Proceedings of the International Congress of Mathematicians Rio de Janeiro 2018 Volume 3 Invited lectures (ICM 2018)

Logic and Foundations
Algebra
Number Theory
Algebraic and Complex Geometry
Geometry
Topology
Lie Theory and Generalizations
Analysis and Operator Algebras
Dynamical Systems and Ordinary Differential Equations
Partial Differential Equations
Mathematical Physics
Probability and Statistics
Combinatorics
Mathematical Aspects of Computer Science
Numerical Analysis and Scientific Computing
Control Theory and Optimization
Mathematics in Science and Technology
Mathematics Education and Popularization of Mathematics
History of Mathematics
Introduction
Notation
Regular families, Tsirelson norms and asymptotic p spaces
Mixed Tsirelson spaces
==========L spaces and Bourgain–Delbaen constructions
Standard BD-spaces
Tsirelson-type estimates for standard BD spaces
Two recent examples
Self-determining subsets and BD augmentations
BD-sets with zero weight
BD-sets with mixed weights
The scalar-plus-compact property and invariant subspaces
Calkin algebras
Indecomposable extensions of Banach spaces with separable duals
A space with the scalar-plus-compact property that contains 1
Conformal complexity and computational consequences
Disks, domes, dogbones, dimension and dendrites
Logarithms, length and Liouville
True trees and transcendental tracts
Introduction
Stein–Tomas–Strichartz and exponential sum estimates on small balls
A first look at decouplings
Fourier analytic decouplings
Applications: Exponential sums on large balls
	Stricharz estimates
	Diophantine inequalities and the Vinogradov Mean Value Theorem
The proof of the decoupling theorem for the parabola
	Parabolic rescaling and linear vs. bilinear decoupling
	A consequence of bilinear Kakeya
	The leap: decoupling from scale  to 2
	Putting everything together
Introduction
Regularization of currents, dynamical degrees and entropies
Super-potentiel theory and equidistribution problems
Theory of densities of currents and periodic points
Some open problems
Introduction
Representations of semigroups
Cancellation properties for semigroups
Examples
Least common multiples
Strings
The spectrum of the semilattice of constructible sets
Ground characters
Subshift semigroups
C*-algebras associated to subshifts
Introduction
Constant mean curvature hypersurfaces
	Rotationally symmetric constant mean curvature surfaces
Constant Nonlocal Mean Curvature hypersurfaces
	Expressions of the NMC of some globally parameterized hypersurfaces
		Without principle value integration
		With principle value integration
	Bounded constant nonlocal mean curvature hypersurfaces
	Unbounded constant nonlocal mean curvature hypersurfaces.
		CNMC hypersurface of revolution
		Near-sphere lattices with CNMC
Serrin's overdetermined problems
Introduction
Preliminaries
	Tracial von Neumann algebras
	Group von Neumann algebras
	Group measure space von Neumann algebras
	Examples of free ergodic p.m.p. actions
	Equivalence relations
	Cartan subalgebras
	Amenability and property (T)
	Popa's intertwining-by-bimodules
Popa's deformation/rigidity theory
	Deformations
	Deformation vs. rigidity
Uniqueness of Cartan subalgebras and W*-superrigidity of Bernoulli actions
	Uniqueness of Cartan subalgebras
	Non-uniqueness of Cartan subalgebras
	W*-superrigidity of Bernoulli actions
Orbit equivalence rigidity
	Cocycle superrigidity
	OE rigidity for actions of non-rigid groups
Structure and rigidity for group von Neumann algebras
	Structural results
	Algebraic rigidity
	W*-superrigidity
Introduction
The diameter of the isomorphism class of a Banach space
Commutators
Counting Ideals in L(Lp)
Spaces that are uniformly homeomorphic to  =====L1 spaces
Weakly null sequences in L1
Subspaces of spaces that have an unconditional basis
Operators on  with dense range
Approximation properties
Introduction
Regularity of solutions to higher order equations
	Regularity of polyharmonic functions in general domains
	Estimates for the Green's function
	The Wiener test
	Weighted integral inequalities
	Historical references
Wave localization
	Dirichlet problem and the birth of the landscape function.
