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1940

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

1940

Aleksandrov introduces exact sequences.

1941

Linnik introduces the large sieve method in number theory.

1941

Abraham Albert starts work on nonassociative algebras.

1942

Steenrod publishes a paper in which "Steenrod squares" are introduced for the first time.

1942

Eilenberg and Mac Lane publish a paper which introduces "Hom" and "Ext" for the first time.

1943

Marshall Hall publishes on projective planes.

1943

Naimark proves the "Gelfand-Naimark theorem" on self-adjoint algebras of operators in Hilbert space.

1944

Von Neumann and Morgenstern publish *Theory of Games and Economic Behaviour*. The theory of games is used in the study of economics.

1944

Artin studies rings with the minimum condition, now called "Artinian rings".

1945

Eilenberg and Mac Lane introduce the terms "category" and "natural transformation".

1946

Weil publishes *Foundations of Algebraic Geometry*.

1947

George Dantzig introduces the simplex method of optimisation.

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.

1948

Shannon invents information theory and applies mathematical methods to study errors in transmitted information. This becomes of vital importance in computer science and communications.

1948

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.

1949

Selberg and Erdös find an elementary proof of the prime number theorem that makes no use of complex function theory.

1950

Carnap publishes *Logical Foundations of Probability*.

1950

Hamming publishes a fundamental paper on error-detecting and error-correcting codes.

1950

Hodge puts forward the "Hodge Conjecture" on projective algebraic varieties.

List of mathematicians alive in 1940.

List of mathematicians alive in 1950.

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