Page 7
Geometry......Page 8
Topology......Page 9
Lie Theory and Generalizations......Page 10
Analysis and Operator Algebras......Page 11
Dynamical Systems and Ordinary Differential Equations......Page 12
Partial Differential Equations......Page 13
Mathematical Physics......Page 14
Probability and Statistics......Page 15
Combinatorics......Page 16
Mathematical Aspects of Computer Science......Page 17
Numerical Analysis and Scientific Computing......Page 18
Mathematics in Science and Technology......Page 19
History of Mathematics......Page 20
Orders of Infinity and Transseries......Page 24
H-Fields......Page 30
H-Field Elements as Germs......Page 32
H-Field Elements as Generalized Transseries......Page 35
Growth Rates as Numbers......Page 38
Introduction and background......Page 48
The theory of reals and ordinals......Page 53
More general sets......Page 56
Introduction......Page 70
 The existential theory of Fq((t))......Page 74
 Axiomatizing Fq((t))......Page 75
 A universal definition for Z in Q......Page 76
 Hilbert's 10th Problem for number rings using elliptic curves......Page 77
Two infinite extensions of Qp......Page 79
Introduction: Proof Theory, Hilbert's Program and Kreisel's `Unwinding of Proofs'......Page 86
Logical metatheorems for bound extractions......Page 88
General types of applications......Page 91
 Strong convergence theorems......Page 92
 Extraction of effective moduli......Page 96
Proof Mining in Convex Analysis......Page 97
Proof Mining in Ergodic Theory......Page 100
Proof Mining in Nonlinear Semigroup Theory......Page 103
The model theoretic point of view......Page 110
Limit points......Page 112
Towards classification theory......Page 113
Ultrapowers......Page 116
In more detail......Page 117
The current picture......Page 118
Introduction......Page 126
 Symmetrizable Cartan matrices and quivers......Page 127
 Bilinear forms, reflections and roots......Page 128
 Orientation and Coxeter elements......Page 129
 Kac-Moody Lie algebras......Page 130
 A class of 1-Iwanaga-Gorenstein algebras......Page 131
 Auslander-Reiten theory and Coxeter functors......Page 133
 Dynkin type......Page 135
 Generalized preprojective algebras......Page 136
 Notation......Page 138
 Varieties of E-filtered modules for......Page 139
 Bundle constructions......Page 141
 Crystals......Page 142
 Constructible functions and Euler characteristic......Page 144
 Convolution algebras as enveloping algebras......Page 145
 =====ME(H) and a dual PBW-basis in the Dynkin case......Page 146
 Semicanonical functions and the support conjecture for =====ME()......Page 148
Introduction......Page 154
Preliminaries......Page 155
 Cohen-Macaulay modules......Page 156
 The triangulated category of Cohen-Macaulay modules......Page 158
 Representation theory......Page 160
Tilting and cluster tilting......Page 164
 Cluster tilting and higher Auslander-Reiten theory......Page 165
 Dimension zero......Page 169
 Dimension one......Page 170
 Classical preprojective algebras......Page 171
 Higher preprojective algebras......Page 173
 d-representation-finite algebras......Page 176
 Preprojective algebras and Coxeter groups......Page 178
 Basic properties......Page 179
 Cohen-Macaulay representations......Page 181
 Geigle-Lenzing projective spaces......Page 183
Negative K-groups of schemes......Page 196
Platification par éclatement......Page 198
Weibel's conjecture......Page 199
Negative K-groups of affinoid algebras......Page 203
Introduction......Page 208
 Basic notions......Page 209
 Quantum groupoids......Page 212
 Fusion categories from quantum groups at roots of 1......Page 213
 Grothendieck ring......Page 214
 Frobenius-Perron dimension......Page 216
 Categorical Morita equivalence......Page 218
 Exact sequences of tensor categories......Page 219
 Hopf algebra extensions......Page 220
 Exact sequences with respect to a module category......Page 221
Fusion categories from finite groups......Page 222
 Group extensions and equivariantization......Page 224
 Weakly group-theoretical fusion categories......Page 226
Braided fusion categories......Page 227
 Modular categories......Page 229
 Tannakian categories and braided G-crossed fusion categories......Page 230
Introduction......Page 241
Principal G-bundles......Page 242
Non-constant case of the conjecture for complex algebraic varieties......Page 244
The conjecture, main results and some corollaries......Page 248