	The Schrödinger equation and the effective potential of localization
	Historical references
Paradoxical equidecompositions and invariant measures
	Remark on amenability
	Equidecompositions using sets of the Baire property
	The Banach–Ruziewicz problem and the measurable Banach–Tarski paradox
	Tarski's circle-squaring problem
	Equidecompositions and perfect matchings in bi-partite graphs
Measurable Banach–Tarski
Laczkovich's circle-squaring
Circle-squaring with measurable functions
Measurable circle-squaring
Open problems
Introduction
Aspects of Multi-Parameter Theory
	Chang-Fefferman BMO
	Little BMO
	Little product BMO
Upper Bounds
	Hilbert transform
	Calderón-Zygmund operators
	Journé operators
Lower Bounds
Weak Factorization
Div-Curl Lemma
Introduction
Completeness problems and spectral gaps
	Beurling-Malliavin, gap and type problems
	Toeplitz kernels
	A formula for the gap characteristic of a set
	Bernstein's weighted uniform approximation
	Type formulas
	Toeplitz order
Inverse spectral problems and truncated Toeplitz operators
	Canonical systems
	De Branges spaces
	Inverse spectral problems
	PW-measures and systems
	Truncated Toeplitz operators
	Examples of inverse spectral problems via truncated Toeplitz operators
Introduction
	The setup
	Approximation and stability
	Metric ultraproducts
Weak soficity and the pro-finite topology
	Connections with the pro-finite topology
	Approximation by classes of finite groups
	Approximability of Lie groups
Approximation by unitary matrices
	The choice of the metric
	Cohomological obstructions to stability
	Cohomology vanishing and examples of n-Kazhdan groups
Applications to group theory
	The basic setup
	Topological methods to prove existence of solutions
Internal approximation: nuclearity and exactness
K-theory, the UCT, and stable uniqueness
External approximation: quasidiagonality
Rosenberg's conjecture: amenability
Toms–Winter regularity
Elliott's programme: classification
Optimization of Birkhoff averages
Optimization of vectorial Birkhoff averages
Optimization of the top Lyapunov exponent
Optimization of all Lyapunov exponents
Introduction
	The Ornstein-Weiss factor
Topological entropy for Z-actions
Topological sofic entropy for actions of free groups
	Examples
		A boring example and asymptotic freeness
		A curious example
		The Ornstein-Weiss example revisited
Sofic groups
An application to Gottschalk's Surjunctivity Conjecture
Measure sofic entropy
Symbolic actions: the topological case
	Symbolic actions: the measure case
	Bernoulli shifts
	The f-invariant and RS-entropy
Classification of Bernoulli shifts
Bernoulli factors
Rokhlin entropy
Algebraic actions
	Topological versus measure entropy
		Principal algebraic actions
		Yuzvinskii's addition formula
		Pinsker algebra
Geometry of model spaces
The critical orbit conjecture
Critical orbit relations
Proof strategy: heights and equidistribution
What is known
Arbitrary points
The transitive and nonhyperbolic setting
	Paradigmatic examples
	Weak forms of hyperbolicity
	Oseledecs' theorem and nonhyperbolicity
	Nonhyperbolic settings
	Robustly nonhyperbolic transitive diffeomorphisms
	Hyperbolic flavors in nonhyperbolic dynamics
Robustly nonhyperbolic dynamics
	Blenders
	Heterodimensional cycles
	Homoclinic relations and classes
	Weak forms of hyperbolicity and homoclinic classes
Tools to build nonhyperbolicity
	GIKN method
	The flip-flop method
Building nonhyperbolic measures and sets
	Applications of the GIKN-method
		Step skew-products
		Multiple zero exponents in step skew-products
		Nonhyperbolic ergodic measures in homoclinic classes
		Ergodic measures with multiple zero exponents
	Applications of the flip-flop method: robust zeros
Arithmetic conditions
Diffeomorphisms of the circle and the torus
Pseudo-rotations of the disc
	Birkhoff rigidity conjecture
	Rigidity times, mixing and entropy
Hamiltonian systems
	Topological stability
	Beyond the classical KAM theory.
