International Congress of Mathematicians 2022 July 6–14 Proceedings: Sections 9-11

Front cover
Front matter
Contents
9. Dynamics
M. Abért: On a curious problem and what it lead to
References
A. Brown: Lattices acting on manifolds
1. Introduction: lattices, group actions, and rigidity
1.1. Rigidity of linear representations
1.2. The general setting
1.3. Actions on manifolds and the Zimmer program
2. Low dimensions and Zimmer's conjecture
2.1. Motivation and Zimmer's conjecture
2.2. Work of Brown, Fisher, and Hurtado
2.3. C^1 actions
2.4. C^0 actions and actions on the circle
2.5. Beyond R-split groups
2.6. Dimension gaps between (3) and (4) of Conjecture 2.2
3. Classification in lowest dimensions and rigidity of projective actions
4. Classification under dynamical and topological hypotheses
4.1. Classification in dimension n
4.2. Toral homeomorphisms and Anosov diffeomorphisms
4.3. Global topological and smooth rigidity of Anosov actions
4.4. Anosov actions in dimension n
5. Tools used in proofs
5.1. Suspension space and induced G-action
5.2. Common themes
5.3. Theme 1 and subexponential growth
5.4. Theme 2 and invariance of measures
5.5. Theme 2: measure and cocycle rigidity; homogeneous structures
5.6. Cohomological versions of Theme 1 and Theme 2
A. Numerology associated with Zimmer's conjecture
References
J. Chaika and B. Weiss: The horocycle flow on translation surfaces
1. Introduction
2. Definitions and background
2.1. Some foundational results
2.2. The analogy with Ratner's work, and the magic wand theorem
3. Behavior of individual horocycle orbits
3.1. Some early results
3.2. Recent results
4. Tremors
5. Some ideas in the proof of Theorem 3.1
5.1. U-orbits of tremored surfaces almost track U-orbits
5.2. From genericity to lack of genericity
6. Questions
References
M. F. Demers: Topological entropy and pressure for finite-horizon Sinai billiards
1. Introduction
2. Preliminaries and statements of main theorems
2.1. Singularities and distortion
2.2. Measure-theoretic pressure for geometric potentials
2.3. Topological entropy and variational principle for t=0
2.4. Topological pressure and variational principle for t>0
2.4.1. Weight function for the topological pressure
2.4.2. Definition of topological pressure
2.4.3. Equilibrium state and variational principle
3. Ideas from the proof of Theorem 2.7
3.1. Growth lemmas and exact exponential growth of Q_n(t)
3.2. Banach spaces adapted to t ∈[t_0, t_1]
3.2.1. Definition of norms
3.2.2. A spectral gap for L_t
3.2.3. An equilibrium state and a variational principle
4. Ideas from the proof of Theorem 2.3
4.1. Complexity and exact exponential growth of # \mathcal{M}_0^n
4.1.1. Fragmentation lemmas
4.1.2. Uniform bounds on growth
4.2. Banach spaces adapted to t=0
4.2.1. Sparse recurrence to singularities
4.2.2. Definition of norms
4.2.3. Spectrum of L_0 and construction of an invariant measure
4.2.4. Properties of \mu_0
4.2.5. Uniqueness of \mu_0
5. Open questions
References
R. Dujardin: Geometric methods in holomorphic dynamics
1. Geometric currents
1.1. Definitions
1.2. Construction and approximation
1.3. Geometric intersection
1.4. Dynamical applications
1.5. Foliations
2. Bifurcation theory in one and several dimensions
2.1. Bifurcation currents in one-dimensional dynamics
2.2. Stability/bifurcation theory in higher dimension
2.2.1. Polynomial automorphisms of C^2
2.2.2. Semiparabolic implosion and tangencies
2.2.3. Holomorphic maps on P^k
2.3. Robust bifurcations
2.3.1. Polynomial automorphisms
2.3.2. Holomorphic maps on P^k
3. (Non-)Wandering Fatou components
References
D. Fisher: Rigidity, lattices, and invariant measures beyond homogeneous dynamics
1. Introduction
2. Zimmer's conjecture and the Zimmer program
2.1. Zimmer's conjecture
2.2. Measures in the proof of Zimmer's conjecture
2.3. Other results, future directions
3. Totally geodesic manifolds and rank one symmetric spaces
3.1. Other results and open questions
3.2. Dynamics and (non)arithmeticity
References
M. Lemańczyk: Furstenberg disjointness, Ratner properties, and Sarnak’s conjecture
1. Introduction. State-of-the-art
2. Ergodic theory – basic concepts
3. Measure-theoretic dynamical systems – constructions and examples
3.1. Topological dynamics. Subshifts. Invariant measures for homeomorphisms
3.2. Flows, special flows, change of time
4. Ratner's question, MW-strategy, and MOC for smooth time changes of horocycle flows
4.1. Horocycle flows and MOC
4.2. Time changes of horocycle flows and MOC
