New Trends in Mathematical Physics (eBook)

Preface 4
Foreword 6
The Henri Poincaré Prize 11
Contributors 18
Contents 24
Entropy of Eigenfunctions 40
Motivations 40
Main Result 43
Outline of the Proof 46
Definition of the Metric Entropy 46
From Classical to Quantum Dynamical Entropy 48
Connection with Other Quantum Entropies 50
Naive Treatment of the Entropy hn(psih,P q) 51
Entropic Uncertainty Principle 52
Applying the Entropic Uncertainty Principle to the Laplacian Eigenstates 53
Sharp Energy Localization 53
Applying Theorem 7: Step 1 54
Coarse-Grained Unstable Jacobian 56
Applying Theorem 7: Step 2 57
Subadditivity Until the Ehrenfest Time 58
References 60
Stability of Doubly Warped Product Spacetimes 62
Introduction 62
Warped Product Spacetimes 63
Asymptotic Behavior 65
Fuchsian Method 66
Velocity Dominated Equations 67
Velocity Dominated Solution 68
Stability 69
References 70
Rigorous Construction of Luttinger Liquids Through Ward Identities 72
Introduction 72
The Tomonaga Model with Infrared Cutoff 73
The RG Analysis 74
The Dyson Equation 76
The First Ward Identity 78
The Second Ward Identity 79
The Euclidean Thirring Model 80
References 82
New Algebraic Aspects of Perturbative and Non-perturbative Quantum Field Theory 83
Introduction 83
Lie and Hopf Algebras of Feynman Graphs 84
From Hochschild Cohomology to Physics 88
Dyson-Schwinger Equations 89
Feynman Integrals and Periods of Mixed (Tate) Hodge Structures 93
References 95
Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions 97
Six-Vertex Model 97
Phase Diagram of the Six-Vertex Model 100
Izergin-Korepin Determinantal Formula 101
The Six-Vertex Model with DWBC and a Random Matrix Model 101
Asymptotic Formula for the Recurrence Coefficients 103
Previous Exact Results 105
Zinn-Justin's Conjecture 108
Large N Asymptotics of ZN in the Ferroelectric Phase 109
References 109
Mathematical Issues in Loop Quantum Cosmology 111
Introduction 111
Quantum Representation and Dynamical Equations 113
Quantum Reduction 113
Dynamics 114
Quantum Singularity Problem 116
Examples for Properties of Solutions 117
Effective Theory 119
Summary 122
References 122
Boundary Effects on the Interface Dynamics for the Stochastic Allen-Cahn Equation 125
Introduction 125
Results and Strategy of Proofs 127
References 130
Dimensional Entropies and Semi-Uniform Hyperbolicity 132
Introduction 132
Low Dimension 134
Interval Maps 134
Surface Transformations 135
Dimensional Entropies 136
Singular Disks 136
Entropy of Collections of Subsets 137
Definitions of the Dimensional Entropies 138
Other Growth Rates of Submanifolds 139
Volume Growth 139
Resolution Entropies 143
Properties of Dimensional Entropies 144
Link between Topological and Resolution Entropies 144
Gap Between Uniform and Ordinary Dimensional Entropies 145
Continuity Properties 146
Hyperbolicity from Entropies 147
A Ruelle-Newhouse Type Inequality 147
Entropy-Expanding Maps 147
Entropy-Hyperbolicity 149
Examples of Entropy-Hyperbolic Diffeomorphisms 150
Further Directions and Questions 150
Variational Principles 150
Dimensional Entropies of Examples 150
Other Types of Dimensional Complexity 151
Necessity of Topological Assumptions 151
Entropy-Hyperbolicity 151
Generalized Entropy-Hyperbolicity 152
Cr Sizes 152
References 153
The Scaling Limit of (Near-)Critical 2D Percolation 154
Introduction 154
Critical Scaling Limits and SLE 154
Percolation 157
The Critical Loop Process 158
General Features 158
Construction of a Single Loop 159
The Near-Critical Scaling Limit 161
References 162
Black Hole Entropy Function and Duality 164
Introduction 164
Entropy Function and Electric/Magnetic Duality Covariance 165
Application to N=2 Supergravity 167
Duality Invariant OSV Integral 170
References 170
Weak Turbulence for Periodic NLS 172