		Weak transversality conditions.
		Absence of transversality conditions.
	Effective stability
	On invariant tori of convex Hamiltonians
		The ''last invariant curve'' of annulus twist maps.
		On the destruction of all tori
	Birkhoff Normal Forms
Dynamics of quasi-periodic cocycles
	The case G=SL(2,R)
	The symplectic case
	The case G=Diff0(T)
Mixing surface flows
	Spectral type.
	Spectral type of related systems
	Multiple mixing.
Ergodic theory of diagonal actions on the space of lattices and applications to metric Diophantine approximation
	Kronecker sequences
	Higher dimensional actions.
Iteration of a semicontraction on Euclidean space
Horofunctions
Iteration of random semicontractions
Ergodic theory and subadditivity
Introduction
Self-similar sets and measures
	Preliminaries on dimension
	Similarity and Lyapunov dimension
	The overlaps conjecture, and what we know about it
	Some ideas from the proofs
	Higher dimensions
	Parametric families
	Further developments
Bernoulli convolutions
	Bounds on the size of the exceptional parameters
	Mahler measure
	Absolute continuity for algebraic parameters
	Dimension results for other parameters
Projection and slice theorems
	Dimension conservation
	Projections of self-homothetic sets
	Projections of sets and measures with rich symmetries
	Slices
Renormalization Group in Statistical Mechanics and Critical Phenomena
Dynamical Renormalization
Renormalization and Rigidity for Circle Homeomorphisms
Critical behaviour and parameter dependence
Beyond dimension one
Concluding remarks
Introduction
Dynamics under a boundary condition
	Prime ends vs. boundary dynamics
	The boundary condition
	The irrational case
	The rational case
	On Cr-generic area-preserving diffeomorphisms
	The smooth setting
	Homotopical boundedness
Further results
	Vanishing rotation numbers without fixed points
	Rotation sets
	No Birkhoff-like behavior for area-preserving maps
	A Poincaré-like result for decomposable circloids
Dynamically optimal billiard tables and flat surfaces
The list of known examples
Teichmüller curves and variations of Hodge structures
Constructing Veech surfaces and computing the Veech group
Affine invariant manifolds
Finiteness and classification results
Modular forms and Euler characteristics
Orbifold points and other connections to arithmetic geometry
Actions of Kazhdan's groups
Cones and orders on groups
Groups of piecewise-projective homeomorphisms
The spectrum of sharp regularities for group actions
Zero Lebesgue measure for exceptional minimal sets
Ergodicity of minimal actions
Absolute continuity of the stationary measure
Structural stability and the space of representations
Approximation by conjugacy and single diffeomorphisms
Topological invariance of the Godbillon–Vey class
On groups of real-analytic diffeomorphisms
Introduction
Robust dynamics
Examples
	h-Transversalities
	Surgeries
Partial hyperbolicity in other contexts
	Dynamics far from tangencies
	Skew-products
	Discrete subgroups of Lie groups
Strong partial hyperbolicity in 3-manifolds
	Topological obstructions
	Integrability
	Classification
	More questions
Dynamical implications
Thermodynamic formalism, introductory notions
Introduction to dimension 1
Hyperbolic potentials
Non-uniform hyperbolicity in real and complex dimension 1
Geometric pressure and equilibrium states
Other definitions of geometric pressure
Geometric coding trees, limit sets, Gibbs meets Hausdorff
Boundaries, radial growth, harmonic vs Hausdorff
Law of Iterated Logarithm refined versions
Accessibility
Quasi-periodic operators, cocycles and systems
	One dimensional quasi-periodic Schrödinger operators
	Quasi-periodic cocycles and quasi-periodic linear systems
	Relations between operators and dynamical systems.
