International Congress of Mathematicians 2022 July 6–14

Front cover Front matter Foreword Contents Past ICMs Award winners Opening greetings Closing remarks Staus report Photographs The work of the prize winners Hugo Duminil-Copin 1. Introduction 1.1. Bernoulli percolation 1.2. The Ising model 1.3. A general picture 2. (Dis)continuity of phase transitions 3. Triviality of \Phi^4_4 4. Rotational invariance for the critical FK models References June Juh 1. Graphs, chromatic polynomials, and Read's conjecture 1.1. The four-color conjecture and chromatic polynomials 1.2. Read's conjecture 2. Matroids and the Heron–Rota–Welsh conjecture 2.1. Matroids 2.2. From graphs to matroids 2.3. Rank functions, characteristic polynomials, and the Heron–Rota–Welsh conjecture 3. The Dowling–Wilson conjecture 3.1. Background: Theorems by de Bruijn–Erdős, Motzkin, Greene, and Ryser's linear algebraic proof 3.2. The proof of the Dowling–Wilson conjecture 4. The connection with Hodge theory and algebraic geometry 4.1. Three fundamental ideas and other ingredients from the proof of the Heron–Rota–Welsh conjecture 4.2. Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann relations 5. The strong Mason conjecture (on independence numbers), and related developments and applications 5.1. Mason conjecture, regular strength, strong, and ultra-strong 5.2. The Mihail–Vazirani conjecture Conclusion References James Maynard References Maryna Viazovska 1. Introduction 2. The past 3. Modular forms 4. Viazovska's construction for single roots 5. Viazovska's construction for double roots 6. Interpolation and consequences 7. The future References Mark Braverman 1. Communication complexity 2. Information complexity 3. Interactive compression 4. Direct sum 5. Communication complexity of Set-Intersection 6. Parallel repetition of two-prover games 7. Interactive coding theory 8. Lower bounds for bounded-depth circuits 9. Grothendieck's constant vs. Krivine's bound References Barry Mazur 1. Geometric and differential topology 2. Algebraic geometry 3. Arithmetic topology 4. Torsion subgroups of elliptic curves 5. Rational points on modular curves 6. Fermat's Last Theorem 7. Iwasawa main conjectures 8. Elliptic curves and the Birch and Swinnerton-Dyer conjecture 9. The Fontaine–Mazur conjecture 10. Deformations of Galois representations 11. Diophantine geometry 12. Euler systems and related areas 13. Exposition 14. Mentorship References Elliott Lieb 1. Quantum Coulomb systems Stability of matter Existence of the thermodynamic limit for real matter with Coulomb forces Thomas–Fermi theory and density functional theory Lieb–Thirring inequalities The ionization problem Bosonic systems 2. Functional inequalities Lieb's concavity theorem and the strong subadditivity The Brascamp–Lieb inequalities The sharp Hardy–Littlewood–Sobolev inequality 3. Topics not covered References Nikolai Andreev References Prize Lectures Hugo Duminil-Copin 1. Short motivation 2. The first 20 years: a laborious start 2.1. Ising model's prehistory 2.2. Formal definition 2.3. What does the Ising model truly model? 2.4. Peierls' argument 3. Onsager's 1944 revolution and the integrability of the Ising model 3.1. Kramers–Wannier treatment of the Ising model and duality 3.2. Onsager's result 4. The 1950s and 1960s: The Ising model becomes a laboratory for understanding critical phenomena 4.1. Progress in mathematical physics: From perturbative regions of the phase diagram to the vicinity of the critical point 4.1.1. Correlation inequalities 4.1.2. The Ising model with a magnetic field: The Lee–Yang theory 4.2. Revolutionary progress on the physics front 4.2.1. Critical exponents and the success of scaling theory 4.2.2. Kadanoff's block-spin renormalization and