The Abel Prize 2018-2022

Preface
Contents
Part I 2018 Robert P. Langlands
	Citation
	Autobiography
	The work of Robert Langlands
		Foreword
		Contents
		1 Group representations and harmonic analysis
		2 Eisenstein series
		3 L-functions and class field theory
		4 Global Functoriality and its implications
		5 Local Functoriality and early results
		6 Trace formula and first comparison
		7 Base change
		8 Shimura varieties
		9 Motives and Reciprocity
		10 The theory of endoscopy
		11 Beyond Endoscopy
		References
	List of Publications for Robert P. Langlands
	Curriculum Vitae for Robert Phelan Langlands
Part II 2019 Karen K. Uhlenbeck
	Citation
		Minimal surfaces and bubbling analysis
		Gauge theory and Yang–Mills equations
		Integrable systems and harmonic mappings
	Mathematical Meanderings
		Childhood and Education
		Midcareer
		Texas and Beyond
	A journey through the mathematical world of Karen Uhlenbeck
		Contents
		1 Introduction
		2 Nonlinear systems and p-harmonic functions
			2.1 A regularity theorem
			2.2 A differential inequality
			2.3 Outline of proof of Theorem 2.1
		3 Harmonic maps of surfaces
			3.1 Background
			3.2 Bubbling
			3.3 Small energy
			3.4 The stress energy tensor and removal of point singularities
		4 Harmonic maps in higher dimensions
			4.1 Monotonicity of normalised energy
			4.2 Minimising maps
			4.3 Small energy
			4.4 Some further developments
		5 Gauge Theory
			5.1 Background
			5.2 The 1982 papers in Commun. Math. Phys.
			5.3 Applications
		6 The Yang–Mills equations in higher dimensions
			6.1 Hermitian Yang–Mills connections on stable bundles
			6.2 Connections with small normalised energy
			6.3 Removal of codimension 4 singularities
		7 Harmonic maps to Lie groups
			7.1 Harmonic maps, flat connections and loop groups
			7.2 Uniton addition and instantons on R
			7.3 Weak solutions to the harmonic map equation on surfaces
		References
	List of Publications for Karen K. Uhlenbeck
	Curriculum Vitae for Karen Keskulla Uhlenbeck
Part III 2020 Hillel Furstenberg and Grigoriy Margulis
	Citation
	Autobiography
	Autobiography
		Appendix 1. Arithmeticity and superrigidity
			Arithmeticity of non-uniform lattices
			Arithmeticity of uniform lattices
		Appendix 2. Homogeneous dynamics and number theory/diophantine approximation
			Distribution of values of indefinite quadratic forms at integral points
			Diophantine approximation on manifolds
		Appendix 3. Fields Medal
		Appendix 4. Dissertations
	The work of Hillel Furstenberg and its impact on modern mathematics
		Contents
		Introduction
		1 Topological dynamics
		2 Stationary dynamical systems and the Poisson–Furstenberg boundary
		3 Probability, ergodic theory and fractal geometry
		4 Multiple recurrence and applications to combinatorics and number theory
		References
	The work of G. A. Margulis
		Acknowledgments
		Contents
		1 General introduction
		2 Arithmeticity and superrigidity
			2.1 Proof of superrigidity
			2.2 Proof of arithmeticity
		3 Normal subgroup theorem
		4 Expanders, relative property (T) and lattices
		5 Local rigidity of group actions
		6 Dynamical systems on homogeneous spaces: an introduction
		7 Quantitative non-divergence
			7.1 An elementary non-divergence result
			7.2 The general case
			7.3 Applications to Diophantine approximation on manifolds
		8 Conjectures of Oppenheim and Raghunathan
		9 Linearization
			9.1 Non-ergodic measures invariant under a unipotent
			9.2 The theorem of Dani–Margulis on uniform convergence
