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

Contents......Page 6
Section 13 Probability and Statistics......Page 10
2. Euclidean Perturbed......Page 12
3. Unimodular Random Graphs, Uniform Random Triangulations......Page 13
3.1. Scaling limit of Planar maps......Page 15
3.2. QG and GFF......Page 16
3.3. Harmonic measure and recurrence......Page 18
4.1. Vacant sets......Page 19
Acknowledgements......Page 20
References......Page 21
1. Introduction......Page 23
2. A Two-dimensional Growth Model......Page 24
2.2. Macroscopic scale, one-point fluctuations, and local structure......Page 27
2.3. Complex structure and multi-point fluctuations. The Gaussian Free Field......Page 29
2.4. Universality class......Page 30
3.1. More general update rules......Page 31
3.3. Symmetric functions and skew plane partitions......Page 32
3.4. Random growth in 1+1 dimensions......Page 33
References......Page 35
1. Introduction......Page 38
3. Limiting Spectral Distribution and Moments......Page 39
4.1. Link function......Page 40
4.3. Trace formula and circuits......Page 42
4.4. Words......Page 43
4.6. Only pair matched words contribute......Page 44
4.7. Vertex, generating vertex and Carleman’s condition......Page 45
5. The LSD for Some Specific Matrices......Page 46
5.1. Wigner matrix: the semicircular law......Page 47
5.2. Toeplitz and Hankel matrices......Page 49
5.3. The reverse circulant and the palindromic matrices......Page 51
5.4. XX' matrices......Page 52
5.5. Band matrices......Page 56
5.6. Matrices with dependent entries......Page 58
6.2. Tridiagonal matrices......Page 60
7.2. Stieltjes transform and the Wigner and sample covariance matrices......Page 61
8. Discussion......Page 62
References......Page 63
1. Introduction......Page 67
1.1. Background......Page 68
1.2. Continuous-time weakly self-avoiding walk and the main result......Page 70
3. Integral Representation......Page 72
4. Quadratic or Gaussian Approximation......Page 74
5.1. The space N......Page 75
5.2. Local polynomials and localisation......Page 76
6.1. The super-expectation......Page 78
6.2. Finite-range decomposition of covariance......Page 79
7. Perturbation Theory and Flow Equations......Page 80
8.1. Scales and the circle product......Page 82
8.2. The renormalisation group map......Page 83
9. The Inductive Step: Construction of Vj+1......Page 84
10. Norms for K......Page 85
11. The Inductive Step Completed: Existence of Kj+1......Page 87
12. Decay of the Two-point Function......Page 88
References......Page 90
1. Large Deviation Principles......Page 93
1.2. Annealed LDP......Page 94
1.3. Quenched LDP......Page 95
2. Collision Local Time of Two Random Walks......Page 96
3.2. Coupled branching processes......Page 99
4.1. A polymer in a random potential......Page 100
4.2. A polymer pinned at an interface......Page 102
4.3. A copolymer near a selective interface......Page 105
5. Closing Remarks......Page 107
References......Page 108
1. Introduction......Page 110
2. Ingredients......Page 111
3. Rigorous Definition of the Model......Page 115
4. Convergence of the Discrete Generation Model......Page 118
5. Equilibria in General......Page 120
6. Equilibria for Demographic Selective Costs......Page 122
7. Step Profiles and Demographic Selective Costs......Page 124
8. Polynomial Selective Costs......Page 126
References......Page 129
1. Introduction......Page 132
2. Mathematical Models......Page 135
3.1. Static Host Model......Page 136
3.2. Dynamic Host Model......Page 139
4. Experiments......Page 141
References......Page 142
1. KPZ and Asymmetric Exclusion......Page 145
2. Directed Random Polymers......Page 148
3. The t1/3 Law......Page 149
4. Weakly Asymmetric Limit of Simple Exclusion......Page 150
5. KPZ/Stochastic Burgers in Equilibrium: The Method of Second Class Particles......Page 152
6. Tracy-Widom Formula for ASEP......Page 154
7. The Crossover Distributions......Page 155
8. The Intermediate Coupling Regime for Random Polymers......Page 156
References......Page 157
Stein’s Method, Self-normalized Limit Theory and Applications......Page 160
2. Stein’s Method......Page 161
2.1. Stein’s equation......Page 162
2.2. Normal approximation for smooth functions and Berry-Esseen bounds......Page 163
2.3. Cramér type moderate deviations......Page 166