History of the topic......Page 249
Sketch of the proof of Main for an infinite field......Page 251
Sketch of the proof of Main for a finite field......Page 256
Introduction......Page 264
Current Approaches and Recent Results......Page 267
 The McKay conjecture......Page 268
 The Brauer height zero conjecture (BHZ)......Page 269
 Dimensions of irreducible representations and lowdim......Page 270
 Bounds on character values: bounds......Page 274
 Automorphy lifting and adequate groups......Page 277
 The -invariant and Thompson's conjecture......Page 278
 Word maps on simple groups......Page 279
 Random walks, probabilistic generation, and representation varieties of Fuchsian groups......Page 281
Introduction......Page 294
 p-Adic representation theoretic variations.......Page 296
 Relative constructions. p-Adic variations.......Page 297
 The classical construction......Page 298
 Interpolation.......Page 299
  A perfectoid digression.......Page 300
 Eigenvarieties.......Page 302
 The spectral halo......Page 303
 The halo conjecture.......Page 305
p-Adic variation of de Rham automorphic sheaves......Page 306
 p-Adic variations of (HE, Fil).......Page 308
 p-Adic variations of the connection.......Page 309
 The overconvergent projection.......Page 310
 Application: the triple product p-adic L-function in the finite slope case.......Page 311
Higher coherent cohomology......Page 312
 Modular curves and weight one forms......Page 316
 The group GSp4 and potentially modular abelian surfaces......Page 317
The direct summand conjecture......Page 324
 Perfectoid notions......Page 325
 Direct summand conjecture: the case when S[1p] is etale over R[1p].......Page 326
 The perfectoid Abhyankar lemma......Page 327
 Infinite Kummer extensions......Page 329
 Cohen–Macaulay rings and Cohen–Macaulay algebras for the non-Cohen-Macaulay rings......Page 330
 Constructions of Cohen–Macaulay algebras......Page 331
 Intersection conjectures......Page 332
  BookmarkTitle:......Page 334
 Fonctions holomorphes de la variable p......Page 338
 La courbe schématique......Page 340
 La courbe adique......Page 341
 Fibrés vectoriels......Page 342
 Simple connexité géométrique de la courbe......Page 344
 Fibrés et espaces de Banach-Colmez......Page 345
  Faiblement admissible implique admissible......Page 346
  Modifications de fibrés et groupes p-divisibles......Page 348
  -modules sur Ainf et modifications de fibrés......Page 350
 Géométrisation......Page 352
 Interprétation géométrique de certains résultats de Tate......Page 353
 Application aux espaces de périodes p-adiques......Page 354
 La courbe en familles......Page 357
 Résultats de Kedlaya et Liu sur les familles de fibrés......Page 358
 Le champ BunG et la conjecture de géométrisation......Page 359
 Simple connexité des fibres d'une application d'Abel-Jacobi......Page 363
Sketch of the proof of thm3......Page 373
General multiplicative functions......Page 375
The full result......Page 378
Logarithmic Chowla......Page 381
Sarnak's conjecture......Page 384
The shfited convolution problem on average and L-functions......Page 386
 The shifted convolution problem for dk(n)......Page 387
 Moments of L-functions......Page 388
 Gaps between multiplicative sequences......Page 389
The structure of multiplicative functions......Page 391
Introduction......Page 396
The GPY sieve method......Page 397
A modified GPY sieve method......Page 400
Consequences for small gaps between primes......Page 402
Large gaps between primes......Page 404
Limitations......Page 407
Other applications and further reading......Page 409
Recollections on p-adic groups......Page 432
Local models......Page 434
Global theory: Arithmetic models......Page 437
Reductions: Singularities and points......Page 444
Local theory: Formal schemes......Page 446
Introduction......Page 454
 Counting elliptic curves by height......Page 455
 Selmer groups and Shafarevich–Tate groups......Page 456
 The Selmer group as an intersection of maximal isotropic direct summands......Page 457
 The Birch and Swinnerton-Dyer conjecture......Page 458
 Modeling the p-Selmer group......Page 459
 Modeling the p to the e-Selmer group......Page 460
 Modeling the p to the infty-Selmer group and the Shafarevich–Tate group......Page 461
 Modeling the rank of an elliptic curve......Page 463
 Consequences of the model......Page 464
