Sidoravicius New Trends in Mathematical Physics

Selected contributions of the XVth International Congress on Mathematical Physics
2009
ISBN: 978-90-481-2810-5
Verlag: Springer Netherlands
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 872 Seiten

This book collects selected papers written by invited and plenary speakers of the 15th International Congress on Mathematical Physics (ICMP) in the aftermath of the conference. In extensive review articles and expository texts as well as advanced research articles the world leading experts present the state of the art in modern mathematical physics. New mathematical concepts and ideas are introduced by prominent mathematicalphysicists and mathematicians, covering among others the fields of Dynamical Systems, Operator Algebras, Partial Differential Equations, Probability Theory, Random Matrices, Condensed Matter Physics, Statistical Mechanics, General Relativity, Quantum Mechanics, Quantum Field Theory, Quantum Information and String Theory. All together the contributions in this book give a panoramic view of the latest developments in mathematical physics. They will help readers with a general interest in mathematical physics to get an update on the most recent developments in their field, and give a broad overview on actual and future research directions in this fascinating and rapidly expanding area.

Autoren/Hrsg.

Sidoravicius, Vladas

Inhaltsverzeichnis

1;Preface;4
2;Foreword;6
3;The Henri Poincaré Prize;11
4;Contributors;18
5;Contents;24
6;Entropy of Eigenfunctions;40
6.1;Motivations;40
6.2;Main Result;43
6.3;Outline of the Proof;46
6.3.1;Definition of the Metric Entropy;46
6.3.2;From Classical to Quantum Dynamical Entropy;48
6.3.2.1;Connection with Other Quantum Entropies;50
6.3.2.2;Naive Treatment of the Entropy hn(psih,P q);51
6.3.3;Entropic Uncertainty Principle;52
6.3.4;Applying the Entropic Uncertainty Principle to the Laplacian Eigenstates;53
6.3.4.1;Sharp Energy Localization;53
6.3.4.2;Applying Theorem 7: Step 1;54
6.3.4.3;Coarse-Grained Unstable Jacobian;56
6.3.4.4;Applying Theorem 7: Step 2;57
6.3.4.5;Subadditivity Until the Ehrenfest Time;58
6.4;References;60
7;Stability of Doubly Warped Product Spacetimes;62
7.1;Introduction;62
7.2;Warped Product Spacetimes;63
7.2.1;Asymptotic Behavior;65
7.3;Fuchsian Method;66
7.3.1;Velocity Dominated Equations;67
7.3.2;Velocity Dominated Solution;68
7.4;Stability;69
7.5;References;70
8;Rigorous Construction of Luttinger Liquids Through Ward Identities;72
8.1;Introduction;72
8.2;The Tomonaga Model with Infrared Cutoff;73
8.3;The RG Analysis;74
8.4;The Dyson Equation;76
8.5;The First Ward Identity;78
8.6;The Second Ward Identity;79
8.7;The Euclidean Thirring Model;80
8.8;References;82
9;New Algebraic Aspects of Perturbative and Non-perturbative Quantum Field Theory;83
9.1;Introduction;83
9.2;Lie and Hopf Algebras of Feynman Graphs;84
9.3;From Hochschild Cohomology to Physics;88
9.4;Dyson-Schwinger Equations;89
9.5;Feynman Integrals and Periods of Mixed (Tate) Hodge Structures;93
9.6;References;95
10;Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions;97
10.1;Six-Vertex Model;97
10.2;Phase Diagram of the Six-Vertex Model;100
10.3;Izergin-Korepin Determinantal Formula;101
10.4;The Six-Vertex Model with DWBC and a Random Matrix Model;101
10.5;Asymptotic Formula for the Recurrence Coefficients;103
10.6;Previous Exact Results;105
10.7;Zinn-Justin's Conjecture;108
10.8;Large N Asymptotics of ZN in the Ferroelectric Phase;109
10.9;References;109
11;Mathematical Issues in Loop Quantum Cosmology;111
11.1;Introduction;111
11.2;Quantum Representation and Dynamical Equations;113
11.2.1;Quantum Reduction;113
11.2.2;Dynamics;114
11.3;Quantum Singularity Problem;116
11.4;Examples for Properties of Solutions;117
11.5;Effective Theory;119
11.6;Summary;122
11.7;References;122
12;Boundary Effects on the Interface Dynamics for the Stochastic Allen-Cahn Equation;125
12.1;Introduction;125
12.2;Results and Strategy of Proofs;127
