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1930

Van der Waerden's famous work

1930

Hurewicz proves his embedding theorem for separable metric spaces into compact spaces.

1930

Kuratowski proves his theorem on planar graphs.

1931

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.

1931

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.

1931

Von Mises introduces the idea of a sample space into probability theory.

1931

Borsuk publishes his theory of retracts in metric differential geometry.

1932

Haar introduces the "Haar measure" on groups.

1932

Hall publishes *A contribution to the theory of groups of prime power order*.

1932

Magnus proves that the word problem is true for one relator groups.

1932

Von Neumann publishes *Grundlagen der Quantenmechanik* on quantum mechanics. (See this History Topic.)

1933

Kolmogorov publishes *Foundations of the Theory of Probability* which presents an axiomatic treatment of probability.

1934

Gelfond and Schneider solve "Hilbert's Seventh problem" independently. They proved that *a*^{q} is transcendental when *a* is algebraic (≠ 0 or 1) and *q* is an irrational algebraic number.

1934

Leray shows the existence of weak solutions to the Navier-Stokes equations.

1934

Zorn establishes "Zorn's lemma" so named (probably) by Tukey. It is equivalent to the axiom of choice.

1935

Church invents "lambda calculus" which today is an invaluable tool for computer scientists.

1936

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.

1936

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.

1937

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.

1938

Kolmogorov publishes *Analytic Methods in Probability Theory* which lays the foundations of the theory of Markov random processes.

1939

Douglas gives a complete solution to the Plateau problem, proving the existence of a surface of minimal area bounded by a contour.

1939

Abraham Albert publishes *Structure of Algebras*.

1940

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

1940

Aleksandrov introduces exact sequences.

JOC/EFR May 2015
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