		10 Partially hyperbolic flows and Diophantine approximation
			10.1 Exceptional trajectories
			10.2 Cusp excursions
			10.3 Effective equidistribution
		11 A quantitative version of the Oppenheim Conjecture
			11.1 Passage to the space of lattices
			11.2 Margulis functions
			11.3 A system of inequalities
			11.4 Averages over large spheres
			11.5 Signatures (2,1) and (2,2)
		12 Effective estimates
			12.1 Periodic orbits of semisimple groups
			12.2 Effective solution of the Oppenheim Conjecture
			12.3 Power law estimates in dimension at least
		References
	List of Publications for Hillel Furstenberg
	List of Publications for Grigoriy Margulis
	Curriculum Vitae for Hillel Furstenberg
	Curriculum Vitae for Grigoriy Aleksandrovich Margulis
Part IV 2021 László Lovász and Avi Wigderson
	Citation
	Autobiography, mostly mathematical
		Meeting with mathematics
		Moving around in Hungary and in the world
		Tarski’s problem and graph limits
		Perfect graphs and combinatorial optimization
		Algorithmic geometry and cryptography
	Avi Wigderson — a short biography
	The Mathematics of László Lovász
		Contents
		1 Introduction
		2 Logic and Universal Algebra – Homomorphisms and Tarski’s Problem
		3 Coloring Graphs Constructively (on a Way to Expanders)
		4 The Lovász Local Lemma
		5 Coloring Graphs via Topology
		6 Geometric Graphs and Exterior Algebra
		7 Perfect Graphs and Computational Complexity
		8 The Shannon Capacity of a Graph and Orthogonal Representations
		9 The Ellipsoid Method
		10 Oracle-Polynomial Time Algorithms and Convex Bodies
		11 Polyhedra, Low Dimensionality, and the LLL Algorithm
		12 The LLL Algorithm and its Consequences
		13 Cutting Planes and the Solution of Practical Applications
		14 Computing Optimal Stable Sets and Colorings in Perfect Graphs
		15 Submodular Functions
		17 Analysis, Algebra, and Graph Limits
		18 Final Remarks
		References
	On the works of Avi Wigderson
		Contents
		1 Introduction
		2 Cryptography
			2.1 Cryptography under computational assumptions
				2.1.1 Zero knowledge proofs for all languages in NP
				2.1.2 Computationally secure multiparty computation
			2.2 Information-Theoretic Cryptography
				2.2.1 Multi-Prover Zero-Knowledge Interactive Proofs
				2.2.2 Bit commitment scheme in the 2-prover setting
			2.3 The Importance of the Multi-Prover Interactive Proof Model
				2.3.1 Information-Theoretic Secure Multi-Party Computation
				2.3.2 Verifiable Secret Sharing
					Gate-by-Gate Emulation in the Malicious Setting
		3 Pseudorandomness
			3.1 Hardness vs. Randomness
				3.1.1 Motivation
				3.1.2 Wigderson’s Contributions
				3.1.3 Pseudorandom Generators
				3.1.4 The Nisan–Wigderson Generator
				3.1.5 Pseudorandom Generators from Worst-Case Lower Bounds
			3.2 Expanders, Extractors, and Ramsey Graphs
				3.2.1 Expander Graphs
				3.2.2 Randomness Extractors
				3.2.3 Multi-source Extractors and Ramsey Graphs
			3.3 Unconditional derandomization
				3.3.1 Undirected S-T Connectivity
				3.3.2 General Space-Bounded Computation
				3.3.3 Constant-depth Circuits and Iterated Restrictions
		4 Computational Complexity Lower Bounds
			4.1 Boolean Circuit Complexity
			4.2 Communication Complexity
			4.3 Karchmer–Wigderson Games
			4.4 Lower Bounds for the Monotone Depth of ST-Connectivity
			4.5 Lower Bounds for the Monotone Depth of Clique and Matching
			4.6 The KRW Conjecture
			4.7 Communication Complexity of Set-Disjointness
			4.8 Quantum versus Classical Communication Complexity