12.3;References;130
13;Dimensional Entropies and Semi-Uniform Hyperbolicity;132
13.1;Introduction;132
13.2;Low Dimension;134
13.2.1;Interval Maps;134
13.2.2;Surface Transformations;135
13.3;Dimensional Entropies;136
13.3.1;Singular Disks;136
13.3.2;Entropy of Collections of Subsets;137
13.3.3;Definitions of the Dimensional Entropies;138
13.4;Other Growth Rates of Submanifolds;139
13.4.1;Volume Growth;139
13.4.2;Resolution Entropies;143
13.5;Properties of Dimensional Entropies;144
13.5.1;Link between Topological and Resolution Entropies;144
13.5.2;Gap Between Uniform and Ordinary Dimensional Entropies;145
13.5.3;Continuity Properties;146
13.6;Hyperbolicity from Entropies;147
13.6.1;A Ruelle-Newhouse Type Inequality;147
13.6.2;Entropy-Expanding Maps;147
13.6.3;Entropy-Hyperbolicity;149
13.6.4;Examples of Entropy-Hyperbolic Diffeomorphisms;150
13.7;Further Directions and Questions;150
13.7.1;Variational Principles;150
13.7.2;Dimensional Entropies of Examples;150
13.7.3;Other Types of Dimensional Complexity;151
13.7.4;Necessity of Topological Assumptions;151
13.7.5;Entropy-Hyperbolicity;151
13.7.6;Generalized Entropy-Hyperbolicity;152
13.8;Cr Sizes;152
13.9;References;153
14;The Scaling Limit of (Near-)Critical 2D Percolation;154
14.1;Introduction;154
14.1.1;Critical Scaling Limits and SLE;154
14.1.2;Percolation;157
14.2;The Critical Loop Process;158
14.2.1;General Features;158
14.2.2;Construction of a Single Loop;159
14.3;The Near-Critical Scaling Limit;161
14.4;References;162
15;Black Hole Entropy Function and Duality;164
15.1;Introduction;164
15.2;Entropy Function and Electric/Magnetic Duality Covariance;165
15.3;Application to N=2 Supergravity;167
15.4;Duality Invariant OSV Integral;170
15.5;References;170
16;Weak Turbulence for Periodic NLS;172
16.1;Introduction;172
16.2;NLS as an Infinite System of ODEs;174
16.3;Conditions on a Finite Set LambdaZ2 ;175
16.4;Arnold Diffusion for the Toy Model ODE;176
16.5;Construction of the Resonant Set Lambda;177
16.6;References;179
17;Angular Momentum-Mass Inequality for Axisymmetric Black Holes;180
17.1;Introduction;180
17.2;Variational Principle for the Mass;181
17.3;References;184
18;Almost Everything About the Fibonacci Operator;186
18.1;Introduction;186
18.2;The Trace Map;187
18.3;The Cantor Structure and the Dimension of the Spectrum;189
18.4;The Spectral Type;191
18.5;Bounds on Wavepacket Spreading;193
18.6;References;195
19;Entanglement-Assisted Quantum Error-Correcting Codes;197
19.1;Introduction;197
19.2;Notations;198
19.3;Entanglement-Assisted Quantum Error-Correcting Codes;199
19.3.1;The Channel Model: Discretization of Errors;200
19.3.2;The Entanglement-Assisted Canonical Code;201
19.3.3;The General Case;203
19.3.4;Distance;205
19.3.5;Generalized F4 Construction;205
19.3.6;Bounds on Performance;206
19.4;Conclusions;207
19.5;References;207
20;Particle Decay in Ising Field Theory with Magnetic Field;209
20.1;Ising Field Theory;209
20.2;Evolution of the Mass Spectrum;211
20.3;Particle Decay off the Critical Isotherm;212
20.4;Unstable Particles in Finite Volume;218
20.5;References;220
21;Fluctuations and Large Deviations in Non-equilibrium Systems;222
21.1;Introduction;222
21.2;Large Deviation Function of the Density;223
21.3;Free Energy Functional;224
21.4;Simple Exclusion Processes (SSEP);226
21.5;The Large Deviation Function F(rho(x)) for the SSEP;228
21.6;The Matrix Ansatz for the Symmetric Exclusion Process;229
21.7;Additivity as a Consequence of the Matrix Ansatz;232
21.8;Large Deviation Function of Density Profiles;233
21.9;Non-locality of the Large Deviation Functional of the Density and Long Range Correlations;235
21.10;The Macroscopic Fluctuation Theory;237
21.11;Large Deviation of the Current;238
21.12;Generalized Detailed Balance and the Fluctuation Theorem;239
21.13;Current Fluctuations in the SSEP;241
21.14;The Additivity Principle;242
21.15;References;244
22;Robust Heterodimensional Cycles and Tame Dynamics;246
22.1;Robust Heterodimensional Cycles;246
22.1.1;General Setting;246