Almost Reducibility
	Perturbative reducibility
	Non-perturbative reducibility.
	Strong almost reducibility and Quantitative almost reducibility.
	Global reducibility.
Applications to quasi-periodic Schrödinger operators
	Spectrum of quasi-periodic Schrödinger operators
		Cantor spectrum
		All gaps are open
		Estimate of the spectral gaps.
		Homogeneous spectrum.
	The spectral measure, IDS and Lyapunov exponent
		Anderson localization.
		Absolutely continuous spectrum.
		Continuity of Lyapunov exponent and IDS
		Positivity of Lyapunov exponent.
Introduction and physical background
Linear dynamics
Overview of the mathematical results
	Stability result
	Instability result
The nonlinear dynamics: The toy model
Proof of the stability result
	Coordinate transform
	Construction of the toy model norm
	Main energy estimate
Proof of the instability result
	Ideas of the proof
		The choice of data, and setup
		The linearized system
		Linear analysis, and a more precise toy model
		Nonlinear analysis, and the Taylor expansion
	Further discussions
		Asymptotic instability
		Genericity
Introduction
Three-level weights procedure and properties
Compactness and quantitative regularity estimates
	Regularity conditioned by the weights
	Our explicit regularity estimate
Stability and existence of renormalized solutions: Proof of Renormanelastic
	Stability of renormalized solutions
	Existence of renormalized solutions
	Toward the uniqueness of weak solutions to (1)
An anelastic compressible equation coming from fluid mechanics
Introduction
Euler equations
	Splash and Splat Singularities for the Free Boundary
	Splash singularities for the internal wave
	Stationary Splash singularity
Incompressible porous media equation-Darcy's law
Introduction
Structure of singular derivatives
Functionals on measures
Characterization of generalized Young measures
The converse of Rademacher's theorem
Cheeger's conjecture
Sketch of the proof of thm:main
The water waves equations
The question of local existence
Global existence with small decaying data
Quadratic terms. Normal forms
Global existence: bootstrap procedure
Optimal decay estimates
Further results
Introduction
Interpolation inequalities and flows on compact manifolds
	Interpolation inequalities on Sd
	Flows and carré du champ methods on Sd
	Inequalities on compact manifolds
Rigidity on cylinders and sharp symmetry results in critical Caffarelli-Kohn-Nirenberg inequalities
	Three equivalent rigidity results
	Optimal symmetry range in critical Caffarelli-Kohn-Nirenberg inequalities
	Sketch of the proof of theorem:Rigidity-dimension-n
Rigidity and sharp symmetry results in subcritical Caffarelli-Kohn-Nirenberg inequalities
	Subcritical Caffarelli-Kohn-Nirenberg inequalities
	A rigidity result
	Sketch of the proof of Thm:Rigidity
	Considerations on the optimality of the method
Bifurcations and symmetry breaking
	Rigidity and bifurcations
	Bifurcations, reparametrization and turning points
	Symmetry breaking and energy considerations
	An open question
Introduction
First order models
	Explicit solutions
	Asymptotic speed and profile
	Non-coercive case
Second order models
	Models
	Radial case and its generalization
	Lipschitz bounds
	Existence of asymptotic speed
Asymptotic profile
	Limit equations
	Case of intermediate speed
	Asymptotic profile of large level set
Unscaled asymptotic profile
	Large time convergence of a solution
	Some open problems
Introduction
Ways of describing a function
	From ODEs to rough differential equations
	Reconstruction of a coherent germ
	From ODEs to PDEs
	Paraproducts and the paracontrolled Ansatz
	Ambiguities
	Other approaches
Weak universality
	The Hairer–Quastel universality result
	KPZ universality
	A notion of solution for KPZ
	Convergence to KPZ for the growth model
	Other weak universality results
Stochastic quantisation in three dimensions
Other results
Introduction
Microlocal analysis for Anosov and Axiom A flows
	Anosov and Axiom A flows
	Solving transport equations and continuation of the resolvent
	Localisation of the spectrum and decay of correlations
	Dynamical zeta functions
Boundary rigidity and X-ray tomography problems
	Simple metrics
	Cases with trapped set, conjugate points or non-convex boundary
	Closed manifolds
Introduction
The 3D Euler equation and the 2D Boussinesq system: the hyperbolic scenario
The 2D Euler equation
The one-dimensional models
The SQG patch problem: a blow up blueprint
Discussion
Introduction
	Three spheres theorem for wild sets.
