CONTENTS -- Preface -- Congress Committees -- INTERNATIONAL SCIENTIFIC COMMITTEE -- Prize Committees -- Acknowledgments -- Welcome addresses -- Henri Poincaré Prize -- LAUDATIO BY JAKOB YNGVASON -- LAUDATIO BY DOMOKOS SZÁSZ -- LAUDATIO BY JOEL LEBOWITZ -- IAMP Early Career Award and other prizes -- OTHER PRIZES AWARDED AT THE CONGRESS -- IUPAP YOUNG SCIENTIST PRIZE -- SPRINGER BEST POSTER PRIZE -- ANNALES HENRI POINCARÉ PRIZES -- THE AHP PRIZE -- THE AHP DISTINGUISHED PAPER AWARD -- Part A Plenary Talks -- Quantum geometry of 3-dimensional lattices and tetrahedron equation V.V. Bazhanov, V.V. Mangazeev, S.M. Sergeev -- Keywords: -- 1. Introduction -- 2. Discrete differential geometry: "Existence as integrability" -- 2.1. Quadrilateral lattices -- 2.2. Discrete evolution systems: "Existence as integrability" -- 3. Quantization of the 3D circular lattices -- 3.1. Poisson structure of circular lattices -- 3.2. Quantization and the tetrahedron equation -- 4. Solutions of the tetrahedron equation -- 4.1. Fock representation solution -- 4.2. Modular double solution -- 4.2.1. The "interaction-round-a-cube" formulation of the modular double solution -- 4.3. Cyclic representation solution -- 4.3.1. Generalized form of the q-oscillator map -- 4.3.2. Cyclic representations of the q-oscillator algebra -- 4.3.3. Solution of the tetrahedron equation -- 5. Conclusion -- Acknowledgments -- References -- The formation of black holes in general relativity D. Christodoulou -- Keywords: -- References -- Liouville quantum gravity & the KPZ relation: a rigorous perspective B. Duplantier -- Keywords: -- 1. Introduction -- 1.1. Historical perspective -- 1.2. Quantum measure -- 1.3. Euclidean and quantum scaling exponents -- 1.4. Box formulation of Liouville quantum gravity -- 1.5. Statement of KPZ -- 2. GFF regularization -- 2.1. GFF circular average.

2.2. GFF covariance -- 2.3. GFF circular average and Brownian motion -- 2.3.1. Brownian motion -- 3. Random measure and Liouville quantum gravity -- 3.1. Gaussian exponential averages -- 3.2. Liouville weighted GFF measure -- 3.3. Random measure and Brownian motion with drift -- 4. KPZ Proof -- 4.1. Quantum balls -- 4.2. Quantum scaling -- 4.3. Brownian stopping time -- 4.4. Brownian martingale -- 4.5. Probability distribution and GFF thick points -- 5. Boundary KPZ -- 5.1. Boundary quantum measure. -- 5.2. Boundary scaling and KPZ -- 6. Liouville quantum duality -- 6.1. ' = 4/ Liouville duality -- 6.2. Brownian approach to duality [2] -- Conclusion and Perspectives -- Acknowledgements -- References -- Universality of Wigner random matrices L. Erd˝os -- Keywords: -- 1. Introduction -- 2. Local semicircle law, delocalization and level repulsion -- 3. Sine kernel universality -- 4. Dyson Brownian motion -- 5. Local Relaxation Flow -- 6. Proof of Theorem 5.1 -- Acknowledgements -- References -- Uses of free probability in random matrix theory A. Guionnet -- Keywords: -- 1. Introduction -- 2. Free probability theory -- 2.1. Basics of free probability -- 2.2. Free Brownian motion -- 2.3. Free convolution -- 3. Single ring theorem -- 4. Enumeration of maps -- 5. Conclusion -- References -- The physics of decision making: stochastic differential equations as models for neural dynamics and evidence accumulation in cortical circuits P. Holmes, P. Eckho., K.F. Wong-Lin, R. Bogacz, M. Zacksenhouse, J.D. Cohen -- 1. Introduction -- 2. Decision-making models and an optimal speed-accuracy tradeo -- 2.1. Leaky competing accumulators and drift-diffusion processes -- 2.2. A spiking neuron model and its reduction to competing accumulators -- 2.3. An optimal speed-accuracy tradeo -- 3. Behavioral experiments: a prevalent lack of optimality.