5. Sarnak's conjecture and Furstenberg systems
5.1. Characteristic classes and the problem of orthogonality
5.2. Orthogonality to characteristic classes. Veech's conjecture
5.3. Proof of Theorem 5.4
5.4. Some consequences of Theorem 5.4
References
A. Mohammadi: Finitary analysis in homogeneous spaces
1. Introduction
2. Complexity of periodic orbits
3. Effective equidistribution of nilflows
4. Horospherical groups
5. Periodic orbits of semisimple groups
6. Effective unipotent dynamics
6.1. Effective versions of the Oppenheim conjecture
6.2. Linearization of unipotent orbits
6.3. Effective density of unipotent orbits
7. Arithmetic combinatorics and polynomial bounds
7.1. Random walks by toral automorphisms
7.2. Quotients of SL_2(C) and SL_2(R) x SL_2 (R)
Periodic orbits
Totally geodesic planes in hybrid manifolds
References
M. Procesi: Stability and recursive solutions in Hamiltonian PDEs
1. Introduction
2. Long time stability
2.1. Questions and open problems
2.2. An idea of the strategies
3. Quasiperiodic solutions
3.1. Questions and open problems
4. Almost periodic solutions
4.1. Questions and open problems
4.2. An idea of the strategies
References
C. Ulcigrai: Dynamics and “arithmetics” of higher genus surface flows
1. Introduction
2. Flows on surfaces of genus one and classical arithmetic conditions
3. Dynamics of flows on surfaces of higher genus
3.1. Locally Hamiltonian flows
3.2. Linearization and rigidity in higher genus
4. Renormalization and cocycles
5. Diophantine-like conditions in higher genus
5.1. Bounded-type IETs and Lagrange spectra
5.2. Roth-like conditions and type
5.3. Controlled growth Diophantine-like conditions
5.4. Effective Oseledets Diophantine-like conditions
References
P. P. Varjú: Self-similar sets and measures on the line
1. Exponential separation property
2. Bernoulli convolutions
3. Failure of exponential separation
4. IFSs with algebraic contraction factors
5. Homogeneous IFSs of three maps
6. Other developments
6.1. Fourier decay
6.2. Absolute continuity
References
10. Partial Differential Equations
T. Buckmaster, T. D. Drivas, S. Shkoller, and V. Vicol: Singularities in compressible Euler with smooth data
1. Introduction
2. Prior results for Euler shock formation and development
3. Classical vs regular shock solutions
4. Azimuthal symmetry
4.1. Riemann-like variables in azimuthal symmetry
4.2. Rankine–Hugoniot jump and entropy conditions
5. Main results
6. Outline: the formation of the preshock
7. Outline: the development of shocks and weak singularities
References
P. Cardaliaguet and F. Delarue: Selected topics in mean field games
1. The MFG equilibria
1.1. The N-player problem
1.2. The MFG equilibria
2. The master equation and the convergence of the Nash system
2.1. Derivatives of maps defined on the space of probability measures
2.2. The master equation
2.3. Convergence of the Nash system
2.4. Propagation of chaos for the N-player game
3. The long-time behavior
3.1. The ergodic MFG system
3.2. The convergence in the monotone setting
3.3. The long-time behavior without monotonicity
4. Smoothing effect of the common noise
4.1. The linear–quadratic case as a warm-up
4.2. Finite-state mean field games
4.3. Vanishing viscosity
4.4. Complements and open problems
5. Further prospectives and related open problems
5.1. Analysis of the MFG system and of the master equation
5.2. Mean field games of control
5.3. Numerical methods and learning
5.4. Mean field control
References
S. Dyatlov: Macroscopic limits of chaotic eigenfunctions
1. Introduction
2. Semiclassical measures
2.1. Semiclassical quantization
2.2. Semiclassical measures for eigenfunctions
3. Ergodic systems
4. Strongly chaotic systems
4.1. Entropy bounds
4.2. Full support property
5. Quantum cat maps
References
R. Ferreira, I. Fonseca, and R. Venkatraman: Variational homogenization: old and new
1. Introduction
2. An overview of contributions to homogenization
3. Homogenization of quasicrystalline functionals via two-scale-cut-and-project convergence
4. Phase transitions in heterogeneous media
5. Phase field model
5.1. Sharp interface limit
5.2. Bounds on the anisotropic surface tension σ
5.2.1. A geometric framework
5.2.2. Structure of minimizers of the cell formula
5.2.3. The planar metric problem
5.2.4. Bounds on the anisotropic surface tension
References
R. L. Frank: Lieb–Thirring inequalities and other functional inequalities for orthonormal systems
1. Introduction
1.1. The general setup
1.2. Example: the HLS inequality