Generalizations......Page 465
 Abelian varieties......Page 466
Introduction......Page 472
Review of the case G = GLn......Page 473
Pseudocharacters......Page 476
The work of V. Lafforgue......Page 480
An automorphy lifting theorem for G......Page 483
Coxeter parameters......Page 487
Potential automorphy......Page 489
 Classical Results......Page 494
 Functional Analogue......Page 495
 Geometric Formulation......Page 496
  Shimura Varieties......Page 497
  Hodge Structures......Page 498
 Definitions and Introduction......Page 499
  Examples of o-minimal structures......Page 500
 Counting Rational Points in Definable Sets......Page 501
 Fundamental Domains and Definability of Uniformization Maps......Page 502
 Proofs of Functional Transcendence results......Page 503
  Ax-Schanuel......Page 504
 Langs conjecture......Page 505
  Tori......Page 506
  The case of =====Ag......Page 507
 Proof of the André-oort conjecture......Page 508
 The Zilber-Pink Conjecture......Page 509
 Results on integral points......Page 510
Introduction......Page 516
Linear programming bounds for sphere packings in metric spaces......Page 517
Sharp linear programming bounds......Page 520
The sphere packing problem in dimensions 8 and 24......Page 522
Fourier interpolation......Page 524
 Notation......Page 530
 Incidence geometry and its applications......Page 531
 Enter the polynomial method......Page 533
 Polynomial partitioning......Page 534
 Stepanov's method......Page 536
 The inverse sieve problem......Page 537
 The determinant method......Page 538
 Exponential sums and Montgomery's conjecture......Page 540
 Baker's theorem......Page 541
 Structure and randomness......Page 542
 Polynomial partitioning over varieties......Page 543
 Restriction estimates......Page 544
Introduction......Page 552
  Automorphic period......Page 556
  Spherical pairs......Page 557
  The global Gan–Gross–Prasad conjecture......Page 558
  Ichino–Ikeda refinement......Page 560
  Reformulation in terms of spherical characters......Page 562
  BookmarkTitle:......Page 563
  Gan–Gross–Prasad pair......Page 564
  Height pairings......Page 565
  The arithmetic Gan–Gross–Prasad conjecture......Page 566
  Reformulation in terms of spherical characters......Page 567
Shtukas and higher Gross–Zagier formula......Page 569
 The Heegner–Drinfeld cycle......Page 570
 Taylor expansion of L-functions......Page 571
 Comparison with the conjecture of Birch and Swinnerton-Dyer......Page 573
 An overview of RTF......Page 574
 Jacquet–Rallis RTFs......Page 575
  Geometric terms: orbital integrals......Page 576
  Spectral terms: spherical characters......Page 578
 Arithmetic RTF......Page 580
 Geometric RTF (over function fields)......Page 582
The arithmetic fundamental lemma conjecture......Page 583
 The arithmetic fundamental lemma conjecture......Page 584
 Varieties and singularities......Page 592
 Hironaka's theorem......Page 593
  Blowing up a regular subvariety......Page 595
 Embedded resolution......Page 596
 Principalization......Page 597
  Are we working too hard?......Page 598
 Order reduction......Page 599
 Derivatives and order......Page 600
 Derivatives and existence in characteristic 0......Page 601
  Coefficient ideals and the induction scheme......Page 602
 A trouble of exceptional loci......Page 603
 Włodarczyk's functoriality principle......Page 604
  Special automorphisms......Page 605
  Homogenization......Page 606
 The re-embedding principle......Page 607
 Toric varieties......Page 608
 The alteration theorem......Page 609
 Toroidalization......Page 610
 Semistable reduction......Page 611
 Temkin's functoriality principle......Page 612
 Enter toroidal orbifolds......Page 613
 Logarithmic derivatives and logarithmic orders......Page 614
Introduction......Page 618
Foliations......Page 622
Classification of del Pezzo foliations......Page 627
 Canonical divisor......Page 638
 Minimal model program......Page 639
 Pluricanonical systems, Kodaira dimension and Iitaka fibration......Page 641
 Fano varieties, and connection with families, singularities, and termination......Page 642
 Hyperstandard sets......Page 643
 Pairs......Page 644
 Generalised pairs......Page 645
 Complements and anti-pluri-canonical systems......Page 646
 Effective birationality......Page 648
 Boundedness of Fano varieties: BAB......Page 649
 Birational automorphism groups......Page 650
