- •Preface
- •Foreword
- •The Henri Poincaré Prize
- •Contributors
- •Contents
- •Stability of Doubly Warped Product Spacetimes
- •Introduction
- •Warped Product Spacetimes
- •Asymptotic Behavior
- •Fuchsian Method
- •Velocity Dominated Equations
- •Velocity Dominated Solution
- •Stability
- •References
- •Introduction
- •The Tomonaga Model with Infrared Cutoff
- •The RG Analysis
- •The Dyson Equation
- •The First Ward Identity
- •The Second Ward Identity
- •The Euclidean Thirring Model
- •References
- •Introduction
- •Lie and Hopf Algebras of Feynman Graphs
- •From Hochschild Cohomology to Physics
- •Dyson-Schwinger Equations
- •References
- •Introduction
- •Quantum Representation and Dynamical Equations
- •Quantum Singularity Problem
- •Examples for Properties of Solutions
- •Effective Theory
- •Summary
- •Introduction
- •Results and Strategy of Proofs
- •References
- •Introduction
- •Critical Scaling Limits and SLE
- •Percolation
- •The Critical Loop Process
- •General Features
- •Construction of a Single Loop
- •The Near-Critical Scaling Limit
- •References
- •Black Hole Entropy Function and Duality
- •Introduction
- •Entropy Function and Electric/Magnetic Duality Covariance
- •Duality Invariant OSV Integral
- •References
- •Weak Turbulence for Periodic NLS
- •Introduction
- •Arnold Diffusion for the Toy Model ODE
- •References
- •Angular Momentum-Mass Inequality for Axisymmetric Black Holes
- •Introduction
- •Variational Principle for the Mass
- •References
- •Introduction
- •The Trace Map
- •Introduction
- •Notations
- •Entanglement-Assisted Quantum Error-Correcting Codes
- •The Channel Model: Discretization of Errors
- •The Entanglement-Assisted Canonical Code
- •The General Case
- •Distance
- •Generalized F4 Construction
- •Bounds on Performance
- •Conclusions
- •References
- •Particle Decay in Ising Field Theory with Magnetic Field
- •Ising Field Theory
- •Evolution of the Mass Spectrum
- •Particle Decay off the Critical Isotherm
- •Unstable Particles in Finite Volume
- •References
- •Lattice Supersymmetry from the Ground Up
- •References
- •Stable Maps are Dense in Dimensional One
- •Introduction
- •Density of Hyperbolicity
- •Quasi-Conformal Rigidity
- •How to Prove Rigidity?
- •The Strategy of the Proof of QC-Rigidity
- •Enhanced Nest Construction
- •Small Distortion of Thin Annuli
- •Approximating Non-renormalizable Complex Polynomials
- •References
- •Large Gap Asymptotics for Random Matrices
- •References
- •Introduction
- •Coupled Oscillators
- •Closure Equations
- •Introduction
- •Conservative Stochastic Dynamics
- •Diffusive Evolution: Green-Kubo Formula
- •Kinetic Limits: Phonon Boltzmann Equation
- •References
- •Introduction
- •Bethe Ansatz for Classical Lie Algebras
- •The Pseudo-Differential Equations
- •Conclusions
- •References
- •Kinetically Constrained Models
- •References
- •Introduction
- •Local Limits for Exit Measures
- •References
- •Young Researchers Symposium Plenary Lectures
- •Dynamics of Quasiperiodic Cocycles and the Spectrum of the Almost Mathieu Operator
- •Magic in Superstring Amplitudes
- •XV International Congress on Mathematical Physics Plenary Lectures
- •The Riemann-Hilbert Problem: Applications
- •Trying to Characterize Robust and Generic Dynamics
- •Cauchy Problem in General Relativity
- •Survey of Recent Mathematical Progress in the Understanding of Critical 2d Systems
- •Random Methods in Quantum Information Theory
- •Gauge Fields, Strings and Integrable Systems
- •XV International Congress on Mathematical Physics Specialized Sessions
- •Condensed Matter Physics
- •Rigorous Construction of Luttinger Liquids Through Ward Identities
- •Edge and Bulk Currents in the Integer Quantum Hall Effect
- •Dynamical Systems
- •Statistical Stability for Hénon Maps of Benedics-Carleson Type
- •Entropy and the Localization of Eigenfunctions
- •Equilibrium Statistical Mechanics
- •Short-Range Spin Glasses in a Magnetic Field
- •Non-equilibrium Statistical Mechanics
- •Current Fluctuations in Boundary Driven Interacting Particle Systems
- •Fourier Law and Random Walks in Evolving Environments
- •Exactly Solvable Systems
- •Correlation Functions and Hidden Fermionic Structure of the XYZ Spin Chain
- •Particle Decay in Ising Field Theory with Magnetic Field
- •General Relativity
- •Einstein Spaces as Attractors for the Einstein Flow
- •Loop Quantum Cosmology
- •Operator Algebras
- •From Vertex Algebras to Local Nets of von Neuman Algebras
- •Non-Commutative Manifolds and Quantum Groups
- •Partial Differential Equations
- •Weak Turbulence for Periodic NSL
- •Ginzburg-Landau Dynamics
- •Probability Theory
- •From Planar Gaussian Zeros to Gravitational Allocation
- •Quantum Mechanics
- •Recent Progress in the Spectral Theory of Quasi-Periodic Operators
- •Recent Results on Localization for Random Schrödinger Operators
- •Quantum Field Theory
- •Algebraic Aspects of Perturbative and Non-Perturbative QFT
- •Quantum Field Theory in Curved Space-Time
- •Lattice Supersymmetry From the Ground Up
- •Analytical Solution for the Effective Charging Energy of the Single Electron Box
- •Quantum Information