universality 5. The 1960s and 1970s: Emergence of the probabilistic interpretation 5.1. The random geometry of the spin configuration 5.2. Boundary conditions and the Gibbs formalism 5.3. Phase coexistence and Wulff shape 6. The 1970s and 1980s: the Ising model and field theory 6.1. Constructive quantum field theory 6.2. Reflection positivity 6.3. The random current revolution 6.4. Triviality in dimension d>4 6.5. Rigorous renormalization group in 4D Ising 6.6. Forty years later: The random current strikes back 7. The last 50 years: Ising model and percolation 7.1. Percolation interpretation of random currents 7.2. Fortuin–Kasteleyn percolation 7.3. The broader impact of the Ising model on dependent percolation models 8. Over the last ten years: Conformal invariance of the Ising model 8.1. What is conformal invariance? 8.2. Conformal invariance of the 2D Ising model 8.3. Towards universality of the 2D Ising model 8.4. Conformal bootstrap in 3D Ising model 9. A tail to this story References June Huh 1. Introduction 2. Lorentzian polynomials 3. Intersection cohomology of matroids References James Maynard 1. Introduction 2. Multiplicative number theory 2.1. Primes and zeros 2.2. Zero density estimates 2.3. Limits to multiplicative techniques 3. Sieve methods 3.1. Arranging the large prime factors 3.2. Limitations of sieve methods and the parity phenomenon 4. Side-stepping limitations of sieve methods 5. Primes in arithmetic progressions and extending the level of distribution 6. Bilinear estimates 6.1. Type I/II ranges to primes 7. Primes in thin sets 8. Further arithmetic information 9. Choice of lift and comparison sets 10. Abelian quadratic limitations References Maryna Viazovska 1. Introduction 1.1. Construction of a discrete Fourier uniqueness set 2. Auxiliary results from Fourier analysis 3. Auxiliary results from the theory of modular forms 4. Proof of Theorem 3.2 5. Proof of Theorem 1.4 References Mark Braverman 1. Computational complexity theory 1.1. Upper and lower bounds 1.2. Abstraction and complexity classes 1.3. Reductions and conditional lower bounds 1.4. Unconditional lower bounds: some attack routes 1.5. Shannon's information theory and one-way communication 2. Communication complexity 3. Information complexity 3.1. Direct sum for information and amortized communication 3.2. Direct sum and direct product for communication 3.3. Exact communication complexity of set disjointness 3.4. Some other connections 4. Challenges and next steps References Nikolai Andreev Marie-France Vignéras 1. Introduction 2. Notation 3. Change of basic field 4. Change of coefficient ring 5. Parabolic induction 6. Admissible representations and duality 7. Supercuspidal support 8. Hecke algebras 9. Representations over a field of characteristic different from p 10. Bernstein blocks 11. Satake isomorphism 12. Pro-p Iwahori Hecke ring 13. Modules of pro-p Iwahori Hecke algebras over a field in characteristic p 14. Representations over a field of characteristic p 15. Local Langlands correspondences for GL(n,F) 16. Gelfand–Kirillov Dimension References Popular scientific expositions by A. Okounkov The Ising model in our dimension and our times 1. Mathematics and physics 2. The Ising model 2.1. Stuff fluctuates in space 2.2. A lattice in space 2.3. Signs on a lattice 2.4. Probabilities and energy 2.5. Energy vs. entropy 2.6. Interactions in the Ising model 2.7. Clusters and interfaces 3. Gibbs measures 3.1. Definition 3.2. High temperature 3.2.1. 3.2.2. 3.2.3. 3.2.4. 3.2.5. 3.2.6. 3.2.7. 3.2.8. 3.3. Low temperature 3.3.1. 3.3.2. 3.3.3. 3.3.4. 3.3.5. 3.3.6. 3.4. Critical temperature 4. What happens at T=T_c? 4.1. Critical Gibbs measures 4.2. The Potts model 4.3. Theorems 4.4. Contours of proofs, seen in the distance 4.4.1. 