1.3. The duality argument
1.4. A generalization
1.5. Appendix: proof of Theorem 1
2. Sobolev-type inequalities for orthonormal functions
2.1. Bessel-potential bounds
2.2. The Lieb–Thirring inequality
2.3. A more general Lieb–Thirring inequality
3. Fourier restriction inequalities for orthonormal functions
3.1. Strichartz inequality for orthonormal functions
3.2. Stein–Tomas inequality for orthonormal functions
3.3. Spectral cluster bounds
3.4. Kenig–Ruiz–Sogge inequalities
References
A. D. Ionescu and H. Jia: On the nonlinear stability of shear flows and vortices
1. Introduction
1.1. Monotonic shear flows
1.1.1. Remarks
1.2. Point vortices
1.2.1. Adapted polar coordinates and precise results
1.2.2. Remarks
1.3. Organization
2. Main ideas
2.1. Renormalization and the new equations
2.2. Energy functionals and imbalanced weights
2.2.1. The weights A_NR, A_R, A_k
2.3. The auxiliary nonlinear profile
2.4. Control of the full profile
2.5. Analysis of the linearized flow
2.6. Energy functionals and the bootstrap proposition
3. An unstable model: the generalized SQG equation
References
M. Lewin: Mean-field limits and nonlinear Gibbs measures
1. Introduction
2. The N-particle quantum model
3. Mean-field limit to the Gross–Pitaevskii equation
4. Derivation of nonlinear Gibbs measures
Appendix: An elementary proof of Theorem 1
References
K. Nakanishi: Global dynamics around and away from solitons
1. Introduction
2. Ground states as the dynamical threshold
2.1. Below the ground states
2.2. Above the threshold
2.3. Higher energy
3. Transition between solitons
3.1. Threshold dynamics
3.2. Higher mass
4. Transition between multisolitons
4.1. 3-solitons and soliton merger
References
A. I. Nazarov: Variety of fractional Laplacians
References
G. Perelman: Formation of singularities in nonlinear dispersive PDEs
1. Introduction
2. Overview of the well-posedness theory for the NLS equation
3. Mass critical focusing NLS
3.1. Global existence and scattering below the ground state
3.2. Near ground state blow-up dynamics
4. Mass supercritical, energy subcritical NLS
4.1. Self-similar blow-up
4.2. Standing sphere and contracting sphere blow-up solutions
5. Type II blow-up in the energy critical models
5.1. Blow-up for Schrödinger maps from R^2 to S^2
5.2. Energy critical NLS
5.3. Radial multibubble dynamics
5.4. Further generalizations
6. Finite time blow-up for the energy supercritical defocusing NLS
References
G. Tarantello: On the asymptotics for minimizers of Donaldson functional in Teichmüller theory
1. Introduction
2. Blow-up at collapsing zeroes: local analysis
3. Asymptotics for minimizers of the Donaldson functional
References
D. Wei and Z. Zhang: Hydrodynamic stability at high Reynolds number
1. Introduction
2. Linear inviscid damping for shear flows
3. Linear enhanced dissipation for Kolmogorov flow
4. Transition threshold problem for the 3D Couette flow
References
11. Mathematical physics – Special lecture
P. Hintz and G. Holzegel: Recent progress in general relativity
1. Introduction
1.1. The Einstein equations
1.2. The Cauchy problem and generalised harmonic gauges
1.3. Double null gauge and the characteristic initial value problem
1.4. Explicit solutions
1.4.1. Maximally symmetric solutions
1.4.2. The Schwarzschild manifold
1.4.3. Further spherically symmetric spacetimes
1.4.4. The Kerr metric and related metrics
1.5. Matter models
2. The stability of black hole solutions
2.1. Prelude: stability of maximally symmetric solutions
2.1.1. Λ=0
2.1.2. Λ>0
2.1.3. Λ<0
2.2. The formulation of the stability problem and overview of the results
2.3. Toy stability
2.3.1. Λ=0. The classical vector field approach
2.3.2. Λ=0: the extremal case
2.3.3. Λ=0: sharp asymptotics
2.3.4. Λ=0: spectral theoretic approach
2.3.5. Λ>0: exponential decay and quasinormal mode expansions
2.3.6. Λ<0: stable trapping and logarithmic decay
2.4. Linear stability
2.4.1. The double null approach
2.4.2. Generalized harmonic gauge
2.4.3. Other approaches: outgoing radiation gauge
2.5. Nonlinear stability
2.5.1. The case Λ=0
2.5.2. The case Λ>0
2.5.3. The case Λ<0
3. Singularities
3.1. The interior of black holes
3.2. Naked singularities
4. Further topics
4.1. Black hole gluing
4.2. Inverse problems
5. Conclusions and outlook
References
11. Mathematical physics
R. Bauerschmidt and T. Helmuth: Spin systems with hyperbolic symmetry
1. Introduction
2. Hyperbolic spin systems
3. Physical background: Anderson transition
4. Linearly reinforced walks and H^{2|2}