 Lc thresholds of R-linear systems.......Page 651
 Lc thresholds of R-linear systems with bounded degree.......Page 652
 Sketch of proof of boundedness of complements......Page 654
 Sketch of proof of effective birationality......Page 655
 Sketch of proof of boundedness of lc thresholds......Page 657
 Sketch of proof of BAB......Page 658
 Fano fibrations......Page 659
 Stable Fano varieties......Page 660
 Other topics......Page 661
Convex functions on spaces of norms......Page 668
 Finsler geometry on the space of norms......Page 669
 Convergence to non-Archimedean norms......Page 670
 Slopes at infinity of a convex function......Page 672
 Kähler metrics with constant curvature......Page 673
 Energy functionals......Page 675
 The Mabuchi L2-metric and weak geodesics......Page 677
 Lp-geometry in the space of Kähler potentials......Page 678
 Energy functionals on =====E1......Page 680
 Variational characterization of cscK metrics......Page 681
 Non-Archimedean pluripotential theory......Page 683
 From geodesic rays to non-Archimedean potentials......Page 685
 Non-Archimedean energy functionals......Page 687
 A version of the Yau–Tian–Donaldson conjecture......Page 689
 Algebraic transformations......Page 696
  Projective spaces......Page 697
 Plan......Page 698
 Automorphisms......Page 699
 Linear groups and their Zariski closure......Page 700
 Real algebraic variation......Page 702
 Dynamical degrees......Page 703
 Entropy......Page 704
 Invariant fibrations......Page 705
 Degree growth......Page 706
 Minimal transformations......Page 707
  Fatou components on K3 surfaces......Page 708
  Dynamics on real K3 surfaces......Page 709
Introduction......Page 714
 Jacobians, Picard schemes and Néron models......Page 715
 Combinatorics of Néron models......Page 716
 Compactifying Néron models......Page 717
 Moduli spaces of algebraic and tropical curves......Page 719
 Skeleta and tropicalizations......Page 721
 Algebraic and tropical admissible covers......Page 722
 Compactifying Jacobians over Mg......Page 724
 Compactified Jacobians in degree g-1......Page 725
 Jacobians in degree g: Néron compactified Jacobians......Page 727
Introduction......Page 734
Theory of weighted baskets......Page 736
 Kollár's method......Page 738
 The case of rX2......Page 739
 On irregular 3-folds of general type......Page 741
 Lower bound of the anti-canonical volume......Page 742
 The anti-pluricanonical birationality......Page 743
 Weighted complete intersections......Page 744
 On quasi-polarized threefolds......Page 745
 Projective varieties with very large canonical volumes......Page 746
 Geography of varieties of general type......Page 747
Introduction......Page 754
Log Calabi–Yau varieties......Page 755
 Birational geometric description of cluster varieties......Page 757
 Combinatorial data for toric model of a cluster variety......Page 759
 The tropicalization of a log Calabi–Yau variety.......Page 760
 The Strominger–Yau–Zaslow conjecture......Page 761
 Homological mirror symmetry......Page 763
 The Fock–Goncharov mirror of a cluster variety......Page 767
 Definitions and algorithmic construction of scattering diagrams.......Page 769
 Initial scattering diagram for cluster variety......Page 771
 Reduction to irreducible case and sign convention.......Page 772
 Reduction to the case of linearly independent mi: Universal deformation and universal torsor.......Page 773
 Mutation invariance of the support of the scattering diagram and cluster complex......Page 774
 Broken lines......Page 775
Q-homology Projective Planes and Montgomery-Yang problem......Page 784
 Trichotomy: KS ample, -ample, numerically trivial......Page 785
 The Maximum Number of Quotient Singularities......Page 786
 Smooth S1-actions on Sm......Page 787
 Pseudofree S1-actions on S2n-1......Page 788
 Algebraic Montgomery-Yang Problem......Page 789
 Progress on Algebraic Montgomery-Yang Problem......Page 790
 Gorenstein Q-homology projective planes......Page 792
 The differentiable case......Page 793
Fake projective planes......Page 794
 Picard group of a fake projective plane......Page 795
 Quotients of fake projective planes......Page 796
 Vanishing theorem for some fake projective planes......Page 797
 Exceptionl collections of line bundles......Page 798
 Explicit equations of fake projective planes......Page 800
Introduction......Page 808
Arc-symmetric sets and arc-analytic functions......Page 810