4.4.2. 4.4.3. 4.4.4. 4.4.5. 4.4.6. 4.4.7. 4.4.8. 4.4.9. 5. Further reading A. The universal attraction of the Ising model A.1. Universality A.2. Models like the Ising model A.2.1. A.2.2. A.2.3. A.3. Critical points A.3.1. A.3.2. A.3.3. A.3.4. A.3.5. References Combinatorial geometry takes the lead 1. Points, lines, and planes 2. Points, lines, planes, etc. 3. Matching flats to flats 4. Rank and matroids 5. Some examples of matroids 5.1. Points in F^d, where F is a field 5.2. Projective spaces 5.3. Field extensions 5.4. Tropical realization of matroids 6. Graded Möbius algebra 6.1. Algebras 6.2. Graded algebras 6.3. Hard Lefschetz property 6.4. The graded Möbius algebra, finally 7. The big induction 8. Inspirations from topology 8.1. Cohomology 8.2. Multiplication and Poincare duality 8.3. The hard Lefschetz property 9. Further reading A. A rice bowl of linear algebra A.1. Linear equations A.2. Linear maps A.3. Abstract linear spaces A.4. Kernel, image, and quotient A.5. Dual vector spaces A.6. Rank and rank B. Determinant B.1. Formula B.2. Permutations B.3. The N=2 case and the cohomology of the torus B.4. The general case C. Tropical lines, planes, etc. C.1. C.2. C.3. C.4. C.5. C.6. References Rhymes in primes 1. The ancient sieve 2. Last digits of primes 3. The Chinese remainder theorem 4. Infinity and limits 5. The density of primes 6. The prime number theorem 7. Inclusion–exclusion 8. The first challenge for sieves 9. Patterns in primes 10. Closing the gap 11. Further reading 12. A glimpse into the argument 12.1. Being prime on average 12.2. Looking for ρ, part I 12.3. Looking for ρ, part II 12.4. Primes in arithmetic progressions, on average A. Limits B. Mellin transform and the density of primes References The magic of 8 and 24 1. Spheres keep their distance 1.1. Spheres in a d-dimensional space 1.2. Sphere packings in R^2 1.3. Contact number in R^3 1.4. The densest packings in R^3 2. Beyond the 3-space 2.1. 4, 5, 6, 7, 8, ... 2.2. Fluid diamond in d=9 2.3. Stars align in E_8 2.3.1. Roots 2.3.2. Reflections 2.3.3. ADE classification 2.3.4. Discriminant 2.3.5. Codes 2.3.6. The Coxeter plane 2.4. Very large dimensions 3. Upper bounds on packing density 3.1. Positive definite forms and functions 3.1.1. 3.1.2. 3.1.3. 3.1.4. 3.1.5. 3.1.6. 3.1.7. 3.1.8. 3.2. The fundamental bound 3.2.1. 3.2.2. 3.2.3. 3.2.4. 3.2.5. 3.2.6. 3.2.7. 3.2.8. 3.2.9. 3.2.10. 3.2.11. 4. Viazovska's magic function 4.1. Lattice packings that saturate the bound 4.1.1. 4.1.2. 4.1.3. 4.1.4. 4.2. The wait is over 4.3. Interpolation 4.3.1. 4.3.2. 4.3.3. 4.3.4. 5. Further reading A. Inner products A.1. A.2. A.3. A.4. A.5. B. Groups and positive definite functions B.1. B.2. B.3. B.4. B.5. B.6. B.7. B.8. C. Fourier series C.1. C.2. C.3. C.4. C.5. C.6. C.7. C.8. C.9. C.10. D. Modular forms D.1. The space of lattices D.2. D.3. D.4. D.5. D.6. D.7. E. The volume of a d-dimensional ball E.1. E.2. E.3. E.4. E.5. E.6. E.7. F. More on E_8 and regular m-gons F.1. F.2. F.3. F.4. References Summaries of prize winners’ work by A. Jackson Mark Braverman Barry Mazur The ``Mazur Swindle'' The Lure of Algebraic Geometry Deforming Galois Representations Beyond Mathematics Elliott H. Lieb Different Fields, Different Goals Square Ice Stability of Matter Bose-Einstein Condensate Shaping Decades of Research Nikolai Andreev An Unusual Approach A Potato Chip, A Sausage, A Sheet of Paper From Multimedia to Print Reaching Across Barriers List of contributors Back cover Front cover Front matter Contents Special plenary lectures K. Buzzard: What is the point of computers? A question for pure mathematicians 1. Introduction 2. Overview of the paper 3. A brief history of formally verified theorems 3.1. The 20th century 3.2. The prime number