Introduction 172
NLS as an Infinite System of ODEs 174
Conditions on a Finite Set LambdaZ2 175
Arnold Diffusion for the Toy Model ODE 176
Construction of the Resonant Set Lambda 177
References 179
Angular Momentum-Mass Inequality for Axisymmetric Black Holes 180
Introduction 180
Variational Principle for the Mass 181
References 184
Almost Everything About the Fibonacci Operator 186
Introduction 186
The Trace Map 187
The Cantor Structure and the Dimension of the Spectrum 189
The Spectral Type 191
Bounds on Wavepacket Spreading 193
References 195
Entanglement-Assisted Quantum Error-Correcting Codes 197
Introduction 197
Notations 198
Entanglement-Assisted Quantum Error-Correcting Codes 199
The Channel Model: Discretization of Errors 200
The Entanglement-Assisted Canonical Code 201
The General Case 203
Distance 205
Generalized F4 Construction 205
Bounds on Performance 206
Conclusions 207
References 207
Particle Decay in Ising Field Theory with Magnetic Field 209
Ising Field Theory 209
Evolution of the Mass Spectrum 211
Particle Decay off the Critical Isotherm 212
Unstable Particles in Finite Volume 218
References 220
Fluctuations and Large Deviations in Non-equilibrium Systems 222
Introduction 222
Large Deviation Function of the Density 223
Free Energy Functional 224
Simple Exclusion Processes (SSEP) 226
The Large Deviation Function F(rho(x)) for the SSEP 228
The Matrix Ansatz for the Symmetric Exclusion Process 229
Additivity as a Consequence of the Matrix Ansatz 232
Large Deviation Function of Density Profiles 233
Non-locality of the Large Deviation Functional of the Density and Long Range Correlations 235
The Macroscopic Fluctuation Theory 237
Large Deviation of the Current 238
Generalized Detailed Balance and the Fluctuation Theorem 239
Current Fluctuations in the SSEP 241
The Additivity Principle 242
References 244
Robust Heterodimensional Cycles and Tame Dynamics 246
Robust Heterodimensional Cycles 246
General Setting 246
Basic Definitions 248
Robust Cycles at Heterodimensional Cycles 249
Questions and Consequences 251
Cycles and Non-hyperbolic Tame Dynamics 252
Setting 252
Tangencies, Heterodimensional Cycles, and Examples 253
Robust Homoclinic Tangencies, Non-dominated Dynamics, and Heterodimensional Cycles 255
Ingredients of the Proof of Theorem 2 257
Strong Homoclinic Intersections of Saddle-Nodes 258
Model Blenders 260
References 262
Hamiltonian Perturbations of Hyperbolic PDEs: from Classification Results to the Properties of Solutions 265
Introduction 265
Towards Classification of Hamiltonian PDEs 269
Deformation Theory of Integrable Hierarchies 272
Frobenius Manifolds and Integrable Hierarchies of the Topological Type 283
Critical Behaviour in Hamiltonian PDEs, the Dispersionless Case 298
Universality in Hamiltonian PDEs 303
References 307
Lattice Supersymmetry from the Ground Up 311
References 318
Convergence of Symmetric Trap Models in the Hypercube 319
Introduction 319
The Model 320
Convergence to the K Process 321
Proof of Theorem 1 322
Coupling of Xd,M and YM 322
Coupling of Xd and Xd,M 323
Conclusion of proof of Theorem 1 325
The REM-Like Trap Model and the Random Hopping Times Dynamics for the REM 328
The REM-Like Trap Model 328
Random Hopping Times Dynamics for the REM 329
References 330
Spontaneous Replica Symmetry Breaking in the Mean Field Spin Glass Model 332
Introduction 332
The Mean Field Spin Glass Model. Basic Definitions 335
The Thermodynamic Limit 337
The Parisi Representation for the Free Energy 338
Conclusion and Outlook for Future Developments 342
References 343
Surface Operators and Knot Homologies 345
Introduction 345
Gauge Theory and Categorification 348
Incorporating Surface Operators 350
Braid Group Actions 352
