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

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
Computing Julia Sets
Information Complexity
Mechanism Design
Problem-Solving Prowess and Theoretical Insight
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
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