theorem 3.3. The four color theorem 3.4. The odd order theorem 3.5. The Kepler conjecture 3.6. Perfectoid spaces 3.7. Condensed mathematics 3.8. Other results 4. mathlib 5. A brief guide to type theory 5.1. What is a type? 5.2. Inductive types 5.3. Dependent types 5.4. Examples 5.5. Foundations 6. The future 6.1. A new kind of mathematical document 6.2. Semantic search in a mathematical database 6.3. Checking proofs 6.4. Teaching 6.5. Other ideas References F. Calegari: Reciprocity in the Langlands program since Fermat’s Last Theorem 1. Introduction 1.1. The Fontaine–Mazur conjecture 1.2. R = \mathbf{T} theorems 2. The early years 2.1. The work of Diamond and Fujiwara 2.2. Integral p-adic Hodge Theory, part I: Conrad–Diamond–Taylor 2.3. Integral p-adic Hodge Theory, part II: Breuil–Conrad–Diamond–Taylor 2.4. Higher weights, totally real fields, and base change 3. Reducible representations: Skinner–Wiles 4. The Artin conjecture 5. Potential modularity 6. The work of Kisin 6.1. Local deformation rings at v = p 6.2. Kisin's modification of Taylor–Wiles 7. p-adic local Langlands 7.1. The Breuil–Mézard conjecture 7.2. Local–global compatibility for completed cohomology 8. Serre's conjecture 8.1. Ramakrishna lifting 8.2. The Khare–Wintenberger method 9. Higher dimensions 9.1. Construction of Galois representations, part I: Clozel–Kottwitz 9.2. The Sato–Tate conjecture, part I 9.3. Taylor's trick: Ihara avoidance 9.4. The Sato–Tate conjecture, part II 9.5. Big image conditions 9.6. Potentially diagonalizable representations 9.7. Even Galois representations 9.8. Modularity of higher symmetric powers 10. Beyond self-duality and Shimura varieties 10.1. The Taylor–Wiles method when l_0 > 0, part I: Calegari–Geraghty 10.2. Construction of Galois representations, part II 10.3. Construction of Galois representations, part III: Scholze 10.4. The Taylor–Wiles method when l_0 > 0, part II: DAG 11. Recent progress 11.1. Avoiding conjectures involving torsion I: the 10-author paper 11.2. Avoiding conjectures involving torsion II: abelian surfaces 12. The depths of our ignorance References F. Pretorius: A survey of gravitational waves 1. Introduction 2. Einstein gravity 2.1. Gravitational waves 2.1.1. Basic properties of gravitational waves in the weak field limit 2.1.2. Weak field emission from a compact object binary 2.1.3. Strong field gravitational wave emission 2.1.4. Strong field emission from a compact object binary 2.1.5. The ringdown 3. Gravitational wave observational landscape 4. Survey of what has been observed to date 5. The future of gravitational wave astronomy References Plenary lectures M. Bestvina: Groups acting on hyperbolic spaces—a survey 1. Introduction 2. Hyperbolic groups 2.1. Classification of elements 2.2. The Rips complex 2.3. Subgroups 2.4. Boundary 2.5. Asymptotic dimension 2.6. JSJ decomposition 2.7. The combination theorem 2.8. Random groups are hyperbolic 2.9. \mathbb{R}-trees and applications 2.10. Hyperbolic spaces degenerate to \mathbb{R}-trees 2.11. The Rips machine 2.12. Applications 2.12.1. Automorphisms of hyperbolic groups 2.12.2. Local connectivity of ∂G 2.12.3. Thurston's compactness theorem 3. Mapping class groups 3.1. The boundary of the curve complex 3.2. WPD, acylindrically hyperbolic groups, quasimorphisms 3.3. Subsurface projections 4. Projection complexes 5. Group \operatorname{Out}(F_{n}) 5.1. Outer space 5.2. The boundary of Outer space 5.3. Lipschitz metric and train-track maps 5.4. Hyperbolic complexes 5.4.1. The free splitting complex \operatorname{FS}_n 5.4.2. The cyclic splitting complex \operatorname{FZ}_n 5.4.3. The free factor complex \operatorname{FF}_n 5.5. Subfactor projections 5.6. Questions References B. Bhatt: Algebraic geometry in mixed