22.1.2;Basic Definitions;248
22.1.3;Robust Cycles at Heterodimensional Cycles;249
22.1.4;Questions and Consequences;251
22.2;Cycles and Non-hyperbolic Tame Dynamics;252
22.2.1;Setting;252
22.2.2;Tangencies, Heterodimensional Cycles, and Examples;253
22.3;Robust Homoclinic Tangencies, Non-dominated Dynamics, and Heterodimensional Cycles;255
22.4;Ingredients of the Proof of Theorem 2;257
22.4.1;Strong Homoclinic Intersections of Saddle-Nodes;258
22.4.2;Model Blenders;260
22.5;References;262
23;Hamiltonian Perturbations of Hyperbolic PDEs: from Classification Results to the Properties of Solutions;265
23.1;Introduction;265
23.2;Towards Classification of Hamiltonian PDEs;269
23.3;Deformation Theory of Integrable Hierarchies;272
23.4;Frobenius Manifolds and Integrable Hierarchies of the Topological Type;283
23.5;Critical Behaviour in Hamiltonian PDEs, the Dispersionless Case;298
23.6;Universality in Hamiltonian PDEs;303
23.7;References;307
24;Lattice Supersymmetry from the Ground Up;311
24.1;References;318
25;Convergence of Symmetric Trap Models in the Hypercube;319
25.1;Introduction;319
25.1.1;The Model;320
25.2;Convergence to the K Process;321
25.2.1;Proof of Theorem 1;322
25.2.1.1;Coupling of Xd,M and YM;322
25.2.1.2;Coupling of Xd and Xd,M;323
25.2.1.3;Conclusion of proof of Theorem 1;325
25.3;The REM-Like Trap Model and the Random Hopping Times Dynamics for the REM;328
25.3.1;The REM-Like Trap Model;328
25.3.2;Random Hopping Times Dynamics for the REM;329
25.4;References;330
26;Spontaneous Replica Symmetry Breaking in the Mean Field Spin Glass Model;332
26.1;Introduction;332
26.2;The Mean Field Spin Glass Model. Basic Definitions;335
26.3;The Thermodynamic Limit;337
26.4;The Parisi Representation for the Free Energy;338
26.5;Conclusion and Outlook for Future Developments;342
26.6;References;343
27;Surface Operators and Knot Homologies;345
27.1;Introduction;345
27.2;Gauge Theory and Categorification;348
27.2.1;Incorporating Surface Operators;350
27.2.2;Braid Group Actions;352
27.3;Surface Operators and Knot Homologies in N=2 Gauge Theory;355
27.3.1;Donaldson-Witten Theory and the Equivariant Knot Signature;355
27.3.2;Seiberg-Witten Theory;358
27.4;Surface Operators and Knot Homologies in N=4 Gauge Theory;362
27.5;References;372
28;Conformal Field Theory and Operator Algebras;376
28.1;Introduction;376
28.2;Conformal Quantum Field Theory;377
28.3;Representation Theory;380
28.4;Classification Theory;381
28.5;Moonshine Conjecture;383
28.6;References;385
29;Diffusion and Mixing in Fluid Flow: A Review;388
29.1;Introduction;388
29.2;The Heart of the Matter;394
29.3;Open Questions;398
29.4;References;399
30;Random Schrödinger Operators: Localization and Delocalization, and All That;401
30.1;Random Schrödinger Operators;401
30.2;Basic Examples of Random Schrödinger Operators;402
30.2.1;The Anderson (Tight-Binding) Model;403
30.2.2;The (Continuum) Anderson Hamiltonian;403
30.2.3;The Random Landau Hamiltonian;403
30.2.4;The Poisson Hamiltonian;404
30.3;The Metal-Insulator Transition;404
30.4;The Spectral Metal-Insulator Transition;405
30.4.1;Anderson Localization;405
30.4.1.1;Anderson Localization in the One-Dimensional Case;406
30.4.1.2;Anderson Localization in the Multi-Dimensional Case;406
30.4.1.3;Multiplicity of Eigenvalues in Intervals of Anderson Localization;406
30.4.2;Absolutely Continuous Spectrum;407
30.4.3;The Spectral Metal-Insulator Transition for the Anderson Model on the Bethe Lattice;407
30.5;The Dynamical Metal-Insulator Transition;408
30.5.1;Dynamical Localization;408
30.5.2;Transport Exponents;409
30.5.3;The Dynamical Spectral Regions;410
30.5.4;The Region of Complete Localization;411
30.6;The Dynamical Transition in the Random Landau Hamiltonian;412
30.7;References;414
31;Unifying R-Symmetry in M-Theory;419
31.1;Introduction;419
31.2;Kinematics;422
31.2.1;Definition of e10 and K(e10);422
31.2.2;Level Decompositions for D=11, IIA and IIB;423
31.2.3;Representations of K(e10);424
31.3;Dynamics;426
31.4;Discussion;428