4.1. Hyperbolic symmetry and the VRJP
4.2. Random Schrödinger representation and STZ field
4.3. Phase diagram of the VRJP
4.4. Further discussion
5. The arboreal gas and \H^{0|2}
5.1. Phase transitions for the arboreal gas
5.2. Further discussion
6. Concluding remarks
References
F. Bonetto, E. Carlen, and M. Loss: Kac model
1. Introduction
2. Kac's conjecture
3. Approach to equilibrium in entropy
4. A quantum Kac model
4.1. Example
4.2. Propagation of chaos
4.3. Equilibrium states
References
S. Fournais and J. P. Solovej: On the energy of dilute Bose gases
1. Introduction
2. Quantum many-particle Hamiltonians with 2-body interactions
3. The 2-particle case and the scattering length
4. Bogolubov's theory of superfluidity
4.1. Second quantized formalism
4.2. The Bogolubov's approximation
5. Rigorous proof of the Lee–Huang–Yang formula
References
A. Giuliani: Scaling limits and universality of Ising and dimer models
1. Universality in Statistical Mechanics: a mathematical challenge
2. The scaling limit of nonplanar Ising models
2.1. The nearest neighbor case
2.2. The nonplanar case
3. The scaling limit of interacting dimer models
3.1. The non-interacting case
3.2. Interacting dimer models
4. Methods and ideas behind the proofs
5. Further results, perspectives, and open problems
5.1. Non-planar Ising models
5.2. Interacting dimer models
References
M. B. Hastings: Gapped quantum systems
1. Introduction
1.1. Lattice quantum systems
1.2. Outline of results and notation
2. Exponential decay of connected correlation functions
3. Higher-dimensional Lieb–Schultz–Mattis
3.1. Review of one-dimensional Lieb–Schultz–Mattis theorem
3.2. Higher-dimensional extensions: physics
3.3. Quasiadiabatic continuation
3.4. Sketched proof of higher-dimensional Lieb–Schultz–Mattis theorem
4. Hall conductance quantization
4.1. Introduction
4.2. Results
A. Lieb–Robinson bounds
A.1. Lieb–Robinson bound
References
K. K. Kozlowski: Bootstrap approach to 1+1-dimensional integrable quantum field theories
1. Introduction
1.1. Scattering matrices for quantum integrable field theories
1.2. The operator content and the bootstrap program
1.2.1. The basic operators
1.2.2. The bootstrap program for the zero particle sector
1.2.3. The bootstrap program for the multiparticle sector
1.2.4. The road towards the bootstrap program
2. Solving the bootstrap program
2.1. The 2-particle sector solution
2.2. The n-particle sector solution
3. Towards physical observables and the convergence problem
3.1. The well-poised series expansion for two-point functions
3.2. Convergence of series representation for two-point functions
4. The proof of the convergence of the form factor series
4.1. An simpler upper bound
4.2. Energetic bounds
4.3. Characterization of the minimizer and a lower-bound minimizer
4.4. Singular integral equation characterization of the minimizer σ_{eq}^{(N)}
4.5. The Riemann–Hilbert based inversion of the operator
4.6. Estimation of the minimum
5. Conclusion
References
J. Luk: Singularities in general relativity
1. Introduction
1.1. The Cauchy problem in general relativity
1.2. Singularities and black hole spacetimes
1.3. Trapped surfaces and Penrose's incompleteness theorem
1.4. The cosmic censorship conjectures
2. Construction of singularities
2.1. Spacelike singularities
2.2. Null singularities
2.3. κ-self-similar singularities and naked singularities
3. Black hole interiors and the strong cosmic censorship conjecture
3.1. Spherically symmetric model problems
3.2. C^0-stability of the Kerr Cauchy horizon
3.3. Further problems concerning black hole interiors
3.3.1. Breakdown of weak null singularities
3.3.2. Other singularities in the presence of matter
3.3.3. Extremal black holes
3.3.4. Nonvanishing cosmological constant
4. Gravitational collapse, formation of trapped surfaces, and the weak cosmic censorship conjecture
4.1. Formation of trapped surfaces by focussing of gravitational radiation
4.2. Instability of anti-de Sitter spacetime
4.3. The bounded L^2-curvature theorem and beyond
4.4. Weak cosmic censorship conjecture
References
Y. Ogata: Classification of gapped ground state phases
1. Introduction
2. Finite-dimensional quantum mechanics
3. Quantum spin systems
4. Paths of automorphisms generated by time-dependent interactions
5. The classification of gapped ground state phases
6. Symmetry protected topological (SPT) phases
7. Anyons in topological phases
References
List of contributors
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