characteristic 1. Introduction 2. Prisms and relative prismatic cohomology 3. Absolute prismatic cohomology 3.1. Definition and key examples 3.2. Hodge–Tate crystals 3.3. The Nygaard filtration 3.4. Galois representations 4. Algebraic K-theory 5. Commutative algebra and birational geometry 5.1. Vanishing theorems in commutative algebra 5.2. Birational geometry 6. p-adic Riemann–Hilbert References T. Bodineau et al.: Dynamics of dilute gases: a statistical approach 1. Aim: providing a statistical picture of dilute gas dynamics 1.1. A very simple physical model 1.2. Three levels of averaging 1.3. A probabilistic approach 2. Typical dynamical behavior 2.1. Boltzmann's great intuition 2.2. Lanford's theorem 2.3. Heuristics of the proof 2.4. Some elements of proof 2.5. On the irreversibility 3. Correlations and fluctuations 3.1. From instability to stochasticity 3.2. Defects in the chaos assumption 3.3. Higher-order correlations and exponential moments 3.4. A complete statistical picture for short times 4. Beyond Lanford's time 4.1. Main difficulties 4.2. Close to equilibrium 4.3. Some elements of the proof of Theorem 4.2 5. Open problems and perspectives 5.1. Long time behavior for dilute gases 5.2. The role of microscopic interactions 5.3. Nonequilibrium stationary states 5.4. A realm of kinetic limits References A. Braverman and D. Kazhdan: Automorphic functions on moduli spaces of bundles on curves over local fields: a survey 1. Introduction 1.1. Langlands correspondence over functional fields 1.2. Hecke eigenfunctions on moduli spaces of bundles over local fields 1.3. Relation of the archimedian case to geometric Langlands correspondence and conformal field theory 1.4. Notations 1.5. Organization of the paper 2. Smooth sections of line bundles on varieties and stacks 2.1. Smooth sections on varieties 2.2. Smooth sections on stacks 2.3. Functoriality 2.4. An example: stacks over \mathcal{O}_{F} 2.5. Nice and excellent stacks 3. The case of \operatorname{Bun}_{G}: preliminaries 4. Affine Grassmannian and Hecke operators: the case of finite field 4.1. The affine Grassmannian 4.2. Satake isomorphism 4.3. Hecke operators 4.4. Langlands conjectures 5. The affine Grassmannian and Hecke operators: the case of local field 5.1. More on formal discs 5.2. Line bundles on \operatorname{Gr}_{G} 5.3. Hecke algebra over local field 5.4. Hecke operators for curves over local fields: the first approach 5.5. Hecke operators for curves over local fields: the second approach 5.6. Example 5.7. Parabolic bundles 5.8. More spaces with Hecke action 5.8.1. The map E_{\mathcal{Y},\kappa ,n} 5.8.2. Commutation with Hecke operators 5.8.3. Eigenfunctions and cuspidal functions: the idea 5.9. The case of G=\operatorname{GL}_{2} 5.9.1. The constant term in the usual case 5.10. Main question 6. The case F=\mathbb{C} 6.1. From Hecke operators to differential operators: the idea 6.2. Opers 6.3. Opers and differential operators 6.4. Differential operators and Hecke operators 6.5. Eigenvalues of Hecke operators 6.6. Parabolic bundles: results 7. The case F=\mathbb{R} 7.1. Real groups, L-groups, and all that 7.2. L-systems 7.3. Connection to Gaudin model References T. H. Colding: Evolution of form and shape 1. Introduction 2. Uniqueness of blowups in geometry 3. Regularity of singular set 4. Harmonic functions with polynomial growth 5. Ancient caloric functions with polynomial growth 6. Growth of drift equations 6.1. Growth of drift equations 7. Minimal surfaces 8. Motion by mean curvature 8.1. Gaussian surface area and entropy 8.2. Second variation and stability 8.3. Higher codimension 8.4. A new approach to dealing with the gauge group 8.5. Bounding the growth of gauge transformations 8.6. Applications 8.7. Minimal graphs