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Hurewicz proves his embedding theorem for separable metric spaces into compact spaces.
Kuratowski proves his theorem on planar graphs.
G D Birkhoff proves the general ergodic theorem. This will transform the Maxwell-Boltzmann kinetic theory of gases into a rigorous principle through the use of Lebesgue measure.
Gödel publishes Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme (On Formally Undecidable Propositions in Principia Mathematica and Related Systems). He proves fundamental results about axiomatic systems showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system. In particular the consistency of the axioms cannot be proved.
Von Mises introduces the idea of a sample space into probability theory.
Borsuk publishes his theory of retracts in metric differential geometry.
Haar introduces the "Haar measure" on groups.
Hall publishes A contribution to the theory of groups of prime power order.
Magnus proves that the word problem is true for one relator groups.
Von Neumann publishes Grundlagen der Quantenmechanik on quantum mechanics. (See this History Topic.)
Kolmogorov publishes Foundations of the Theory of Probability which presents an axiomatic treatment of probability.
Gelfond and Schneider solve "Hilbert's Seventh problem" independently. They proved that aq is transcendental when a is algebraic (≠ 0 or 1) and q is an irrational algebraic number.
Leray shows the existence of weak solutions to the Navier-Stokes equations.
Zorn establishes "Zorn's lemma" so named (probably) by Tukey. It is equivalent to the axiom of choice.
Church invents "lambda calculus" which today is an invaluable tool for computer scientists.
Turing publishes On Computable Numbers, with an application to the Entscheidungsproblem which describes a theoretical machine, now known as the "Turing machine". It becomes a major ingredient in the theory of computability.
Church publishes An unsolvable problem in elementary number theory. "Church's Theorem", which shows there is no decision procedure for arithmetic, is contained in this work.
Vinogradov publishes Some theorems concerning the theory of prime numbers in which he proves that every sufficiently large odd integer can be expressed as the sum of three primes. This is a major contribution to the solution of the Goldbach conjecture.
Kolmogorov publishes Analytic Methods in Probability Theory which lays the foundations of the theory of Markov random processes.
Douglas gives a complete solution to the Plateau problem, proving the existence of a surface of minimal area bounded by a contour.
Abraham Albert publishes Structure of Algebras.
Baer introduces the concept of an injective module, then begins studying group actions in geometry.
Aleksandrov introduces exact sequences.
List of mathematicians alive in 1940.
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University of St Andrews, Scotland