Surface Operators and Knot Homologies in N=2 Gauge Theory 355
Donaldson-Witten Theory and the Equivariant Knot Signature 355
Seiberg-Witten Theory 358
Surface Operators and Knot Homologies in N=4 Gauge Theory 362
References 372
Conformal Field Theory and Operator Algebras 376
Introduction 376
Conformal Quantum Field Theory 377
Representation Theory 380
Classification Theory 381
Moonshine Conjecture 383
References 385
Diffusion and Mixing in Fluid Flow: A Review 388
Introduction 388
The Heart of the Matter 394
Open Questions 398
References 399
Random Schrödinger Operators: Localization and Delocalization, and All That 401
Random Schrödinger Operators 401
Basic Examples of Random Schrödinger Operators 402
The Anderson (Tight-Binding) Model 403
The (Continuum) Anderson Hamiltonian 403
The Random Landau Hamiltonian 403
The Poisson Hamiltonian 404
The Metal-Insulator Transition 404
The Spectral Metal-Insulator Transition 405
Anderson Localization 405
Anderson Localization in the One-Dimensional Case 406
Anderson Localization in the Multi-Dimensional Case 406
Multiplicity of Eigenvalues in Intervals of Anderson Localization 406
Absolutely Continuous Spectrum 407
The Spectral Metal-Insulator Transition for the Anderson Model on the Bethe Lattice 407
The Dynamical Metal-Insulator Transition 408
Dynamical Localization 408
Transport Exponents 409
The Dynamical Spectral Regions 410
The Region of Complete Localization 411
The Dynamical Transition in the Random Landau Hamiltonian 412
References 414
Unifying R-Symmetry in M-Theory 419
Introduction 419
Kinematics 422
Definition of e10 and K(e10) 422
Level Decompositions for D=11, IIA and IIB 423
Representations of K(e10) 424
Dynamics 426
Discussion 428
Remarks 428
Outlook 429
References 430
Stable Maps are Dense in Dimensional One 432
Introduction 432
Density of Hyperbolicity 433
Quasi-Conformal Rigidity 434
How to Prove Rigidity? 434
The Strategy of the Proof of QC-Rigidity 435
Enhanced Nest Construction 436
Small Distortion of Thin Annuli 438
Approximating Non-renormalizable Complex Polynomials 440
References 441
Large Gap Asymptotics for Random Matrices 442
References 448
On the Derivation of Fourier's Law 449
Introduction 449
Coupled Oscillators 450
Closure Equations 452
Kinetic Limit 455
References 459
Noncommutative Manifolds and Quantum Groups 460
Introduction 460
The Algebras and the Representations 462
The Algebras of Functions and of Symmetries 462
The Equivariant Representation of A(SUq(2)) 465
The Spin Representation 466
The Equivariant Dirac Operator 469
The Real Structure 471
The Tomita Operator of the Regular Representation 471
The Real Structure on Spinors 472
The Local Index Formula for SUq(2) 474
The Cosphere Bundle and the Dimension Spectrum 475
The Local Index Formula for 3-Dimensional Geometries 477
The Pairing Between HC1 and K1 479
References 481
Topological Strings on Local Curves 483
Introduction 483
Topological Strings on Local Curves 485
A Model 485
Relation to Hurwitz Theory 486
Mirror Symmetry from Large Partitions 488
Higher Genus and Matrix Models 490
Phase Transitions, Critical Behavior and Double-Scaling Limit 491
Review of Phase Transitions in Topological String Theory 491
Phase Transitions for Local Curves 493
Non-perturbative Effects and Large Order Behavior 495
References 498
Repeated Interaction Quantum Systems 500
Introduction 500
Literature 502
Deterministic Systems 502
Mathematical Description 502
Results 506
Asymptotic State 506
Correlations & Reconstruction of Initial State
Random Systems 507
Dynamics and Random Matrix Products 507
Results 509
An Example: Spins 511
Some Proofs 515
Outline of the Proof of Theorem 3 515
Outline of the Proof of Theorem 9 516
References 519
String-Localized Quantum Fields, Modular Localization, and Gauge Theories 520