and the helicoid 8.8. Multivalued graphs, spiral staircases, double spiral staircases 8.9. Structure of embedded minimal disks 8.10. Two key ideas behind the proof of the structure theorem for disks 8.11. Uniqueness theorems 9. Embedded minimal surfaces are automatically proper 9.1. Proper embeddings 9.2. The Calabi–Yau conjectures; the statements and examples References C. De Lellis: The regularity theory for the area functional (in geometric measure theory) 1. Introduction 2. Plateau's problem, criticality, and stability 2.1. Plateau's problem: two general approaches 2.2. Examples of set-theoretic approaches 2.3. Functional-analytic frameworks 2.4. Varifolds and the calculus variations ``in the large'' 3. Monotonicity formula and tangent cones 3.1. Tangent cones 4. Invariant spaces and strata 5. Interior \varepsilon-regularity at multiplicity 1 points 6. Boundary \varepsilon-regularity at multiplicity \frac{1}{2} points 7. Interior regularity theory: minimizing integral hypercurrents 8. Interior regularity theory: minimal sets 9. Interior regularity theory: stable hypersurfaces and stable hypervarifolds 10. Interior regularity theory: minimizing integral currents in higher codimension 11. Interior regularity theory: minimizing currents mod p 12. Boundary regularity theory: minimizing integral hypercurrents 13. Boundary regularity theory: minimizing integral currents with smooth boundaries of multiplicity 1 14. Boundary regularity theory: minimizing integral currents with smooth boundaries of higher multiplicity 15. Uniqueness of tangent cones 16. Open problems 16.1. Stationary and stable varifolds 16.2. Singularities of area-minimizing integral hypercurrents 16.3. Singularities of area-minimizing integral currents in codimension higher than 1 16.4. Singularities of area-minimizing currents mod p 16.5. Boundary regularity of area-minimizing integral currents at multiplicity 1 boundaries 16.6. Boundary regularity of area-minimizing integral currents at boundaries with higher multiplicity 16.7. Uniqueness of tangent cones References W. E: Mathematical machine learning 1. Introduction 2. Deep learning-based algorithms for problems in scientific computing 2.1. Control problems 2.2. High-dimensional partial differential equations 2.3. Parametrizing solutions of differential equations 2.4. Molecular dynamics 2.5. Multiscale modeling 2.6. The many-electron Schrödinger equation 2.7. Purely data-driven methods 3. Mathematical theory of neural network-based machine learning models 3.1. An overview of approximation theory 3.2. General remarks about high-dimensional problems 3.3. Approximation theory for the random feature model 3.4. Approximation theory for two-layer neural networks 3.5. Approximation theory for residual neural networks 3.6. The generalization gap 3.7. A priori estimates of the population risk for regularized models 3.8. The loss function and the loss landscape 3.9. Training dynamics 3.10. Other results 4. Machine learning from a continuous viewpoint 4.1. Integral transform-based representation 4.2. Flow-based representation 5. Some perspectives and concluding remarks References C. Gentry: Homomorphic encryption: a mathematical survey 1. Introduction 2. Some simple homomorphic encryption systems 2.1. Goldwasser–Micali: HE starting from the Legendre symbol 2.2. ElGamal: HE starting from a linear homomorphism 3. General results about homomorphic encryption 3.1. Formal definition of HE 3.2. General approach to key generation and encryption 3.3. Getting to functional completeness 3.4. Homomorphic encryption unbound: Recryption and bootstrapping 3.5. Computational hardness, cryptanalysis, and learning 4. Noisy constructions of fully homomorphic encryption 4.1. Overview of the noisy