2.4. Non-normal approximation via exchangeable pairs approach......Page 167
2.5. Randomized concentration inequalities......Page 168
3. Self-normalized Limit Theory......Page 172
3.1. Self-normalized saddlepoint approximations......Page 174
3.2. A universal self-normalized moderate deviation......Page 175
3.3. Self-normalized Cramér type moderate deviations for the maximum of sums......Page 177
3.4. Studentized U-statistics......Page 178
4. Applications......Page 180
References......Page 182
1. Introduction......Page 186
2. An Oracle Inequality in the Linear Model......Page 189
3. An Oracle Inequality for General Convex Loss......Page 191
4. Compatibility and Restricted Eigenvalues......Page 194
5.2. Variable selection......Page 196
5.3. The adaptive Lasso......Page 198
6. The Lasso with Within Group Structure......Page 199
7. Conclusion......Page 202
References......Page 203
1. Introduction......Page 205
2. Gaussian Process Priors......Page 208
3. Sparsity......Page 214
4. An Abstract Theorem......Page 216
References......Page 219
Section 14 Combinatorics......Page 222
1. Introduction: Face Enumeration in Convex Polytopes......Page 224
1.1. Simplicial polytopes......Page 225
2. Eulerian Posets and the cd-index......Page 226
2.1. Flag enumeration in graded posets......Page 227
2.2. Eulerian posets and the cd-index......Page 228
2.3. Inequalities for flags in polytopes and spheres......Page 230
3.1. The convolution product and derived inequalities......Page 232
3.2. Relations on flag numbers and the enumeration algebra......Page 233
3.3. Quasisymmetric function of a graded poset......Page 235
3.4. Peak functions and Eulerian posets......Page 237
4. Bruhat Intervals in Coxeter Groups......Page 239
4.1. R-polynomial and Kazhdan-Lusztig polynomial......Page 240
4.2. The complete quasisymmetric function of a Bruhat interval and the complete cd-index......Page 241
4.3. Kazhdan-Lusztig polynomial and the complete cd-index......Page 244
5. Epilog: Combinatorial Hopf Algebras......Page 246
References......Page 247
1.1. Genetics of the regular figures......Page 251
1.2. Exceptional symmetry: E8 and the Leech lattice......Page 253
1.3. Energy minimization......Page 254
1.4. Packing and information theory......Page 255
2.1. Sphere packing in low and high dimensions......Page 256
2.2. Lattices and periodic packings......Page 258
2.3. Packing problems in other spaces......Page 260
3.1. Physics on surfaces......Page 262
3.2. Varying the potential function......Page 263
3.3. Universal optimality......Page 264
4.1. Constraints on the pair correlation function......Page 266
4.2. Zonal spherical harmonics......Page 268
4.3. Linear programming bounds......Page 269
4.4. Semidefinite programming bounds......Page 270
5.1. Linear programming bounds in Euclidean space......Page 271
5.2. Apparent optimality of E8 and the Leech lattice......Page 272
6. Future Prospects......Page 274
Acknowledgments......Page 275
References......Page 276
Hurwitz Numbers: On the Edge Between Combinatorics and Geometry......Page 279
1.1. Simple and general Hurwitz numbers......Page 280
1.2. Topological interpretation......Page 281
1.3. Cut-and-join equation of Goulden and Jackson......Page 283
1.4. Certain formulas for rational Hurwitz numbers......Page 285
2.1. Grassmannian embeddings and Plücker equations......Page 286
2.3. The boson-fermion correspondence......Page 287
2.4. Semi-infinite Grassmannian and the KP equations......Page 288
2.5. Action of the diagonal matrices......Page 289
2.6. Symmetric group representations......Page 291
2.7. Application: enumeration of maps and hypermaps......Page 293
3.1. The ELSV formula......Page 295
3.2. Linear Hodge integrals as coefficients of a solution to KP......Page 296
3.3. Witten’s conjecture......Page 297
4.1. Completed cycles......Page 298
4.2. r-Hurwitz numbers and generalized Witten’s conjecture......Page 299
4.3. Geometry of Hurwitz spaces and universal characteristic classes......Page 301
References......Page 302
1. Introduction: Two Problems in Lie Theory......Page 306
2. Cluster Algebras......Page 308
3. The Cluster Structure of C[N]......Page 309
4. The Preprojective Algebra......Page 310
5. The Dual Semicanonical Basis S......Page 312
6. Rigid Λ-modules......Page 313