3.1. A preference for accuracy? -- 3.2. Robust decisions in the face of uncertainty? -- 3.3. Biophysical constraints? -- 4. Discussion and conclusions -- Acknowledgments -- References -- New technologies in the hunt for new physics D.A. Kosower -- Keywords: -- 1. Introduction -- 2. On-Shell Methods -- 2.1. Integrals -- 2.2. Unitarity -- 2.3. Factorization and On-Shell Recursion Relations -- 2.4. Numerical Methods -- 3. N = 4 Supersymmetry: A Theoretical Laboratory -- Acknowledgments -- References -- Operator algebras and noncommutative geometric aspects in conformal field theory R. Longo -- Kinetic transport in crystals J. Marklof -- 1. Introduction -- 2. The periodic Lorentz gas -- 3. Why "a generalization" of the linear Boltzmann equation? -- 4. Joint distribution of path segments -- 5. A limiting random flight process -- 6. The distribution of free path lengths -- 7. The space of lattices -- 8. Equidistribution in the space of lattices -- 9. Asymptotics -- 10. Outlook -- Acknowledgements -- References -- Ising models, universality and the non renormalization of the quantum anomalies V. Mastropietro -- Keywords: -- 1. Introduction -- 2. Ising, Vertex and Ashkin-Teller models -- 3. Quantum spin chain and 1D Fermi systems -- 4. Universal relations -- 5. Renormalization Group analysis for coupled Ising models -- 6. Proof of universality and anomaly non renormalization -- 7. Renormalization Group for Quantum spin chain and 1D Fermi systems -- 8. Universality for quantum spin chains and 1D Fermi systems -- 9. Conclusions -- References -- The infrared problem in nonrelativistic QED A. Pizzo -- 1. Introduction -- 2. Spectroscopy in nonrelativistic QED -- 2.1. Atoms and molecules -- 2.1.1. Spectral analysis -- 2.1.2. Asymptotic completeness, metastable states, and Bohr's frequency condition. -- 2.2. Infraparticles and Čerenkov radiation.

2.2.1. Dressed electron states -- 2.2.2. Infraparticle scattering states -- Appendix A -- Appendix B -- Acknowledgments -- References -- Parking in the city: an example of limited resource sharing P. Šeba -- 1. Introduction -- 2. The parking process -- 3. The parking maneuver -- 4. The measured data -- References -- Hot topics in cold gases R. Seiringer -- Keywords: -- 1. Introduction -- 1.1. The Bose Gas: A Quantum Many-Body Problem -- 1.2. Quantities of Interest -- 2. Homogeneous Systems in the Thermodynamic Limit -- 2.1. The Ground State Energy of Homogeneous Bose Gases -- 2.2. Homogeneous Bose Gas at Positive Temperature -- 2.3. Critical Temperature for BEC -- 3. Trapped Bose Gases -- 3.1. The Gross-Pitaevskii Equation -- 3.2. Ground State Energy of Dilute Trapped Gases -- 3.3. BEC for Rotating Trapped Gases -- 3.4. Rapid Rotation -- Acknowledgments -- References -- Vortex patterns in Ginzburg-Landau minimizers S. Serfaty, E. Sandier -- Keywords: -- 1. Introduction -- 1.1. The model -- 1.2. Formal look at the solutions, vortices and critical fields -- 1.2.1. Types of solutions -- 1.2.2. Vortices and their benefit -- 1.2.3. Critical fields -- 2. Main results for global minimization at the leading order -- 2.1. The vorticity measures -- 2.2. Global minimizers of G" close to Hc1 -- 2.3. Global minimizers in the intermediate regime -- 2.4. Global minimizers in the regime n" proportional to hex -- 2.5. Global minimizers in the regime

2.1. Twisted chiral ring and quantum integrability -- 2.2. Topological field theory -- 2.3. Yang-Yang function and quantum spectrum -- 2.4. Quantization from four dimensions -- 3. Four dimensional gauge theory -- 3.1. The -background and twisted masses -- 3.1.1. Four dimensional theory on R2ε -- 3.1.2. Calculation of the twisted superpotential -- 4. Integrable systems -- 4.1. The classical story -- 4.1.1. The t-deformation -- 4.1.2. Quantization -- 5. Examples -- 5.1. The periodic Toda chain -- 5.1.1. The classical system -- 5.1.2. The gauge theory -- 5.1.3. The quantum system -- 5.2. Elliptic Calogero-Moser system -- 5.2.1. The classical system -- 5.2.2. The gauge theory -- 5.2.3. The quantum system -- 5.3. Hitchin system -- 5.3.1. The quantum system and the gauge theory -- 5.4. Relativistic systems -- 6. Superpotential/Yang-Yang function W(a, ε -- q) -- 6.1. Thermodynamic Bethe Ansatz -- 6.2. The examples -- 6.2.1. The elliptic Calogero-Moser system -- 6.2.2. Periodic Toda -- 6.2.3. Ruijsenaars-Schneider model -- 6.2.4. The spectrum of observables -- 6.2.5. On the relation between the type A and the type B models -- 6.2.6. More details and the origin of integral equation -- 6.3. The sum over partitions -- 7. Discussion -- 8. Acknowledgments -- References -- Changing views of quantum field theory S. Weinberg -- Effective Field Theory: Past and Future -- References -- Part B Topical Sessions -- Dynamical Systems (including integrable systems and Hamiltonian stability) -- Almost dense orbit on energy surface V. Kaloshin, K. Zhang, Y. Zheng -- 1. Introduction -- 2. Choice of F -- 3. A proof of existence of a d-dense orbit using a variational problem with constraints -- 3.1. Single resonance case -- 3.2. Double resonance -- 4. Competition between order of resonance and distance to a KAM torus -- References.

Dissipative perturbations of KdV S.B. Kuksin.