approach 4.2. Learning with noise problems 4.3. Encryption based on LWE 4.4. Bootstrappable encryption construction 4.5. Reflections on the overall FHE system 5. New directions in homomorphic encryption A. What does it mean for an encryption system to be secure? B. Hybrid argument for leveled FHE References A. Guionnet: Rare events in random matrix theory 1. Introduction 1.1. Introduction to random matrix theory 1.1.1. The Gaussian ensembles 1.1.2. Typical events 1.1.3. Fluctuations 1.1.4. Rare events 1.2. Motivations 1.2.1. Bernoulli matrices 1.2.2. The BBP transition 1.2.3. The complexity of random functions 1.2.4. Random matrices and the enumeration of maps 1.2.5. Beta-ensembles 1.2.6. Multimatrix models and the enumeration of maps 1.2.7. Multimatrix models and Voiculescu's entropy 1.3. Extensions 1.3.1. Beta-ensembles and quantum physics 1.3.2. Random tilings 1.3.3. Zeroes of random polynomials 1.3.4. Longest increasing subsequence and discrete polynuclear growth 1.3.5. Sum rules 1.3.6. Gibbs ensembles for Toda lattice 2. One matrix models 2.1. Beta-ensembles 2.2. Wigner matrices 3. Matrix models with an external field 4. Multimatrix models 4.1. Setup 4.2. Large deviations and Voiculescu's entropies 4.3. Free convolution 4.4. Multimatrix models References L. Guth: Decoupling estimates in Fourier analysis 1. Introduction 1.1. Restriction theory 1.2. Analytic number theory 1.3. Influence of the proof 1.4. Outline of the rest of the article 2. The statement of decoupling 3. Induction on scales 4. Ideas of the proof 4.1. Orthogonality 4.2. Multiple scales 4.3. Wave packets 4.4. Transversality 4.5. Induction on scales and transversality together 4.6. Final comments 5. The Kakeya conjecture 5.1. Multilinear Kakeya 6. Applications of decoupling in harmonic analysis 6.1. The helical maximal function 6.2. Pointwise convergence for the Schrödinger equation 6.3. The local smoothing problem 7. Frustrations, limitations, and open problems 7.1. Too much induction 7.2. What does decoupling say about the shapes of superlevel sets? 7.3. Limitations of the information used in the proof References S. Jitomirskaya: Quasiperiodic operators 1. Spectral theory meets (dual) dynamics 2. Aubry duality and higher-dimensional cocycles 3. Avila's global theory and classification of analytic one-frequency cocycles 4. Dual Lyapunov exponents or global theory demystified 5. Precise analysis of small denominators 6. Exact asymptotics and universal hierarchical structure of eigenfunctions 6.1. Frequency resonances 6.2. Phase resonances 6.3. Universality and extensions References I. Krichever: Abelian pole systems and Riemann–Schottky-type problems 1. Introduction 2. Riemann–Schottky problem 3. Welter's conjecture 4. The problem of characterization of Prym varieties 5. Abelian solutions of the soliton equations 6. The Baker–Akhiezer functions—general scheme 7. Key idea and steps of the proofs 8. Characterizing Jacobian of curves with involution 9. Nonlocal generating problem References A. Kuznetsov: Semiorthogonal decompositions in families 1. Introduction 2. New results in homological projective duality 2.1. Noncommutative HPD 2.2. Categorical joins 2.3. Nonlinear HPD theorem 2.4. Categorical cones and quadratic HPD 2.5. Other results 3. Residual categories 3.1. Serre compatibility and rotation functors 3.2. Mirror symmetry interpretation 3.3. The conjectures 3.4. Residual categories of homogeneous varieties 3.5. Residual categories of hypersurfaces 3.6. Residual categories of complete intersections 4. Simultaneous categorical resolutions of singularities 4.1. General results 4.2. Nodal singularities 4.3. Application to nodal degenerations of cubic fourfolds 5. Absorption of singularities 5.1. Absorption and deformation absorption 5.2. \mathbb{CP}^{\infty}-objects 5.3. Absorption of nodal singularities 5.4. Fano threefolds References S. Sheffield: What is a random surface? 