The Notion of String-Localized Quantum Fields 520
Modular Localization and the Construction of Free String-Localized Fields 522
Results on Free String-Localized Fields 524
Fields and Two-Point Functions 524
Massless Infinite Spin Particles 524
Vector Potentials for Photons 525
Massive Vector Bosons 526
Massive Bosonic Particles with Arbitrary Spin 527
Tensor Potentials for Linearized Gravitons 527
Feynman Propagators 528
Outlook: Interacting String-Localized Fields 529
References 532
Kinks and Particles in Non-integrable Quantum Field Theories 534
Introduction 534
A Semiclassical Formula 538
Symmetric Wells 540
Asymmetric Wells 544
Conclusions 547
References 548
Exponential Decay Laws in Perturbation Theory of Threshold and Embedded Eigenvalues 549
Introduction 549
The Basic Formula 552
The Results 554
Properly Embedded Eigenvalues 554
Threshold Eigenvalues 555
A Uniqueness Result 557
Examples 558
Example 1: One Channel Case, nu=-1 558
Example 2: Two Channel Case, nu=-1,1 559
Example 3: Two Channel Radial Case, nu> =3
References 561
Energy Diffusion and Superdiffusion in Oscillators Lattice Networks 563
Introduction 563
Conservative Stochastic Dynamics 565
Diffusive Evolution: Green-Kubo Formula 568
Kinetic Limits: Phonon Boltzmann Equation 569
Levy's Superdiffusion of Energy 570
References 570
Trying to Characterize Robust and Generic Dynamics 572
Introduction 572
Robust Transitivity: Hyperbolicity, Partial Hyperbolicity and Dominated Splitting 574
Hyperbolicity 576
Partial Hyperbolicity 577
Dominated Splitting 578
A General Question About ``Weak Form of Hyperbolicity'' 580
Robust Transitivity and Mechanisms: Heterodimensional Cycle 580
Wild Dynamics 581
Wild Dynamic and Homoclinic Tangency 581
Surfaces Diffeomorphisms and Beyond 582
Generic Dynamics: Mechanisms and Phenomenas 583
References 584
Dynamics of Bose-Einstein Condensates 587
Introduction 587
Heuristic Derivation of the Gross-Pitaevskii Equation 589
Main Results 593
General Strategy of the Proof and Previous Results 596
Convergence to the Infinite Hierarchy 598
Uniqueness of the Solution to the Infinite Hierarchy 602
Higher Order Energy Estimates 603
Expansion in Feynman Graphs 605
References 611
Locality Estimates for Quantum Spin Systems 612
Introduction 612
Lieb-Robinson Bounds 614
Quasi-Locality of the Dynamics 619
Exponential Clustering 622
The Lieb-Schultz-Mattis Theorem 625
The Result and Some Words on the Proof 626
A More Detailed Outline of the Proof 628
Constructing the Trial State 628
Locality and the Trial State 631
The Estimates 631
Generalizations 632
References 635
On Resolvent Identities in Gaussian Ensembles at the Edge of the Spectrum 636
Introduction 636
Proof of Theorems 1 and 3 643
Proof of Theorems 4 and 5 645
Non-Gaussian Case 646
References 647
Energy Current Correlations for Weakly Anharmonic Lattices 649
Introduction 649
Anharmonic Lattice Dynamics 650
Energy Current Correlations 653
The Linearized Collision Operator 657
Gaussian Fluctuation Theory 659
References 660
Metastates, Translation Ergodicity, and Simplicity of Thermodynamic States in Disordered Systems: an Illustration 662
Introduction 662
The Sherrington-Kirkpatrick Model and the Parisi Replica Symmetry Breaking Solution 663
Open Problems 664
Metastates 664
Invariance and Ergodicity 665
A Strategy for Rigorous Studies of Spin Glasses 666
Summary 670
References 670
Random Matrices, Non-intersecting Random Walks, and Some Aspects of Universality 672
Introduction 672
Selected Basic Facts from Random Matrix Theory 673
The Karlin-McGregor Formula 673
The Models 674
Longest Increasing Subsequence of a Random Permutation 674
ABC-Hexagon 676
Last Passage Percolation 676
Non-intersecting Brownian Motion 679
Universality 680
Last Passage Percolation 681