7. Finite-dimensional Representations of Uq(Lg)......Page 314
8. The Subcategories Cl......Page 316
9. The Cluster Algebras Al......Page 317
10. An Intriguing Relation......Page 319
References......Page 320
1. Introduction......Page 324
2. Sparse Graphs......Page 326
3. Dense Graphs......Page 330
References......Page 334
Sparse Combinatorial Structures: Classification and Applications......Page 337
1. Introduction......Page 338
2.1. Graphs vs Structures......Page 340
2.2. Homomorphism order......Page 343
2.3. Sparsity via Resolution in Time......Page 345
2.4. The Nowhere Dense – Somewhere Dense Dichotomy......Page 346
3. Trichotomy for Binary Structures......Page 347
3.1. Classification by Edge Densities......Page 348
4.1. Classification by Decomposition — Chromatic Numbers......Page 349
4.2. Classification by Independence......Page 351
4.3. Classification by Counting......Page 352
5.1. Sub-exponential ω-expansion......Page 353
6.1. Property testing......Page 354
6.2. Weakly hyperfinite classes......Page 355
2. Simplicial Graphs......Page 356
8. Bounded Expansion Classes......Page 357
9. Restricted Dualities — a Characterization......Page 358
References......Page 359
1. Introduction......Page 365
2. Interpolation Functions......Page 369
3. Biorthogonal Functions......Page 380
References......Page 387
1. Introduction......Page 390
2. The Classical Models......Page 391
3. Models of Real-world Networks......Page 394
4. Inhomogeneous Graphs and Branching Process......Page 395
5. Metrics on Dense Graphs......Page 398
6. Percolation on Dense Graph Sequences......Page 402
7. More General Sparse Models......Page 403
8. Sparse Quasi-random Graphs......Page 405
9. Models and Metrics......Page 407
References......Page 409
1. Introduction......Page 414
2. Ramsey Theory......Page 415
2.1. Hypergraphs......Page 417
2.2. Almost monochromatic subsets......Page 419
3. Graph Ramsey Theory......Page 420
3.1. Linear Ramsey numbers......Page 421
3.2. Sparse graphs......Page 422
3.3. Maximizing the Ramsey number......Page 423
3.4. Methods......Page 424
4.1. Classical results......Page 425
4.2. Bipartite graphs......Page 426
4.3. Subgraph multiplicity......Page 428
5.1. Local density......Page 429
5.2. Graphs with large minimum degree......Page 431
5.3. Spectral Turán theorem......Page 432
6. Turán-type Problems for Hypergraphs......Page 433
6.1. Hypergraphs and arithmetic progressions......Page 435
References......Page 436
Section 15 Mathematical Aspects of Computer Science......Page 442
1. Introduction......Page 444
2. Conic Condition Numbers......Page 447
3.1. Renegar’s condition number......Page 450
3.2. Average and smoothed analysis......Page 452
3.3. Grassmann condition number......Page 454
4.1. Smale’s 17th problem......Page 456
4.2. Average and smoothed analysis......Page 458
4.3. A near solution to Smale’s 17th problem......Page 460
4.4. Some ideas of the proofs......Page 461
References......Page 464
1. Introduction......Page 469
1.1. Additional Related Work......Page 472
2. Additional Definitions......Page 473
3.1. Small to Moderate Numbers of Counting Queries......Page 474
3.2. Generalization Bounds......Page 475
3.3. Boosting for Queries......Page 477
5. Conclusions and Future Work......Page 480
References......Page 481
1. Introduction......Page 483
1.1. Modeling errors: Shannon vs. Hamming......Page 484
1.2. List decoding......Page 485
2. Decoding From Erasures......Page 487
2.2. Algebraic-geometric codes......Page 488
2.3. Binary codes and list decoding from erasures......Page 489
3.1. Random errors......Page 491
3.2. Worst-case errors......Page 492
3.3. Large alphabets......Page 493
4.1. Unique decoding RS codes......Page 494
4.2. Reed-Solomon list decoding......Page 495
5.1. Encoding multiple polynomials......Page 499
5.2. Parvaresh-Vardy codes......Page 500
5.3. Folded RS codes......Page 501
6.1. Binary codes......Page 504
6.2. Optimal rate list-decodable codes over fixed alphabets......Page 505
7. Alternate Bridges Between Worst-case and Random Errors......Page 506
References......Page 507
1. Introduction......Page 511
Approximation Algorithms and Reductions......Page 512
The PCP Theorem......Page 513
The Framework......Page 514
The Unique Games Conjecture......Page 516
3. Max-3Lin and Linearity Test with Perturbation......Page 517
