International Congress of Mathematicians 2022 July 6–14 Proceedings: Plenary Lectures

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