 Arc-symmetric sets......Page 811
 Arc-analytic maps......Page 813
 Blow-Nash and blow-analytic functions......Page 814
 Functions regular on smooth algebraic arcs......Page 815
 Introducing regulous functions......Page 816
 Curve-regulous and arc-regulous functions......Page 819
 Constructible topology and k-regulous functions......Page 820
 Approximation by regular maps......Page 822
 Approximation by regulous maps......Page 826
 Algebraic versus topological vector bundles......Page 828
 Regulous versus topological vector bundles......Page 831
Introduction......Page 840
 Definitions, notations.......Page 841
 Series associated with the divisorial filtration.......Page 842
 Quasipolynomials and the periodic constants associated with series.......Page 843
 Reductions of =====P.......Page 844
 The series Z(t).......Page 845
 Surgery formulae for the normalized Seiberg–Witten invariants.......Page 846
Topological invariants. The lattice cohomology of M......Page 847
 Lattice cohomology of M.......Page 848
 Rational singularities.......Page 849
 `Bad' vertices, AR graphs.......Page 850
 Relation with Seiberg–Witten invariant.......Page 851
 Reductions.......Page 852
 Z(t) and ===eu in terms of weighted cubes......Page 853
 The fundamental group of M.......Page 854
 Graded roots.......Page 855
Analytic – topological connections. The Seiberg–Witten Invariant Conjecture......Page 856
 Regarding the validity of SWIC......Page 857
 The path lattice cohomology bound of pg.......Page 858
 Superisolated singularities revisited.......Page 860
 Coverings.......Page 862
 Newton non–degenerate germs revisited.......Page 863
 The =====P(t)=Z(t) identity.......Page 864
 A closer look at splice quotient singularities.......Page 865
 The analytic semigroup of (X,o).......Page 866
 Linear subspace arrangements, as `lifts' of the series =====P(t) and Z(t).......Page 867
Introduction......Page 876
V-filtration, Bernstein-Sato polynomial, and birational invariants......Page 878
Hodge filtration on localizations......Page 881
Review of Hodge ideals for reduced divisors......Page 883
Hodge ideals for arbitrary Q-divisors......Page 885
(Non)triviality criteria......Page 890
Global setting and vanishing theorem......Page 892
Example of application: pluri-theta divisors on abelian varieties......Page 894
V-filtration and microlocal log-canonical threshold......Page 896
Introduction......Page 904
 Dual complex as PL-homeomorphism invariant......Page 906
 Dual complex of log Calabi-Yau pairs......Page 909
ACC of log canonical thresholds......Page 910
Klt singularities and K-stability......Page 915
 Geometry of the minimiser......Page 917
 The volume of a klt singularity......Page 923
Introduction......Page 930
 Symmetric spaces......Page 931
 Examples......Page 932
Hodge theory......Page 933
 Matsushima's formula......Page 934
  Compact quotients of the symmetric space associated to PU (p,q)......Page 935
  Compact quotients of the symmetric space associated to SO0 (p,q)......Page 937
 Asymptotics of Betti numbers of locally symmetric manifolds......Page 939
Cycle theory......Page 941
 Modular and symmetric cycle classes......Page 942
  Quotients of the symmetric space associated to PU(p,q)......Page 944
 Hodge type theorems......Page 945
  BookmarkTitle:......Page 946
  BookmarkTitle:......Page 947
 Restriction to special cycles......Page 948
 Another type of Lefschetz property......Page 949
 Some refined analogies with specific projective varieties: Abelian varieties......Page 950
 Comparison with projective varieties......Page 951
 Back to cycles......Page 952
Some regrets......Page 953
Introduction......Page 960
 The basic results......Page 964
 The Suita conjecture and L2-extension......Page 970
The space of Kähler metrics......Page 973
 Variation of complex structure......Page 977
Positivity of direct image sheaves......Page 980
Kleinian groups and limit sets......Page 988
 Fuchsian and Quasi-Fuchsian Groups......Page 989
 Degenerate groups and the Ending Lamination Theorem......Page 991
 The main theorem for closed surface Kleinian groups......Page 994
 Geometric Group Theory......Page 996
 3-manifolds fibering over the circle......Page 997
 The key tool: hyperbolic ladder......Page 998
Kleinian surface groups: Model Geometries......Page 1000
 i-bounded Geometry......Page 1001
 Split Geometry......Page 1003