4. Max-kCSP and Gowers Uniformity......Page 518
5. Graph Partitioning and Bourgain’s Noise Sensitivity Theorem......Page 520
Connection to Metric Embeddings......Page 521
6. Majority Is Stablest and Borell’s Theorem......Page 522
7. Max-Cut Problem......Page 524
8. Independent Set and the It Ain’t Over Till It’s Over Theorem......Page 525
Inapproximability Result......Page 526
9. Kernel Clustering and the Propeller Problem......Page 527
10. Conclusion......Page 529
References......Page 530
1. Introduction......Page 533
2. Cuts in Graphs......Page 534
4. Cheeger’s Inequality......Page 535
6. Random Matrix Theory......Page 536
7. Spanning Trees......Page 537
4. Resistor Networks......Page 538
3. Solving Linear Equations in Laplacian Matrices......Page 539
3.1. Direct Methods......Page 540
3.2. Iterative Methods......Page 542
3.3. Preconditioned Iterative Methods......Page 543
4. Approximation by Sparse Graphs......Page 544
4.1. Sparsifiers with a linear number of edges......Page 545
4.2. Nearly-linear time computation......Page 546
5. Subgraph Preconditioners and Support Theory......Page 547
6. Low-stretch Spanning Trees......Page 548
7. Ultra-sparsifiers......Page 550
References......Page 552
1. Introduction......Page 558
2. The Framework......Page 560
3. List-decodable Codes......Page 561
4. Samplers......Page 563
5. Expander Graphs......Page 565
6. Randomness Extractors......Page 567
7. Hardness Amplifiers......Page 570
8. Pseudorandom Generators......Page 575
References......Page 578
Section 16 Numerical Analysis and Scientific Computing......Page 582
1. Introduction......Page 584
2.1. The method......Page 586
2.2. The stabilization mechanism......Page 588
2.3. Convergence properties......Page 589
2.4. The RKDG methods......Page 591
3.1. The HDG methods......Page 592
3.3. Convergence properties......Page 595
3.4. Comparison with other finite element methods......Page 598
4.1. The HDG methods......Page 599
4.2. The stabilization mechanism......Page 602
4.3. Convergence properties......Page 603
5. Conclusion and Ongoing Work......Page 606
References......Page 607
Numerical Analysis of Schrödinger Equations in the Highly Oscillatory Regime......Page 611
1. Introduction......Page 612
2. Schrödinger-type Equations, Observables and Wigner Transforms......Page 613
3. Finite Difference Schemes......Page 617
4. Time-splitting Spectral Approximations......Page 621
6. The Emergence of Bloch Bands......Page 626
6.1. Recapitulation of Bloch's decomposition method......Page 627
6.2. Numerical computation of the Bloch bands......Page 629
7. Bloch Decomposition Based Algorithm......Page 631
References......Page 634
1. Introduction......Page 640
2.1. Definition and Properties of Bisection......Page 643
2.2. Complexity of Bisection......Page 645
3.1. Quasi-interpolation......Page 646
3.2. Principle of Error Equidistribution......Page 647
3.3. Thresholding......Page 648
4.1. Modules of AFEM......Page 651
4.2. Basic Properties of AFEM......Page 652
5.1. Piecewise Constant Data......Page 654
5.2. General Data......Page 655
6.1. The Total Error......Page 657
6.2. Approximation Classes......Page 658
6.3. Quasi-Optimal Cardinality: Vanishing Oscillation......Page 659
6.4. Quasi-Optimal Cardinality: General Data......Page 662
7. Extensions and Limitations......Page 665
References......Page 666
1. Introduction......Page 669
2.1. A characterization......Page 671
2.2. Tight wavelet frame generated from MRA......Page 673
2.3. Other extension principles......Page 680
3. Frame Based Image Restoration......Page 682
3.1. Balanced approach for image inpainting......Page 686
3.2. Role of the redundancy......Page 689
3.3. Accelerated algorithm......Page 690
3.4. Some simulation results......Page 691
References......Page 692
Role of Computational Science in Protecting the Environment: Geological Storage of CO2......Page 699
1. Introduction......Page 700
2. Compositional Flow Model......Page 701
2.1. The Equation of State and Flash Implementation......Page 702
2.2. Iterative IMPEC Implementation......Page 703
3.2. Time-Split Scheme......Page 705
4. Numerical Results......Page 706
5.1. Geomechanics, Faults and Fractures......Page 708
5.2. Phase Behavior and Fluid Properties......Page 710
6.1. Multiscale Temporal and Spatial Discretization......Page 711
