Landscape of 21st Century Mathematics: Selected Advances, 2001–2020

Preface
Acknowledgements
Contents
Acronyms
1 Number Theory
	1.1 Gaps Between Primes
	1.2 Prime Values of Polynomials
	1.3 Primes in Sets of Integers
	1.4 Finding Integer Solutions to Equations and Systems
	1.5 Counting Solutions to Equations and Inequalities
	1.6 Special Integer Sequences
	1.7 Multiplicative Functions
	1.8 Diophantine Approximation
	1.9 Elliptic Curves
	1.10 Other Topics in Number Theory
2 Combinatorics
	2.1 Ramsey Theory on the Integers
	2.2 Patterns in Sets of Integers
	2.3 Patterns in Lattices
	2.4 Sum-Product Phenomena
	2.5 Other topics in Combinatorial Number Theory
	2.6 Colouring of Graphs
	2.7 The Ramsey Theory of Graphs
	2.8 Expander Graphs
	2.9 Other Topics in Graph Theory
	2.10 Counting Problems
	2.11 Topological Combinatorics
	2.12 Design Theory
3 Analysis
	3.1 Discrete Time Dynamical Systems
	3.2 Continuous Time Dynamical Systems
	3.3 Functions of Complex Variables and Generalizations
	3.4 Properties of Polynomials
	3.5 The Riemann Zeta Function and Other Special Functions
	3.6 Inequalities in Analysis
	3.7 Differentiability
	3.8 Differential and Integral Equations
	3.9 Transformations of Functions, Operators
	3.10 Harmonic Analysis
	3.11 Banach Spaces
	3.12 The Embedding Theory of Metric Spaces
	3.13 The Kakeya Conjecture
	3.14 Approximation of Functions
	3.15 Other Topics in Analysis
4 Algebra
	4.1 Linear Algebra
	4.2 Growth in Groups
	4.3 Simple Groups
	4.4 Existence of Groups with Special Properties
	4.5 Other Topics in Group Theory
	4.6 Rings
	4.7 Fields
5 Geometry and Topology
	5.1 Packing Problems
	5.2 Combinatorial Geometry
	5.3 The Isoperimetric Inequality and Beyond
	5.4 Topics in Plane Geometry
	5.5 Topics in Space Geometry
	5.6 Knot Theory
	5.7 Minimal Surfaces
	5.8 General Surfaces
	5.9 Topology in Higher Dimensions
6 Probability and Statistics
	6.1 Random Graphs
	6.2 Random Matrices
	6.3 From Independence to Convex Sets
	6.4 Discrete Time Stochastic Processes, Random Walks
	6.5 Continuous Time Stochastic Processes
	6.6 Bernoulli Convolutions
	6.7 Percolation Theory
	6.8 Compressed Sensing and Related Problems
	6.9 Other Theorems Related to Probability
7 Algorithms and Complexity
	7.1 Exact Algorithms
	7.2 Approximation Algorithms and Complexity
	7.3 Linear Programming and Polytopes
	7.4 Numerical Algorithms
	7.5 Average-Case Algorithms and Complexity
	7.6 The Complexity Zoo
	7.7 Decidable and Undecidable Problems
8 Logic and Set Theory
	8.1 Squaring the Circle with and Without the Axiom of Choice
	8.2 Defining Z in Q
	8.3 Comparing Cardinalities of Sets
	8.4 Characterization of Random Real Numbers
References
Author Index
Subject Index