 Cannon-Thurston maps for degenerate manifolds......Page 1004
 Primitive Stable Representations......Page 1006
 Radial and horospherical limit sets......Page 1007
 Motions of limit sets......Page 1008
  Normal subgroups and trees......Page 1010
 Point pre-images: Laminations......Page 1012
 Relative hyperbolicity......Page 1013
  Higher dimensional Kleinian groups......Page 1014
  Surface groups in higher rank......Page 1015
  Geometric group theory......Page 1016
Introduction......Page 1024
Acylindrically hyperbolic groups......Page 1025
Hyperbolically embedded subgroups......Page 1032
Group theoretic Dehn filling......Page 1035
Small cancellation theory and its applications......Page 1039
Introduction......Page 1048
Existence results for global surfaces of section......Page 1049
Global surfaces of section applied to systolic geometry......Page 1056
The planar circular restricted three-body problem......Page 1062
Transverse foliations......Page 1066
A Poincaré-Birkhoff theorem for tight Reeb flows on S3......Page 1068
Closed geodesics on a Riemannian two-sphere......Page 1072
Introduction......Page 1078
 Surfaces......Page 1079
 Higher-dimensional smooth manifolds......Page 1080
 Symplectic manifolds......Page 1081
 Monodromy......Page 1082
 Definitions......Page 1084
 Properties......Page 1086
 Classification of objects......Page 1089
 Measure and growth......Page 1090
 K3 surfaces......Page 1091
 Quiver threefolds......Page 1093
 Cubic four-folds......Page 1095
 Clusters......Page 1096
Introduction......Page 1104
Gromov–Hausdorff limits......Page 1106
Singularities......Page 1111
Moduli spaces......Page 1114
Optimal degenerations......Page 1117
Introduction......Page 1126
Classical Teichmüller space......Page 1128
What is higher Teichmüller theory?......Page 1129
 Hitchin components......Page 1130
 Anosov representations......Page 1131
Topology of the representation variety......Page 1132
Geometric Structures......Page 1134
Relation to the moduli space of Riemann surfaces......Page 1136
Positivity......Page 1137
Coordinates and Cluster structures......Page 1139
Symplectic geometry and dynamics......Page 1141
Geodesic flows and entropy......Page 1142
Compactifications......Page 1143
Arithmetics......Page 1145
Higher dimensional higher Teichmüller spaces......Page 1146
The Formulation of the Farrell-Jones Conjecture......Page 1158
Actions on compact spaces......Page 1159
Flow spaces......Page 1165
Covers at infinity......Page 1168
 Mathematical aspects of knot contact homology......Page 1180
 Physical aspects of knot contact homology......Page 1181
 Background notions......Page 1183
 Coefficients in chains on the based loop space......Page 1184
 Coefficients in relative homology......Page 1186
A complete knot invariant......Page 1187
 Filtered quotients and a product......Page 1188
 String topology and the cord algebra......Page 1189
 Partially wrapped Floer cohomology and Legendrian surgery......Page 1190
Augmentations and the Gromov–Witten disk potential......Page 1191
 Augmentations of non-exact Lagrangians and disk potentials......Page 1192
Legendrian SFT and open Gromov–Witten theory......Page 1194
  A 4-chain with boundary twice LK......Page 1195
 Generalized holomorphic curves and the SFT-potential......Page 1196
 Compactification of 1-dimensional moduli spaces......Page 1198
 The SFT equation......Page 1201
  The unknot......Page 1202
  The trefoil......Page 1203
 Overview......Page 1208
 Intuitive description of the main construction......Page 1209
 Axioms for quasi-tree of metric spaces and projection complex......Page 1210
 Asymptotic dimension......Page 1212
 Some basic notions......Page 1213
 Projection complex axioms......Page 1214
 Quasi-tree of metric spaces......Page 1217
Application to mapping class groups of surfaces......Page 1218
  Curve graph......Page 1219
  Subsurface projection......Page 1220
  Embedding MCG......Page 1221
  Asymptotic dimension......Page 1222
  Promoting actions to CAT(0) spaces or trees......Page 1223
  Uniform uniform exponential growth of MCG()......Page 1224
  Stable commutator length......Page 1225
Other applications......Page 1226
 Contracting geodesics and WPD......Page 1227
 Acylindrically hyperbolic groups and hyperbolically embedded subgroups......Page 1229
 Bounded cohomology and QFA......Page 1230
 Out(Fn)......Page 1231
 Farrell-Jones conjecture for MCG......Page 1232