6.2. A Posteriori Error Estimates......Page 712
6.3. Multiphysics Couplings and Time-stepping......Page 713
Acknowledgments......Page 715
References......Page 716
Fast Poisson-based Solvers for Linear and Nonlinear PDEs......Page 721
1. Introduction......Page 722
2. The FASP and AMG Methods......Page 724
2.1. FASP: Fast Auxiliary Space Preconditioning......Page 725
2.2. AMG for discrete Poisson equations and variants......Page 726
2.3. A FASP AMG method based on the auxiliary grid......Page 727
2.4. Building blocks: Fast solvers for Poisson-like systems......Page 728
3. Solver-friendly Systems......Page 729
3.1. H(grad), H(curl), and H(div) systems......Page 730
3.2. Mixed finite element methods......Page 731
3.4. Darcy–Stokes–Brinkman model......Page 732
3.5. Plate models......Page 733
4. Solver-friendly Eulerian–Lagrangian Method......Page 734
5. Non-Newtonian Flows......Page 737
5.2. A solver-friendly fully discrete scheme......Page 738
6. Magnetohydrodynamics......Page 740
7. Concluding Remarks......Page 742
References......Page 743
Section 17 Control Theory and Optimization......Page 748
1. Introduction......Page 750
2. Lipschitz Dependence of Viable Trajectories on Initial States and Inverse Mapping Theorems......Page 757
3. Value Function and Optimal Synthesis......Page 763
4. Value Function and Maximum Principle......Page 765
5. Regularity of Optimal Trajectories and Controls......Page 769
References......Page 772
1. Introduction......Page 778
Set Cover Function......Page 781
Entropy Functions......Page 782
3. Associated Polyhedra and Discrete Convexity......Page 783
4. Submodular Function Minimization......Page 784
5. Symmetric Submodular Function Minimization......Page 786
6. Submodular Function Maximization......Page 788
7. Submodular Function Approximation......Page 790
8. Submodular Cost Set Cover......Page 791
9. Submodular Partition......Page 793
References......Page 794
1. Introduction......Page 799
2. Primal-dual Subgradient Methods......Page 802
1. Discrete minimax......Page 803
3. Polynomial-time Interior-point Methods......Page 804
4. Smoothing Technique......Page 808
5. Conclusion......Page 810
References......Page 812
1. Introduction......Page 814
2. Asymptotic Analysis......Page 817
3. Multistage Problems......Page 820
4. Estimates of Stochastic Complexity......Page 822
5. Multistage Complexity......Page 825
6. Approximations of Multistage Stochastic Programs......Page 826
7. Concluding Remarks......Page 828
References......Page 829
1. Mixed Integer Cutting Planes and Lattice Point Free Sets......Page 831
2. Complexity and Closures of Split Polyhedra......Page 834
3. Cutting Plane Proofs......Page 836
4. Cutting Plane Generation from Split Polyhedra......Page 838
5. Integer Points in an Affine Cone......Page 839
References......Page 842
1. Introduction......Page 843
2. Main Differences Between the Known Theories......Page 845
3. The Deterministic Case......Page 848
3.2. Pointwise weighted identity......Page 849
3.3. Controllability/Observability of Linear PDEs......Page 851
3.4. Controllability of Semi-linear PDEs......Page 853
3.5. Controllability of Quasilinear PDEs......Page 856
3.6. Stabilization of hyperbolic equations and further comments......Page 858
4.1. Stochastic Parabolic Equations......Page 861
4.2. Stochastic Hyperbolic Equations......Page 864
4.3. Further comments......Page 865
References......Page 866
Section 18 Mathematics in Science and Technology......Page 870
1. Introduction......Page 872
2.1. The model......Page 874
2.2. Solution of the ODE system......Page 877
2.3. Underlying linearity......Page 878
3.1. The model......Page 880
3.2. Connecting stochastic and deterministic models......Page 881
4. Discrete Time......Page 883
5. Concluding Remarks and Outlook......Page 886
References......Page 887
BSDE and Risk Measures......Page 889
1. Introduction......Page 896
2. First Order Augmented Lagrangian Method......Page 897
3. Semi-smooth Newton Method in Function Spaces......Page 903
4. Optimal Dirichlet Boundary Control......Page 907
5. Sparse Controls......Page 909
6. Time Optimal Control......Page 912
7. L1-data Fitting......Page 916
8. Mathematical Programming......Page 920
References......Page 924
1. Biological Background......Page 926