1. Brownian sphere: a random metric measure space 1.1. Continuum random trees (Brownian trees) 1.2. Planar map bijections that motivate Brownian sphere and peanosphere 1.3. Unconstrained variant: when Q is both pointed and rooted 1.4. Passing to the continuum 1.5. An axiomatic approach 2. Peanosphere: a random mating of trees 2.1. Basic definition 2.2. Percolation and other simple kinds of decoration 2.3. Decorated random planar maps 2.4. Computing the scaling exponent 2.5. Schramm–Loewner evolution 2.6. Conformal loop ensembles 3. Liouville quantum gravity sphere: a random Riemannian geometry 3.1. The Gaussian free field 3.2. Conformal parameterizations 3.3. LQG surfaces 3.4. Constructing the LQG sphere 3.5. Polyakov's infinite measure on embedded LQG surfaces 3.5.1. Semi-Gaussian measures 3.5.2. Embedded Polyakov sphere 3.6. More history 3.7. Computing the scaling exponent 3.8. Random surfaces embedded in d-dimensional space 4. Conformal field theory and multipoint correlations 4.1. Gaussian case 4.2. Incorporating the Liouville term or area conditioning 5. Relationships 6. Gauge theory References K. Soundararajan: The distribution of values of zeta and L-functions 1. Values at the edge of the critical strip 2. Selberg's central limit theorem 3. Analogues of Selberg's theorem in families of L-functions 4. Moments of zeta and L-functions 5. Conjectures for the asymptotics of moments 6. Progress towards understanding the moments 7. Extreme values 8. Fyodorov–Hiary–Keating conjecture References C. Stroppel: Categorification: tangle invariants and TQFTs 1. Introduction 2. Four approaches to the Jones polynomial 3. Four approaches to categorifications 3.1. Ad I: Khovanov homology 3.2. Ad II: Triply graded link homology 3.2.1. Interlude: Hecke categories 3.3. Ad IV: Categorification of the web calculus and its tangle invariant 3.3.1. Towards 2-representation theory: categorical actions 3.3.2. Tensor product categorifications 3.4. Ad III: Categorified colored tangle invariants and projectors 4. Two proposals toward 4-TQFTs References M. Van den Bergh: Noncommutative crepant resolutions, an overview 1. Introduction 1.1. Notation and conventions 1.2. Crepant resolutions and derived equivalences 1.3. Noncommutative rings 1.4. Bridgeland's result 1.4.1. Flops 1.4.2. Maps with fibers of dimension ≤1 2. Noncommutative (crepant) resolutions 2.1. Generalities 2.2. Relation with crepant categorical resolutions 3. Constructions of noncommutative crepant resolutions 3.1. Quotient singularities 3.2. Crepant resolutions with tilting complexes 3.3. Resolutions with partial tilting complexes 3.4. Three-dimensional affine toric varieties 3.5. Mutations 4. Quotient singularities for reductive groups 4.1. NCCRs via modules of covariants 4.2. NCCRs via crepant resolutions obtained by GIT 4.3. Local systems, the SKMS, and schobers 5. NCCRs and stringy E-functions References A. Wigderson: Interactions of computational complexity theory and mathematics 1. Introduction 2. Number theory 3. Combinatorial geometry 4. Operator theory 5. Metric geometry 6. Group theory 7. Statistical physics 8. Analysis and probability 9. Lattice theory 10. Invariant theory (and more) 10.1. Geometric complexity theory 10.2. Simultaneous conjugation 10.3. Left–right action 10.4. Nullcones, moment polytopes, geodesic convexity, and noncommutative optimization 10.5. Symbolic determinants, varieties, and circuit lower bounds References List of contributors Back cover Front cover Front matter Contents 1. Logic G. Binyamini and D. Novikov: Tameness in geometry and arithmetic: beyond o-min