Non-intersecting Random Walks 681
References 683
Homogenization of Periodic Differential Operators as a Spectral Threshold Effect 686
Introduction 686
Periodic DO's. The Effective Matrix 687
Homogenization of Periodic DO's. Principal Term of Approximation for the Resolvent 689
More Accurate Approximation for the Resolvent in the L2-Operator Norm 690
(L2-> H1)-Approximation of the Resolvent. Approximation of the Fluxes in L2
The Method of Investigation 693
Some Applications 697
On Further Development of the Method 700
References 701
ABCD and ODEs 703
Introduction 703
Bethe Ansatz for Classical Lie Algebras 706
The Pseudo-Differential Equations 707
An-1 models 707
Dn models 708
Bn models 709
Cn models 710
Conclusions 711
References 712
Nonrational Conformal Field Theory 714
Introduction 714
Constraints from Conformal Symmetry 716
Motivation: Chiral Factorization of Physical Correlation Functions 716
Vertex Algebras 717
Representations of Vertex Algebras 718
Conformal Blocks 718
Definition 718
Insertions of the Vacuum Representation 719
Deformations of the Complex Structure of X 719
Correlation Functions vs. Hermitian Forms 720
Behavior Near the Boundary of Moduli Space 721
Gluing of Riemann Surfaces 722
Gluing of Conformal Blocks 725
Conformal Ward Identities 726
Decorated Marking Graphs 726
Correlation Functions 727
First Requirement: Factorization of Conformal Blocks 728
Second Requirement: Factorization of the Hermitian form HSigma 728
Conformal Blocks as Matrix Elements 729
Chiral Vertex Operators 729
From one Boundary to Another 730
The Modular Groupoid 731
Generators 732
Relations 733
Relations supported on surfaces of genus zero. 733
Relations supported on surfaces of genus one. 733
Representation of the Generators on Spaces of Conformal Blocks 734
Representation of the Relations on Spaces of Conformal Blocks 735
Insertions of the Vacuum 735
Notion of a Stable Modular Functor 736
Towers of Representations of the Modular Groupoid 736
Unitary Modular Functors 738
Similarity of Modular Functors 739
Friedan-Shenker Modular Geometry 739
Example of a Nonrational Modular Functor 740
Unitary Positive Energy Representations of the Virasoro Algebra 741
Free Field Representation 741
Construction of Virasoro Conformal Blocks in Genus Zero 742
Free Field Construction of Chiral Vertex Operators 742
Factorization Property 743
The Hilbert Space Structure 744
Extension to Higher Genus 745
Remarks 746
Existence of a Canonical Scalar Product? 746
Existence of a Canonical Hermitian Form from the Factorization Property 747
Unitary Fusion 749
Associativity of Unitary Fusion 750
Discussion 752
Outlook 752
Modular Functors from W-algebras and Langlands Duality 752
Boundary CFT 753
Nonrational Verlinde Formula? 753
References 755
Kinetically Constrained Models 757
References 767
The Distributions of Random Matrix Theory and their Applications 769
Random Matrix Models: Gaussian Ensembles 769
Largest Eigenvalue Distributions Fbeta. Painlevé II Representations 770
Tail behavior of Fbeta 772
Numerical evaluation of Fbeta 773
Next-Largest, Next-Next Largest, Etc. Eigenvalue Distributions 773
Universality Theorems 773
Invariant Ensembles 773
Wigner Ensembles 775
Multivariate Statistical Analysis 775
Principal Component Analysis (PCA) 776
Testing the Null Hypothesis 776
Spiked Populations: BBP Phase Transition 777
Conclusions 779
References 779
Hybrid Formalism and Topological Amplitudes 782
Introduction 782
Compactified String Theory in RNS and Hybrid Variables 783
Hybrid Variables 783
RNS Variables 787
Field Redefinition from RNS to Hybrid Variables 788
Physical State Conditions and N=4-embeddings 791
Massless Vertex Operators 793
Amplitudes and Correlation Functions 795
Amplitudes 795
Correlation Functions of Chiral Bosons 797