31.4.1;Remarks;428
31.4.2;Outlook;429
31.5;References;430
32;Stable Maps are Dense in Dimensional One;432
32.1;Introduction;432
32.2;Density of Hyperbolicity;433
32.3;Quasi-Conformal Rigidity;434
32.4;How to Prove Rigidity?;434
32.4.1;The Strategy of the Proof of QC-Rigidity;435
32.5;Enhanced Nest Construction;436
32.6;Small Distortion of Thin Annuli;438
32.7;Approximating Non-renormalizable Complex Polynomials;440
32.8;References;441
33;Large Gap Asymptotics for Random Matrices;442
33.1;References;448
34;On the Derivation of Fourier's Law;449
34.1;Introduction;449
34.2;Coupled Oscillators;450
34.3;Closure Equations;452
34.4;Kinetic Limit;455
34.5;References;459
35;Noncommutative Manifolds and Quantum Groups;460
35.1;Introduction;460
35.2;The Algebras and the Representations;462
35.2.1;The Algebras of Functions and of Symmetries;462
35.2.2;The Equivariant Representation of A(SUq(2));465
35.2.3;The Spin Representation;466
35.3;The Equivariant Dirac Operator;469
35.4;The Real Structure;471
35.4.1;The Tomita Operator of the Regular Representation;471
35.4.2;The Real Structure on Spinors;472
35.5;The Local Index Formula for SUq(2);474
35.5.1;The Cosphere Bundle and the Dimension Spectrum;475
35.5.2;The Local Index Formula for 3-Dimensional Geometries;477
35.5.3;The Pairing Between HC1 and K1;479
35.6;References;481
36;Topological Strings on Local Curves;483
36.1;Introduction;483
36.2;Topological Strings on Local Curves;485
36.2.1;A Model;485
36.2.2;Relation to Hurwitz Theory;486
36.2.3;Mirror Symmetry from Large Partitions;488
36.2.4;Higher Genus and Matrix Models;490
36.3;Phase Transitions, Critical Behavior and Double-Scaling Limit;491
36.3.1;Review of Phase Transitions in Topological String Theory;491
36.3.2;Phase Transitions for Local Curves;493
36.4;Non-perturbative Effects and Large Order Behavior;495
36.5;References;498
37;Repeated Interaction Quantum Systems;500
37.1;Introduction;500
37.1.1;Literature;502
37.2;Deterministic Systems;502
37.2.1;Mathematical Description;502
37.2.2;Results;506
37.2.3;Asymptotic State;506
37.2.4;Correlations & Reconstruction of Initial State;507
37.3;Random Systems;507
37.3.1;Dynamics and Random Matrix Products;507
37.3.2;Results;509
37.4;An Example: Spins;511
37.5;Some Proofs;515
37.5.1;Outline of the Proof of Theorem 3;515
37.5.2;Outline of the Proof of Theorem 9;516
37.6;References;519
38;String-Localized Quantum Fields, Modular Localization, and Gauge Theories;520
38.1;The Notion of String-Localized Quantum Fields;520
38.2;Modular Localization and the Construction of Free String-Localized Fields;522
38.3;Results on Free String-Localized Fields;524
38.3.1;Fields and Two-Point Functions;524
38.3.1.1;Massless Infinite Spin Particles;524
38.3.1.2;Vector Potentials for Photons;525
38.3.1.3;Massive Vector Bosons;526
38.3.1.4;Massive Bosonic Particles with Arbitrary Spin;527
38.3.1.5;Tensor Potentials for Linearized Gravitons;527
38.3.2;Feynman Propagators;528
38.4;Outlook: Interacting String-Localized Fields;529
38.5;References;532
39;Kinks and Particles in Non-integrable Quantum Field Theories;534
39.1;Introduction;534
39.2;A Semiclassical Formula;538
39.3;Symmetric Wells;540
39.4;Asymmetric Wells;544
39.5;Conclusions;547
39.6;References;548
40;Exponential Decay Laws in Perturbation Theory of Threshold and Embedded Eigenvalues;549
40.1;Introduction;549
40.2;The Basic Formula;552
40.3;The Results;554
40.3.1;Properly Embedded Eigenvalues;554
40.3.2;Threshold Eigenvalues;555
40.4;A Uniqueness Result;557
40.5;Examples;558
40.5.1;Example 1: One Channel Case, nu=-1 ;558
40.5.2;Example 2: Two Channel Case, nu=-1,1 ;559
40.5.3;Example 3: Two Channel Radial Case, nu>=3;560
40.6;References;561
41;Energy Diffusion and Superdiffusion in Oscillators Lattice Networks;563
41.1;Introduction;563
41.2;Conservative Stochastic Dynamics;565
41.3;Diffusive Evolution: Green-Kubo Formula;568
41.4;Kinetic Limits: Phonon Boltzmann Equation;569
41.5;Levy's Superdiffusion of Energy;570
41.6;References;570