	Preceding results.
	Remez type inequality.
	Propagation of smallness from sets with big Hausdorff dimension.
	Propagation of smallness for gradients.
	Open questions.
	Estimates for Laplace eigenfunctions
Preliminaries
	Hausdorff content
	Three spheres theorem for wild sets.
	Doubling index
Auxiliary lemmas
	Estimates of the zero set
	Estimate for sub-level sets
	Main tool
Proof of the Main result
	Reformulations of th:m
	l:base,pr:2 imply pr:1.
	Proof of pr:2
Propagation of smallness for the gradients of solutions
	Formulation of the result
	Modifications of the proof.
	Outline of changes.
	Proof of l:base2
Introduction
Orlicz spaces: basic facts and useful tools
Linear estimates
Local well-posedness
Non-existence
Global Existence
Introduction
Integrable equations
	KdV solitons and multi-solitons
	Decomposition into solitons for KdV
	One dimensional cubic NLS
	The sine-Gordon equation
	Other integrable models and nearly integrable models
	Formal works and numerical simulations
Nonlinear models with solitary waves
	The generalized Korteweg-de Vries equation
	The nonlinear Schrödinger equation
	The 4 equation
	The energy critical nonlinear wave equation
Asymptotic stability
	Asymptotic stability for gKdV solitons
	Asymptotic stability for NLS equations
	Asymptotic stability of the 4 kink
	Blow up profile for the critical wave equation
Asymptotic multi-solitons
	Multi-solitons with weak interactions
	Multi-solitons with strong interactions
	Soliton interaction with the background
Decomposition into solitons for the energy critical wave equation
Collision problem
	Collision for the quartic gKdV equation I
	Collision for the quartic gKdV equation II
	Collision for the quartic gKdV equation III
	Collision for the perturbed integrable NLS equation
	Collision for the 5D energy critical wave equation
Introduction
	Kinetic theory
	Main equations of kinetic theory
	Open problems and conjectures
		The Cauchy problem
		Study of a priori solutions
		Regularity conjectures for long-range interactions
De Giorgi–Nash–Moser meet Hörmander
	The resolution of Hilbert 19-th problem
	The theory of hypoellipticity
	Extending the DGNM theory to hypoelliptic settings
Conditional regularity of the Landau equation
	Previous works and a conjecture
	DGNM theory and local Hölder regularity
	Maximum principles and pointwise bounds
	Schauder estimates and higher regularity
Conditional regularity of the Boltzmann equation
	Previous works and a conjecture
	Maximum principle and pointwise L bound
	Weak Harnack inequality and local Hölder regularity
	Maximum principle and decay at large velocities
Introduction
	Scattering
	Resonances
	Semiclassical regime
		Semiclassical evolution of wavepackets
		Introducing the trapped set
	Hyperbolicity
Examples of hyperbolic flows
	A single hyperbolic periodic orbit
	Fully developed chaos: fractal hyperbolic trapped set
	An interesting class of examples: hyperbolic surfaces of infinite area
Fractal Weyl upper bounds
	Counting long living resonances
	Complex deformation of Ph: turning resonances into eigenvalues
	Resonances vs. classical trapped set
		Case of a nontrapping dynamics
		Fractal hyperbolic trapped set
		Sketch of the proof of the Fractal Weyl upper bound
		Improved fractal upper bounds on hyperbolic surfaces
Dynamical criteria for resonance gaps
	Evolution of an individual wavepacket
		Hyperbolic dispersion of a wavepacket
		Introducing a quantum partition
	Evolving a general state