 Geometric structures......Page 1238
 Classifying geometric structures......Page 1240
 Representations of discrete groups......Page 1241
 Flag varieties and boundary maps......Page 1242
 Goal of the paper......Page 1243
Holonomy representations......Page 1244
 Convex cocompact locally symmetric structures in rank one......Page 1245
 Convex projective structures: divisible convex sets......Page 1246
 AdS quasi-Fuchsian representations......Page 1248
 The definition......Page 1250
 Important properties and examples......Page 1252
 Higher Teichmüller spaces of Anosov representations......Page 1253
 Characterizations......Page 1254
 Revisiting the examples of sec:dyn-properties-hol......Page 1255
 Cocompact domains of discontinuity......Page 1256
  Hitchin representations for even d=2n......Page 1257
  Maximal representations......Page 1258
 Proper actions on full homogeneous spaces......Page 1259
 Convex cocompactness in higher real rank......Page 1260
 Convex cocompactness in pseudo-Riemannian hyperbolic spaces......Page 1261
 Strong projective convex cocompactness......Page 1262
 Projective convex cocompactness in general......Page 1264
 Brief overview: understanding complete affine manifolds......Page 1265
 Complete AdS manifolds......Page 1266
 Complete affine manifolds......Page 1267
Concluding remarks......Page 1269
Introduction......Page 1280
 Linear and cubical diagrams......Page 1282
 Spatial refinements of diagrams of abelian groups......Page 1283
 Lax spatial refinements......Page 1285
 Framed flow categories......Page 1286
 Speculative digression: matrices of framed cobordisms......Page 1288
 The cube and the Burnside category......Page 1289
 The Khovanov cube......Page 1290
 The stable homotopy type......Page 1291
 Properties and applications......Page 1293
 Speculation......Page 1295
Triangulations of manifolds......Page 1304
The homology cobordism group......Page 1307
Seiberg-Witten theory......Page 1308
Heegaard Floer homology and its involutive refinement......Page 1313
Variations......Page 1316
Preliminaries......Page 1322
 Profinite completion......Page 1323
 Profinite Topology......Page 1324
 Inducing the full profinite topology......Page 1325
Two simple examples......Page 1326
 Some 3-manifold topology......Page 1327
 Profinite completions of 3-manifold groups after Agol and Wise......Page 1329
 Profinite flexibility of 3-manifold groups......Page 1330
 Profinite completions of 3-manifold groups and geometric structures......Page 1332
 Profinite rigidity amongst 3-manifold groups......Page 1333
 A profinitely rigid Kleinian group......Page 1335
 Some restricted genus results......Page 1336
 Nilpotent and polycyclic groups......Page 1337
 Grothendieck's Problem......Page 1338
Final remarks and further questions......Page 1339
Equivariant unitary bordism for families of subgroups......Page 1347
 Complex representations......Page 1350
 Eequivariant complex bundles......Page 1351
 Classifying spaces......Page 1353
The evenness conjecture for finite abelian groups......Page 1356
 Cyclic p-groups......Page 1358
 General p-groups......Page 1360
 The general case......Page 1366
The equivariant unitary bordism groups for non abelian groups......Page 1367
Conclusion......Page 1369
Introduction......Page 1372
Graph complexes......Page 1375
 A variant with external legs......Page 1380
 Rational homotopy theory of operads......Page 1382
 Formality and intrinsic formality of En operads......Page 1384
 Kontsevich's graphical model for En......Page 1386
 A sketch of a proof of thm:intr......Page 1388
Mapping spaces and long knots......Page 1390
Outlook, extensions and open problems......Page 1391
Introduction......Page 1396
Vertex algebras......Page 1397
 W-algebras......Page 1400
Representation theory of W-algebras......Page 1402
BRST reduction of associated varieties......Page 1404
The rationality of minimal series principal W-algerbas......Page 1406
Four-dimensional N=2 superconformal algebras, Higgs branch conjecture and the class =====S chiral algebras......Page 1407
 Contents......Page 1418
Geometric Satake equivalence......Page 1419
 The general case......Page 1420
 Beilinson-Drinfeld slices......Page 1421
 The affine group and its Langlands dual......Page 1422
 The quest......Page 1423
 (Hyperbolic) stalks......Page 1424
 General setup......Page 1425
 Flavor symmetry......Page 1426
 B