42;Trying to Characterize Robust and Generic Dynamics;572
42.1;Introduction;572
42.2;Robust Transitivity: Hyperbolicity, Partial Hyperbolicity and Dominated Splitting;574
42.2.1;Hyperbolicity;576
42.2.2;Partial Hyperbolicity;577
42.2.3;Dominated Splitting;578
42.2.4;A General Question About ``Weak Form of Hyperbolicity'';580
42.2.5;Robust Transitivity and Mechanisms: Heterodimensional Cycle;580
42.3;Wild Dynamics;581
42.3.1;Wild Dynamic and Homoclinic Tangency;581
42.3.2;Surfaces Diffeomorphisms and Beyond;582
42.4;Generic Dynamics: Mechanisms and Phenomenas;583
42.5;References;584
43;Dynamics of Bose-Einstein Condensates;587
43.1;Introduction;587
43.2;Heuristic Derivation of the Gross-Pitaevskii Equation;589
43.3;Main Results;593
43.4;General Strategy of the Proof and Previous Results;596
43.5;Convergence to the Infinite Hierarchy;598
43.6;Uniqueness of the Solution to the Infinite Hierarchy;602
43.6.1;Higher Order Energy Estimates;603
43.6.2;Expansion in Feynman Graphs;605
43.7;References;611
44;Locality Estimates for Quantum Spin Systems;612
44.1;Introduction;612
44.2;Lieb-Robinson Bounds;614
44.3;Quasi-Locality of the Dynamics;619
44.4;Exponential Clustering;622
44.5;The Lieb-Schultz-Mattis Theorem;625
44.5.1;The Result and Some Words on the Proof;626
44.5.2;A More Detailed Outline of the Proof;628
44.5.2.1;Constructing the Trial State;628
44.5.2.2;Locality and the Trial State;631
44.5.2.3;The Estimates;631
44.5.2.4;Generalizations;632
44.6;References;635
45;On Resolvent Identities in Gaussian Ensembles at the Edge of the Spectrum;636
45.1;Introduction;636
45.2;Proof of Theorems 1 and 3;643
45.3;Proof of Theorems 4 and 5;645
45.4;Non-Gaussian Case;646
45.5;References;647
46;Energy Current Correlations for Weakly Anharmonic Lattices;649
46.1;Introduction;649
46.2;Anharmonic Lattice Dynamics;650
46.3;Energy Current Correlations;653
46.4;The Linearized Collision Operator;657
46.5;Gaussian Fluctuation Theory;659
46.6;References;660
47;Metastates, Translation Ergodicity, and Simplicity of Thermodynamic States in Disordered Systems: an Illustration;662
47.1;Introduction;662
47.2;The Sherrington-Kirkpatrick Model and the Parisi Replica Symmetry Breaking Solution;663
47.3;Open Problems;664
47.4;Metastates;664
47.5;Invariance and Ergodicity;665
47.6;A Strategy for Rigorous Studies of Spin Glasses;666
47.7;Summary;670
47.8;References;670
48;Random Matrices, Non-intersecting Random Walks, and Some Aspects of Universality;672
48.1;Introduction;672
48.1.1;Selected Basic Facts from Random Matrix Theory;673
48.1.2;The Karlin-McGregor Formula;673
48.2;The Models;674
48.2.1;Longest Increasing Subsequence of a Random Permutation;674
48.2.2;ABC-Hexagon;676
48.2.3;Last Passage Percolation;676
48.2.4;Non-intersecting Brownian Motion;679
48.3;Universality;680
48.3.1;Last Passage Percolation;681
48.3.2;Non-intersecting Random Walks;681
48.4;References;683
49;Homogenization of Periodic Differential Operators as a Spectral Threshold Effect;686
49.1;Introduction;686
49.2;Periodic DO's. The Effective Matrix;687
49.3;Homogenization of Periodic DO's. Principal Term of Approximation for the Resolvent;689
49.4;More Accurate Approximation for the Resolvent in the L2-Operator Norm;690
49.5;(L2->H1)-Approximation of the Resolvent. Approximation of the Fluxes in L2;692
49.6;The Method of Investigation;693
49.7;Some Applications;697
49.8;On Further Development of the Method;700
49.9;References;701
50;ABCD and ODEs;703
50.1;Introduction;703
50.2;Bethe Ansatz for Classical Lie Algebras;706
50.3;The Pseudo-Differential Equations;707
50.3.1;An-1 models;707
50.3.2;Dn models;708
50.3.3;Bn models;709
50.3.4;Cn models;710
50.4;Conclusions;711
50.5;References;712
51;Nonrational Conformal Field Theory;714
51.1;Introduction;714
51.2;Constraints from Conformal Symmetry;716
51.2.1;Motivation: Chiral Factorization of Physical Correlation Functions;716
51.2.2;Vertex Algebras;717
51.2.3;Representations of Vertex Algebras;718
51.2.4;Conformal Blocks;718
51.2.4.1;Definition;718