2. Continuum Modelling Approaches......Page 928
3. Hybrid Modelling Approaches......Page 930
5. Potential Applications II: Exercise......Page 935
References......Page 938
On Markov State Models for Metastable Processes......Page 940
1. Introduction......Page 941
2. Setting the Scene......Page 943
3.1. Core sets and committors......Page 945
3.2. Jump statistics of milestoning process......Page 946
3.3. Invariant measure and self-adjointness......Page 949
4.1. Galerkin projection and eigenvalues......Page 950
4.2. Estimating the eigenvalues from trajectories......Page 953
5.1. Double well potential with diffusive transition region......Page 956
5.2. Two core sets......Page 957
5.3. Estimation from data......Page 959
5.4. Full partition of state space......Page 961
Conclusion......Page 963
References......Page 964
1. Introduction......Page 967
2. Review of Standard Backward SDEs......Page 968
2.1. The linear case......Page 969
2.2. Wellposedness of Backward SDEs......Page 970
2.3. Markov BSDEs......Page 971
2.4. Numerical implications......Page 972
3.1. Hedging under Gamma constraints......Page 973
3.2. Non-uniqueness in L2.......Page 974
3.4. Intuition from uncertain volatility models......Page 975
4.1. A nondominated family of singular measures......Page 977
4.2. The nonlinear generator......Page 978
4.4. Definition......Page 979
5. Wellposedness of Second Order BSDEs......Page 981
6. A Probabilistic Scheme for Fully Nonlinear PDEs......Page 983
References......Page 984
Data Modeling: Visual Psychology Approach and L1/2 Regularization Theory......Page 986
1. Introduction......Page 987
2. Visual Psychology Approach......Page 989
2.1. Scale Space Based Approach......Page 991
2.2. Receptive Field Function Based Approach......Page 995
2.3. Neural Coding Based Approach......Page 1000
3. L1/2 Regularization Theory......Page 1002
3.1. Why L1/2 Regularization?......Page 1005
3.2. How L1/2 Fast Solved?......Page 1007
3.3. What Theory Says?......Page 1008
3.4. How Useful?......Page 1011
4. Concluding Remarks......Page 1013
References......Page 1016
1. Introduction......Page 1020
2.1. Model formulation......Page 1025
2.2. Ill-Posedness......Page 1027
2.3. Solutions......Page 1029
2.4. An example: Two-piece CRRA utility......Page 1032
3.1. The gain part problem......Page 1034
3.2. General solution scheme for quantile formulation......Page 1036
3.3. An example: Goal-reaching model......Page 1038
4. Choquet Minimization: Combinatorial Optimisation in Function Spaces......Page 1040
References......Page 1041
Section 19 Mathematics Education and Popularization of Mathematics......Page 1046
1. Introduction......Page 1048
2. Teaching and Learning Mathematics in South Africa......Page 1049
Example 1: Angle properties of a triangle......Page 1051
Example 2: Polygons and diagonals - or a version of the "mystic rose"......Page 1055
Managing processes and objects in task design and adaptation......Page 1059
Valuing and evaluating diverse learner productions......Page 1060
5. Professional Knowledge Matters in Mathematics Teaching......Page 1061
Acknowledgements......Page 1062
References......Page 1063
Section 20 History of Mathematics......Page 1066
History of Convexity and Mathematical Programming: Connections and Relationships in Two Episodes of Research in Pure and Applied Mathematics of the 20th Century......Page 1068
1. Introduction......Page 1069
Phase 1 - the minimum problem......Page 1070
Phase 2 - investigations of the lattice and associated bodies......Page 1073
Phase 3 - investigations of convex bodies for their own sake......Page 1075
3. From Logistic Problem Solving in the US. Air Force to Mathematical Programming - an Episode of Connections in Applied Mathematics......Page 1077
The Air Force Programming Problem......Page 1078
From problem to theory - the significance of military funding......Page 1081
4. Convexity Meets Mathematical Programming at Princeton - a Mutual Beneficial Relationship......Page 1085
5. Discussion and Conclusion......Page 1086
References......Page 1089
1. Rewriting History......Page 1093
2. Rewriting Historiography......Page 1095
3. Arithmetic Points......Page 1097
4. Holistic Points......Page 1102
5. Generic Points......Page 1111
References......Page 1121
Author Index......Page 1128