Grechuk Bogdan. Landscape of 21st Century Mathematics: Selected Advances, 2001-2020

Springer, 2021. — 437 p. — ISBN 978-3-030-80626-2.Landscape of 21st Century Mathematics offers a detailed cross section of contemporary mathematics. Important results of the 21st century are motivated and formulated, providing an overview of recent progress in the discipline.The theorems presented in this book have been selected among recent achievements whose statements can be fully appreciated without extensive background. Grouped by subject, the selected theorems represent all major areas of mathematics: number theory, combinatorics, analysis, algebra, geometry and topology, probability and statistics, algorithms and complexity, and logic and set theory. The presentation is self-contained with context, background and necessary definitions provided for each theorem, all without sacrificing mathematical rigour. Where feasible, brief indications of the main ideas of a proof are given.Rigorous yet accessible, this book presents an array of breathtaking recent advances in mathematics. It is written for everyone with a background in mathematics, from inquisitive university students to mathematicians curious about recent achievements in areas beyond their own.Number Theory
Gaps Between Primes
Prime Values of Polynomials
Primes in Sets of Integers
Finding Integer Solutions to Equations and Systems
Counting Solutions to Equations and Inequalities
Special Integer Sequences
Multiplicative Functions
Diophantine Approximation
Elliptic Curves
Other Topics in Number Theory
Combinatorics
Ramsey Theory on the Integers
Patterns in Sets of Integers
Patterns in Lattices
Sum-Product Phenomena
Other topics in Combinatorial Number Theory
Colouring of Graphs
The Ramsey Theory of Graphs
Expander Graphs
Other Topics in Graph Theory
Counting Problems
Topological Combinatorics
Design Theory
Analysis
Discrete Time Dynamical Systems
Continuous Time Dynamical Systems
Functions of Complex Variables and Generalizations
Properties of Polynomials
The Riemann Zeta Function and Other Special Functions
Inequalities in Analysis
Differentiability
Differential and Integral Equations
Transformations of Functions, Operators
Harmonic Analysis
Banach Spaces
The Embedding Theory of Metric Spaces
The Kakeya Conjecture
Approximation of Functions
Other Topics in Analysis
Algebra
Linear Algebra
Growth in Groups
Simple Groups
Existence of Groups with Special Properties
Other Topics in Group Theory
Rings
Fields
Geometry and Topology
Packing Problems
Combinatorial Geometry
The Isoperimetric Inequality and Beyond
Topics in Plane Geometry
Topics in Space Geometry
Knot Theory
Minimal Surfaces
General Surfaces
Topology in Higher Dimensions
Probability and Statistics
Random Graphs
Random Matrices
From Independence to Convex Sets
Discrete Time Stochastic Processes, Random Walks
Continuous Time Stochastic Processes
Bernoulli Convolutions
Percolation Theory
Compressed Sensing and Related Problems
Other Theorems Related to Probability
Algorithms and Complexity
Exact Algorithms
Approximation Algorithms and Complexity
Linear Programming and Polytopes
Numerical Algorithms
Average-Case Algorithms and Complexity
The Complexity Zoo
Decidable and Undecidable Problems
Logic and Set Theory
Squaring the Circle with and Without the Axiom of Choice
Defining Z in Q
Comparing Cardinalities of Sets
Characterization of Random Real NumbersAuthor Index
Subject Index