Topological Amplitudes 798
Generalities 798
R-charge (g-1,g-1) 799
R-charge (1-g,1-g) 800
R-charges (g-1,1-g) and (1-g,g-1) 802
Summary of the Amplitude Computation 803
Appendix: Conventions and Notations 804
Spinors and Superspace 804
Hybrid Variables and N =2 Algebra 805
The Integrated Vertex Operator 806
Appendix: Mapping the RNS to the Hybrid Variables 807
Field Redefinition from RNS to Chiral GS Variables 807
Similarity Transformation Relating Chiral GS to Hybrid Variables 807
Hermitian Conjugation of the Hybrid Variables 808
Hermitian Conjugation of the RNS Variables 809
Appendix: Vertex Operators 812
Massless RNS Vertex Operators 812
Universal Massless Multiplets 814
Compactification Dependent Massless Multiplets 814
Kähler Moduli 815
Complex Structure Moduli 816
References 817
Quantum Phases of Cold Bosons in an Optical Lattice 819
Introduction 820
Reflection Positivity 824
Proof of BEC for Small lambda and T 825
Absence of BEC and Mott Insulator Phase 830
The Non-interacting Gas 834
Conclusion 835
References 835
Random Walks in Random Environments in the Perturbative Regime 837
Introduction 837
Local Limits for Exit Measures 839
References 840
Appendix: Complete List of Abstracts 841
Young Researchers Symposium Plenary Lectures 841
Dynamics of Quasiperiodic Cocycles and the Spectrum of the Almost Mathieu Operator 841
Magic in Superstring Amplitudes 841
The Instructive History of Quantum Groups 842
Scaling Limit of Two-Dimensional Critical Percolation 842
Topics in Dynamics and Physics 842
Quantum Dynamics in a Random Environment 843
Geometry of Low Dimensional Manifolds 843
Gauge Theory and the Geometric Langlands Program 843
XV International Congress on Mathematical Physics Plenary Lectures 844
Mathematical Developments Around the Ginzburg-Landau Model in 3D Program 844
The Riemann-Hilbert Problem: Applications 844
Fluctuations and Large Deviations in Non-equilibrium Systems 844
Hamiltonian Perturbations of Hyperbolic PDE's: from Classification of Equations to Properties of Solutions 845
Spontaneous Replica Symmetry Breaking in the Mean Field Spin Glass Model 845
Spectral Properties of Quasi-Periodic Schrödinger Operators: Treating Small Denominators without KAM 845
Conformal Field Theory and Operator Algebras 846
Random Schrödinger Operators, Localization and Delocalization, and all that 846
Perelman's Work on the Geometrization Conjecture 847
Trying to Characterize Robust and Generic Dynamics 847
Cauchy Problem in General Relativity 848
Survey of Recent Mathematical Progress in the Understanding of Critical 2d Systems 848
Random Methods in Quantum Information Theory 848
Gauge Fields, Strings and Integrable Systems 849
XV International Congress on Mathematical Physics Specialized Sessions 849
Condensed Matter Physics 849
Rigorous Construction of Luttinger Liquids Through Ward Identities 849
Edge and Bulk Currents in the Integer Quantum Hall Effect 850
Quantum Phases of Cold Bosons in Optical Lattices 850
Dynamical Systems 850
Statistical Stability for Hénon Maps of Benedics-Carleson Type 850
Entropy and the Localization of Eigenfunctions 851
The Spectrum of the Almost Mathieu Operator in the Subcritical Regime 851
Hyperbolicity Through Entropy 852
Robust Cycles and Non-dominated Dynamics 852
Hyperbolicity in One Dimensional Dynamics 852
Equilibrium Statistical Mechanics 853
The Scaling Limit of (Near-)Critical 2d Percolation 853
Short-Range Spin Glasses in a Magnetic Field 853
Relaxation Times of Kinetically Constrained Spin Models with Glassy Dynamics 854
Non-equilibrium Statistical Mechanics 854
Current Fluctuations in Boundary Driven Interacting Particle Systems 854
Fourier Law and Random Walks in Evolving Environments 855
Asymptotics of Repeated Interaction Quantum Systems 855