51.2.4.2;Insertions of the Vacuum Representation;719
51.2.4.3;Deformations of the Complex Structure of X;719
51.2.5;Correlation Functions vs. Hermitian Forms;720
51.3;Behavior Near the Boundary of Moduli Space;721
51.3.1;Gluing of Riemann Surfaces;722
51.3.2;Gluing of Conformal Blocks;725
51.3.2.1;Conformal Ward Identities;726
51.3.2.2;Decorated Marking Graphs;726
51.3.3;Correlation Functions;727
51.3.3.1;First Requirement: Factorization of Conformal Blocks;728
51.3.3.2;Second Requirement: Factorization of the Hermitian form HSigma;728
51.3.4;Conformal Blocks as Matrix Elements;729
51.3.4.1;Chiral Vertex Operators;729
51.4;From one Boundary to Another;730
51.4.1;The Modular Groupoid;731
51.4.1.1;Generators;732
51.4.1.2;Relations;733
51.4.1.2.1;Relations supported on surfaces of genus zero.;733
51.4.1.2.2;Relations supported on surfaces of genus one.;733
51.4.2;Representation of the Generators on Spaces of Conformal Blocks;734
51.4.3;Representation of the Relations on Spaces of Conformal Blocks;735
51.4.3.1;Insertions of the Vacuum;735
51.5;Notion of a Stable Modular Functor;736
51.5.1;Towers of Representations of the Modular Groupoid;736
51.5.2;Unitary Modular Functors;738
51.5.3;Similarity of Modular Functors;739
51.5.4;Friedan-Shenker Modular Geometry;739
51.6;Example of a Nonrational Modular Functor;740
51.6.1;Unitary Positive Energy Representations of the Virasoro Algebra;741
51.6.1.1;Free Field Representation;741
51.6.2;Construction of Virasoro Conformal Blocks in Genus Zero;742
51.6.2.1;Free Field Construction of Chiral Vertex Operators;742
51.6.3;Factorization Property;743
51.6.4;The Hilbert Space Structure;744
51.6.5;Extension to Higher Genus;745
51.6.6;Remarks;746
51.7;Existence of a Canonical Scalar Product?;746
51.7.1;Existence of a Canonical Hermitian Form from the Factorization Property;747
51.7.2;Unitary Fusion;749
51.7.3;Associativity of Unitary Fusion;750
51.7.4;Discussion;752
51.8;Outlook;752
51.8.1;Modular Functors from W-algebras and Langlands Duality;752
51.8.2;Boundary CFT;753
51.8.3;Nonrational Verlinde Formula?;753
51.9;References;755
52;Kinetically Constrained Models;757
52.1;References;767
53;The Distributions of Random Matrix Theory and their Applications;769
53.1;Random Matrix Models: Gaussian Ensembles;769
53.1.1;Largest Eigenvalue Distributions Fbeta. Painlevé II Representations;770
53.1.1.1;Tail behavior of Fbeta;772
53.1.1.2;Numerical evaluation of Fbeta;773
53.1.2;Next-Largest, Next-Next Largest, Etc. Eigenvalue Distributions;773
53.2;Universality Theorems;773
53.2.1;Invariant Ensembles;773
53.2.2;Wigner Ensembles;775
53.3;Multivariate Statistical Analysis;775
53.3.1;Principal Component Analysis (PCA);776
53.3.2;Testing the Null Hypothesis;776
53.3.3;Spiked Populations: BBP Phase Transition;777
53.4;Conclusions;779
53.5;References;779
54;Hybrid Formalism and Topological Amplitudes;782
54.1;Introduction;782
54.2;Compactified String Theory in RNS and Hybrid Variables;783
54.2.1;Hybrid Variables;783
54.2.2;RNS Variables;787
54.2.3;Field Redefinition from RNS to Hybrid Variables;788
54.2.4;Physical State Conditions and N=4-embeddings;791
54.2.5;Massless Vertex Operators;793
54.3;Amplitudes and Correlation Functions;795
54.3.1;Amplitudes;795
54.3.2;Correlation Functions of Chiral Bosons;797
54.4;Topological Amplitudes;798
54.4.1;Generalities;798
54.4.2;R-charge (g-1,g-1);799
54.4.3;R-charge (1-g,1-g);800
54.4.4;R-charges (g-1,1-g) and (1-g,g-1);802
54.4.5;Summary of the Amplitude Computation;803
54.5;Appendix: Conventions and Notations;804
54.5.1;Spinors and Superspace;804
54.5.2;Hybrid Variables and N =2 Algebra;805
54.5.3;The Integrated Vertex Operator;806
54.6;Appendix: Mapping the RNS to the Hybrid Variables;807
54.6.1;Field Redefinition from RNS to Chiral GS Variables;807
54.6.2;Similarity Transformation Relating Chiral GS to Hybrid Variables;807
54.6.3;Hermitian Conjugation of the Hybrid Variables;808
54.6.4;Hermitian Conjugation of the RNS Variables;809
54.7;Appendix: Vertex Operators;812