			4.9 Partial Derivatives in Arithmetic Circuit Complexity
			4.10 Resolution Made Simple
		5 Complexity, Optimization, and Symmetries
			5.1 Permanent and matrix scaling
				5.1.1 Doubly stochastic matrices and their permanents
				5.1.2 Matrix scaling
			5.2 Noncommutative singularity testing and operator scaling
				5.2.1 Completely positive operator and its capacity
				5.2.2 Operator scaling
				5.2.3 Noncommutative singularity and identity testing
				5.2.4 Brascamp–Lieb constants
				5.2.5 Polynomial capacity
			5.3 Capacity and geodesic convex optimization
				5.3.1 The Riemannian geometry of positive definite matrices and geodesic convexity
				5.3.2 Geodesic convexity of capacity
				5.3.3 Computing the capacity via geodesically convex optimization
			5.4 The null-cone problem, invariant theory, and noncommutative optimization
				5.4.1 Groups, orbits, and invariants
				5.4.2 Capacity and the null cone
				5.4.3 Geodesic convexity, moment map, and noncommutative duality
				5.4.4 Noncommutative optimization under symmetries
		References
	List of Publications for László Lovász
	List of Publications for Avi Wigderson
	Curriculum Vitae for László Lovász
	Curriculum Vitae for Avi Wigderson
Part V 2022 Dennis P. Sullivan
	Citation
	Encounters with Geometry—an Autobiography of Concepts
	Dennis Sullivan’s Work on Dynamics
		Contents
		1 Smooth dynamics
			1.1 From topology to dynamics
			1.2 Rigidity in smooth dynamics
			1.3 Further results
		2 Dynamics and ergodic theory of Kleinian groups
			The Moebius group
			Hyperbolic space
			Kleinian groups
			Hyperbolic manifolds
			Quasi-conformal homeomorphisms
			2.1 Sullivan’s rigidity theorem
			2.2 Conformal densities and Patterson–Sullivan measures
				Conformal densities
				Patterson–Sullivan measures: construction
			2.3 Further results
		3 Holomorphic dynamics
			3.1 The reemergence of holomorphic dynamics in Paris in the 1980s
			3.2 Conformal measures for rational maps
			3.3 The λ-Lemma
			3.4 Density of stable maps
			3.5 Towards the Fatou conjecture: absence of line fields
			3.6 Monotonicity of entropy and the pullback argument
			3.7 Renormalisation theory for interval maps
			3.8 Real and complex bounds
			3.9 Riemann surface laminations and the non-coiling lemma
			3.10 Renormalisation theory for circle maps
			3.11 The Fatou conjecture in the real setting
			3.12 Sullivan’s quasisymmetry rigidity programme
		4 Sullivan’s dictionary
		5 Final words
		References
	Sullivan’s Juvenilia: Surgery and Algebraic Topology
		Contents
		1 Surgery and its classifying spaces
			1.1 Further developments
				Localization
				Bordism and classifying spaces
				KO[1/2] orientation
				Hauptvermutung and triangulation (and beyond)
				Surgery revisited
		2 The Adams Conjecture
			2.1 Background
			2.2 The Statement of the Adams conjecture
			2.3 Comments on Sullivan’s proof
			2.4 Some of its aftermath
			Epilogue: A return to geometry
		3 Rational homotopy theory
			3.1 Sullivan’s model
			3.2 A few words on the proof
			3.3 A few more applications
				3.3.1 Free loop spaces
				3.3.2 The Elliptic/Hyperbolic dichotomy
				3.3.3 Quantitative Homotopy Theory
				3.3.4 Finite primes and Z
		References
	List of Publications for Dennis P. Sullivan
	Curriculum Vitae for Dennis Parnell Sullivan
Part VI Abel Activities 2018–2022
	Photos
	The Abel Committee
	The Niels Henrik Abel Board
	The Abel Lectures
	The Abel Laureate Presenters
	The Interviews with the Abel Laureates
	The Abel Banquet 2003–2022
	Addenda, Errata, and Updates1