Linear Response of Non-equilibrium Steady States for Open Quantum System 855
Derivation of the Gross-Pitaevski Equation for the Dynamics of Bose-Einstein Condensates 856
Energy Transport in One-Dimensional Chains: Predictions from the Phonon Kinetic Equation 856
Exactly Solvable Systems 857
Correlation Functions and Hidden Fermionic Structure of the XYZ Spin Chain 857
Particle Decay in Ising Field Theory with Magnetic Field 857
ABCD-Integrable Models and Ordinary Differential Equations 857
General Relativity 858
Einstein Spaces as Attractors for the Einstein Flow 858
Loop Quantum Cosmology 858
The Red-Shift Effect and Radiation Decay on Black Hole Space-Times 858
Angular Momentum-Mass Inequality for Axisymmetric Black Holes 859
Black Hole Entropy in Supergravity and String Theory 859
Infinite-Dimensional R-Symmetry in Supergravity 859
Operator Algebras 860
From Vertex Algebras to Local Nets of von Neuman Algebras 860
Non-Commutative Manifolds and Quantum Groups 860
L2 Invariants, Free Probability and Operator Algebras 860
Partial Differential Equations 861
Weak Turbulence for Periodic NSL 861
Ginzburg-Landau Dynamics 861
On the Derivation of Furier's Law 861
TBA 862
A Criterion for the Logarithmic Sobolev Inequality 862
Singular Behaviour in Chemotaxis Models 862
Probability Theory 863
Aging in the Infinite Volume REM-Like Trap Model at Low Temperature 863
From Planar Gaussian Zeros to Gravitational Allocation 863
Random Walks in Random Environments in the Perturbative Regime 863
Quantum Mechanics 864
Recent Progress in the Spectral Theory of Quasi-Periodic Operators 864
Recent Results on Localization for Random Schrödinger Operators 864
Quantum Dynamics and Enhanced Diffusion for Passive Scalar 864
Lieb-Thirring Inequalities, Recent Results 865
Exponential Decay Laws in Perturbation Theory of Threshold and Embedded Eigenvalues 865
Homogenization of Periodic Operators of Mathematical Physics as a Spectral Threshold Effect 865
Quantum Field Theory 866
Algebraic Aspects of Perturbative and Non-Perturbative QFT 866
Quantum Field Theory in Curved Space-Time 867
String-Localized Quantum Fields, Modular Localization, and Gauge Theories 867
Quantization of the Teichmüller Spaces: Quantum Field Theoretical Applications 868
2D Quantum Field Theory 868
Lattice Supersymmetry From the Ground Up 868
Analytical Solution for the Effective Charging Energy of the Single Electron Box 868
Breaking Integrability 869
Quantum Information 869
One-and-a-Half Quantum de Finetti Theorems 869
Catalytic Quantum Error Correction 870
Quantum State Transformations and the Schubert Calculus 870
The Local Hamiltonian Problem 870
The Information-Disturbance Tradeoff and the Continuity of Stinespring's Representation 871
Locality Estimates for Quantum Spin Systems 871
Random Matrices 872
Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions 872
Probabilities of a Large Gap in the Scaled Spectrum of Random Matrices 872
Random Matrices, Asymptotic Analysis, and d-bar Problems 873
Central Limit Theorems for Non-intersecting Random Walks 873
On the Distribution of Largest Eigenvalues in Random Matrix Ensembles 873
Non-Intersecting Brownian Excursions 873
Stochastic PDE 874
Degenerately Forced Fluid Equations: Ergodicity and Solvable Models 874
Microscopic Stochastic Models for the Study of Thermal Conductivity 874
Boundary Effects on the Interface Dynamics for the Stochastic Allen-Cahn Equation 875
String Theory 875
Topological Strings and (Almost) Modular Forms 875
Gauge Theory and Link Homologies 875
Non-Geometric String Backgrounds 876
Topological Reduction of Supersymmetric Gauge Theories and S-Duality 876
Phase Transitions in Topological String Theory 876
Hyper-Multiplet Couplings in N=2 Effective Action 877