54.7.1;Massless RNS Vertex Operators;812
54.7.2;Universal Massless Multiplets;814
54.7.3;Compactification Dependent Massless Multiplets;814
54.7.3.1;Kähler Moduli;815
54.7.3.2;Complex Structure Moduli;816
54.8;References;817
55;Quantum Phases of Cold Bosons in an Optical Lattice;819
55.1;Introduction;820
55.2;Reflection Positivity;824
55.3;Proof of BEC for Small lambda and T;825
55.4;Absence of BEC and Mott Insulator Phase;830
55.5;The Non-interacting Gas;834
55.6;Conclusion;835
55.7;References;835
56;Random Walks in Random Environments in the Perturbative Regime;837
56.1;Introduction;837
56.2;Local Limits for Exit Measures;839
56.3;References;840
57;Appendix: Complete List of Abstracts;841
57.1;Young Researchers Symposium Plenary Lectures;841
57.1.1;Dynamics of Quasiperiodic Cocycles and the Spectrum of the Almost Mathieu Operator;841
57.1.2;Magic in Superstring Amplitudes;841
57.1.3;The Instructive History of Quantum Groups;842
57.1.4;Scaling Limit of Two-Dimensional Critical Percolation;842
57.1.5;Topics in Dynamics and Physics;842
57.1.6;Quantum Dynamics in a Random Environment;843
57.1.7;Geometry of Low Dimensional Manifolds;843
57.1.8;Gauge Theory and the Geometric Langlands Program ;843
57.2;XV International Congress on Mathematical Physics Plenary Lectures;844
57.2.1;Mathematical Developments Around the Ginzburg-Landau Model in 3D Program;844
57.2.2;The Riemann-Hilbert Problem: Applications;844
57.2.3;Fluctuations and Large Deviations in Non-equilibrium Systems;844
57.2.4;Hamiltonian Perturbations of Hyperbolic PDE's: from Classification of Equations to Properties of Solutions;845
57.2.5;Spontaneous Replica Symmetry Breaking in the Mean Field Spin Glass Model;845
57.2.6;Spectral Properties of Quasi-Periodic Schrödinger Operators: Treating Small Denominators without KAM;845
57.2.7;Conformal Field Theory and Operator Algebras;846
57.2.8;Random Schrödinger Operators, Localization and Delocalization, and all that;846
57.2.9;Perelman's Work on the Geometrization Conjecture;847
57.2.10;Trying to Characterize Robust and Generic Dynamics;847
57.2.11;Cauchy Problem in General Relativity;848
57.2.12;Survey of Recent Mathematical Progress in the Understanding of Critical 2d Systems;848
57.2.13;Random Methods in Quantum Information Theory;848
57.2.14;Gauge Fields, Strings and Integrable Systems;849
57.3;XV International Congress on Mathematical Physics Specialized Sessions;849
57.3.1;Condensed Matter Physics;849
57.3.1.1;Rigorous Construction of Luttinger Liquids Through Ward Identities;849
57.3.1.2;Edge and Bulk Currents in the Integer Quantum Hall Effect;850
57.3.1.3;Quantum Phases of Cold Bosons in Optical Lattices;850
57.3.2;Dynamical Systems;850
57.3.2.1;Statistical Stability for Hénon Maps of Benedics-Carleson Type;850
57.3.2.2;Entropy and the Localization of Eigenfunctions;851
57.3.2.3;The Spectrum of the Almost Mathieu Operator in the Subcritical Regime;851
57.3.2.4;Hyperbolicity Through Entropy;852
57.3.2.5;Robust Cycles and Non-dominated Dynamics;852
57.3.2.6;Hyperbolicity in One Dimensional Dynamics;852
57.3.3;Equilibrium Statistical Mechanics;853
57.3.3.1;The Scaling Limit of (Near-)Critical 2d Percolation;853
57.3.3.2;Short-Range Spin Glasses in a Magnetic Field;853
57.3.3.3;Relaxation Times of Kinetically Constrained Spin Models with Glassy Dynamics;854
57.3.4;Non-equilibrium Statistical Mechanics;854
57.3.4.1;Current Fluctuations in Boundary Driven Interacting Particle Systems;854
57.3.4.2;Fourier Law and Random Walks in Evolving Environments;855
57.3.4.3;Asymptotics of Repeated Interaction Quantum Systems;855
57.3.4.4;Linear Response of Non-equilibrium Steady States for Open Quantum System;855
57.3.4.5;Derivation of the Gross-Pitaevski Equation for the Dynamics of Bose-Einstein Condensates;856
57.3.4.6;Energy Transport in One-Dimensional Chains: Predictions from the Phonon Kinetic Equation;856
57.3.5;Exactly Solvable Systems;857
57.3.5.1;Correlation Functions and Hidden Fermionic Structure of the XYZ Spin Chain ;857