	Improving the pressure gap
Normally hyperbolic trapped set
	Examples of normally hyperbolic trapped sets
		Examples in chemistry and general relativity
		From classical to quantum resonances
	An explicit resonance gap for normal hyperbolic trapped sets
L. Prandtl and boundary layers
Vortex sheets in ideal fluid flow
	Full plane
	Domains with boundary
Vanishing viscosity limit and convergence criteria
Introduction
The topological domain categroid
The target categroid
The TQFT functor
The volume conjecture for the Teichmüller TQFT
Future perspectives
Introduction
The special geometry on the CY moduli space
Hodge structure on the middle cohomology of the quintic
Hodge structure on the invariant Milnor ring
Oscillatory representation and computation periods ()
Computation of the Kähler potential
Real structure on the cycles
Conclusion
Introduction
Random surfaces
	1+1-dimensional Lorentzian triangulations and (continuous) integrability
	2-dimensional Euclidian tessellations and (discrete) integrability
Lattice models
	The six-vertex model and beyond
Lie algebras, quantum spin chains and CFT
	Whittaker vectors and path models
	Fusion product, Q-system cluster algebra and Macdonald theory
		Graded characters and quantum Q-system
		From Cluster algebra to quantum toroidal and Elliptic Hall algebras
Open problems
Introduction
Algebraic quantum field theory and local conformal nets
Representation theory and superselection sectors
Subfactors and tensor categories
-induction, modular invariants and classification theory
Vertex operator algebras
From a vertex operator algebra to a local conformal net and back
Other types of conformal field theories
Future directions
Potential theory
Dirichlet and Thomson principles
Recurrence of Markov chains
Eyring-Kramers formula for the transition time
Metastability
The problem
The models: general considerations
The models: microscopic dynamics
Invariant measures, reversibility and currents
	Other models
A two steps strategy
	Soft core potentials
	Hard core potential
Partially hyperbolic Fast-slow systems and limit theorems
	The not so simple simplest example
Final considerations
Introduction
Excitation Hamiltonian
Generalized Bogoliubov Transformations
Renormalized Excitation Hamiltonian
Cubic Conjugation
Diagonalization and Excitation Spectrum
Introduction
The idea of the transfer operator approach
Mechanism of the crossover for =====R0
Analysis of =====R1
Analysis of =====R2 for the block RBM
	Sigma-model approximation for =====R2 for the block RBM
	Analysis of =====R2 for block RBM of (1.13)-(1.14)
Introduction
The framework
	QFTs for =====S-structured manifolds
	Point operators of a QFT
	Deformations of a QFT
	G-symmetric QFTs
	Submanifold operators and morphisms of a QFT
	Compactifications of QFTs
	Anomalous and meta QFTs
	Supersymmetric QFTs
Examples
	Honest constructions
	Using path integrals
	Using String/M theory
Four-dimensional =====N=2 supersymmetric theories
	Basic properties
	Higgs branch functor and the slicing
	Examples
	Known overlaps among the examples
	Two other functors
	Relation among the functors
	Consequences
Introduction
Stochastic interface growth
	(2+1)-dimensional growth: KPZ and Anisotropic KPZ (AKPZ) classes
	Mathematical results for Anisotropic KPZ growth models
		Extensions and open problems
		Slow decorrelation along the characteristics
Interface dynamics at thermal equilibrium
	Reversible tiling dynamics, mixing time and hydrodynamic equation