#### 1940

- Baer introduces the concept of an injective module, then begins studying group actions in geometry.
- Aleksandrov introduces exact sequences.

#### 1941

- Linnik introduces the large sieve method in number theory.
- Abraham Albert starts work on nonassociative algebras.

#### 1942

- Steenrod publishes a paper in which "Steenrod squares" are introduced for the first time.
- Eilenberg and Mac Lane publish a paper which introduces "Hom" and "Ext" for the first time.

#### 1943

#### 1944

- Von Neumann and Morgenstern publish
*Theory of Games and Economic Behaviour*. The theory of games is used in the study of economics.
- Artin studies rings with the minimum condition, now called "Artinian rings".

#### 1945

#### 1946

- Weil publishes
*Foundations of Algebraic Geometry*.

#### 1947

#### 1948

- Norbert Wiener publishes
*Cybernetics: or, Control and Communication in the Animal and the Machine*. The term "cybernetics" is due to Wiener. The book details work done on the theory of information control, particularly applied to computers.
- Shannon invents information theory and applies mathematical methods to study errors in transmitted information. This becomes of vital importance in computer science and communications.
- Schwartz publishes
*Généralisation de la notion de fonction, de dérivation, de transformation de Fourier et applications mathématiques et physiques* which is his first important publication on the theory of distributions.

#### 1949

- Mauchly and John Eckert build the Binary Automatic Computer (BINAC). One of the major advances of this machine is that data is stored on magnetic tape rather than on punched cards.
- Selberg and Erdös find an elementary proof of the prime number theorem that makes no use of complex function theory.