57.3.5.2; Particle Decay in Ising Field Theory with Magnetic Field;857
57.3.5.3;ABCD-Integrable Models and Ordinary Differential Equations;857
57.3.6;General Relativity;858
57.3.6.1;Einstein Spaces as Attractors for the Einstein Flow ;858
57.3.6.2; Loop Quantum Cosmology;858
57.3.6.3;The Red-Shift Effect and Radiation Decay on Black Hole Space-Times ;858
57.3.6.4;Angular Momentum-Mass Inequality for Axisymmetric Black Holes;859
57.3.6.5;Black Hole Entropy in Supergravity and String Theory;859
57.3.6.6;Infinite-Dimensional R-Symmetry in Supergravity;859
57.3.7;Operator Algebras;860
57.3.7.1;From Vertex Algebras to Local Nets of von Neuman Algebras;860
57.3.7.2;Non-Commutative Manifolds and Quantum Groups;860
57.3.7.3;L2 Invariants, Free Probability and Operator Algebras;860
57.3.8;Partial Differential Equations;861
57.3.8.1;Weak Turbulence for Periodic NSL;861
57.3.8.2;Ginzburg-Landau Dynamics;861
57.3.8.3;On the Derivation of Furier's Law;861
57.3.8.4;TBA;862
57.3.8.5;A Criterion for the Logarithmic Sobolev Inequality;862
57.3.8.6;Singular Behaviour in Chemotaxis Models;862
57.3.9;Probability Theory;863
57.3.9.1;Aging in the Infinite Volume REM-Like Trap Model at Low Temperature;863
57.3.9.2;From Planar Gaussian Zeros to Gravitational Allocation;863
57.3.9.3;Random Walks in Random Environments in the Perturbative Regime ;863
57.3.10;Quantum Mechanics;864
57.3.10.1;Recent Progress in the Spectral Theory of Quasi-Periodic Operators ;864
57.3.10.2;Recent Results on Localization for Random Schrödinger Operators;864
57.3.10.3;Quantum Dynamics and Enhanced Diffusion for Passive Scalar;864
57.3.10.4;Lieb-Thirring Inequalities, Recent Results;865
57.3.10.5;Exponential Decay Laws in Perturbation Theory of Threshold and Embedded Eigenvalues ;865
57.3.10.6;Homogenization of Periodic Operators of Mathematical Physics as a Spectral Threshold Effect;865
57.3.11;Quantum Field Theory;866
57.3.11.1;Algebraic Aspects of Perturbative and Non-Perturbative QFT;866
57.3.11.2;Quantum Field Theory in Curved Space-Time ;867
57.3.11.3;String-Localized Quantum Fields, Modular Localization, and Gauge Theories;867
57.3.11.4;Quantization of the Teichmüller Spaces: Quantum Field Theoretical Applications;868
57.3.12;2D Quantum Field Theory;868
57.3.12.1;Lattice Supersymmetry From the Ground Up ;868
57.3.12.2;Analytical Solution for the Effective Charging Energy of the Single Electron Box ;868
57.3.12.3;Breaking Integrability;869
57.3.13;Quantum Information;869
57.3.13.1;One-and-a-Half Quantum de Finetti Theorems ;869
57.3.13.2;Catalytic Quantum Error Correction;870
57.3.13.3;Quantum State Transformations and the Schubert Calculus;870
57.3.13.4;The Local Hamiltonian Problem ;870
57.3.13.5;The Information-Disturbance Tradeoff and the Continuity of Stinespring's Representation;871
57.3.13.6;Locality Estimates for Quantum Spin Systems;871
57.3.14;Random Matrices;872
57.3.14.1;Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions ;872
57.3.14.2;Probabilities of a Large Gap in the Scaled Spectrum of Random Matrices;872
57.3.14.3;Random Matrices, Asymptotic Analysis, and d-bar Problems;873
57.3.14.4;Central Limit Theorems for Non-intersecting Random Walks ;873
57.3.14.5;On the Distribution of Largest Eigenvalues in Random Matrix Ensembles;873
57.3.14.6;Non-Intersecting Brownian Excursions ;873
57.3.15;Stochastic PDE;874
57.3.15.1;Degenerately Forced Fluid Equations: Ergodicity and Solvable Models;874
57.3.15.2; Microscopic Stochastic Models for the Study of Thermal Conductivity;874
57.3.15.3;Boundary Effects on the Interface Dynamics for the Stochastic Allen-Cahn Equation ;875
57.3.16;String Theory;875
57.3.16.1;Topological Strings and (Almost) Modular Forms ;875
57.3.16.2;Gauge Theory and Link Homologies;875
57.3.16.3;Non-Geometric String Backgrounds;876
57.3.16.4;Topological Reduction of Supersymmetric Gauge Theories and S-Duality ;876
57.3.16.5;Phase Transitions in Topological String Theory;876
57.3.16.6;Hyper-Multiplet Couplings in N=2 Effective Action;877