- •One-and-a-Half Quantum de Finetti Theorems
- •Catalytic Quantum Error Correction
- •Random Matrices
- •Probabilities of a Large Gap in the Scaled Spectrum of Random Matrices
- •Random Matrices, Asymptotic Analysis, and d-bar Problems
- •Stochastic PDE
- •Degenerately Forced Fluid Equations: Ergodicity and Solvable Models
- •Microscopic Stochastic Models for the Study of Thermal Conductivity
- •String Theory
- •Gauge Theory and Link Homologies
Appendix: Complete List of Abstracts
YRS and XV ICMP
1 Young Researchers Symposium Plenary Lectures
1.0.1Dynamics of Quasiperiodic Cocycles and the Spectrum of the Almost Mathieu Operator
Artur Avila
CNRS artur@math.sunysb.edu
The almost Mathieu operator is the operator H : l2(Z) → l2(Z),
(H u)n = un+1 + un−1 + 2λ cos 2π(θ + nα),
where λ (the coupling), α (the frequency) and θ (the phase) are parameters. Originally introduced and studied in the physics literature, it turned out to also give rise to a rich mathematical theory, where algebra, analysis and dynamical systems interact. We will discuss several conjectures that have focused the developments since 1980, emphasizing the connection with dynamical systems.
1.0.2 Magic in Superstring Amplitudes
Nathan Jacob Berkovits
UNESP, São Paulo nberkovi@ift.unesp.br
In this talk, a chronological review will be given of scattering amplitudes in superstring theory and their remarkable properties. I will start with the Veneziano amplitude of 1968 which led to bosonic string theory, and subsequent developments in the 70’s which led to supersymmetry and superstring theory. I will then discuss the amplitude calculations of Green and Schwarz in the 80’s which led to aspirations of
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unifying the forces, and the non-perturbative dualities discovered in the 90’s which led to the M-theory conjecture. Finally, I will discuss recent developments using a covariant description of the superstring in which some of these magical properties are easier to study.
1.0.3 The Instructive History of Quantum Groups
Ludwig Faddeev
Petersburg Department of Steklov Institute of Mathematics faddeev@euclid.pdmi.ras.ru
I shall try to use this example to show, how concrete problems in Mathematical Physics (here quantum spin chains) can lead to new construction in pure mathematics.
1.0.4 Scaling Limit of Two-Dimensional Critical Percolation
Charles M. Newman
Courant Institute, NYU newman@courant.nyu.edu
We introduce and discuss the continuum nonsimple loop process (joint work with F. Camia). This process, which describes the full scaling limit of 2D critical percolation, consists of countably many noncrossing nonsimple loops in the plane on all spatial scales; it is based on the Schramm Loewner Evolution (with parameter 6) and extends the work of Schramm and Smirnov on the percolation scaling limit. If time permits, we will introduce some ideas associated with the further extension to scaling limits of “near-critical” percolation (joint work with F. Camia and L.R. Fontes).
1.0.5 Topics in Dynamics and Physics
David Ruelle
IHES ruelle@ihes.fr
The study of dynamics (i.e., time evolutions) is central to physics. I shall discuss several questions of mathematical physics connected with differentiable dynamical systems and related, as it happens, to the ideas of Henri Poincare. I shall go from chaos in turbulence and celestial mechanics to the symbolic dynamics of horseshoes and other hyperbolic dynamical systems, to the dynamics underlying nonequilibrium statistical mechanics.
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1.0.6 Quantum Dynamics in a Random Environment
Thomas Spencer
IAS, Princeton spencer@ias.edu
The first part of this talk will review results and conjectures on the quantum and classical dynamics of a particle moving in a random environment. In general, classical and quantum dynamics are qualitatively different. We review a particular network model studied by J. Cardy and others for which classical and quantum motion are equivalent and compare it to the mirror model.
The second part of the talk will describe a supersymmetric approach to quantum evolution generated by band random matrices. The supersymmetric approach converts time evolution into a problem in which the randomness is integrated out. This produces a statistical mechanics model with hyperbolic symmetry. One is lead to study of determinants and Greens functions of nonuniformly elliptic PDE.
1.0.7 Geometry of Low Dimensional Manifolds
Gang Tian
Princeton tian@Math.Princeton.edu
In this talk, I will first discuss Perelman’s work on proving the Poincare conjecture and the geometrization of 3-manifolds. Perelman’s work is based on Hamilton’s fundamental work on Ricci flow. In the end, I will discuss recent progress on geometry and analysis of 4-manifolds and propose some problems.
1.0.8 Gauge Theory and the Geometric Langlands Program
Edward Witten
IAS, Princeton witten@ias.edu
I will explain how electric-magnetic duality in gauge theory can be used to understand a problem in algebraic geometry known as the geometric Langlands program.