# Chronology

### 1920 - 1930

#### 1920

- Takagi publishes his fundamental paper on class field theory.
- Hasse discovers the "local-global" principle.
- Siegel's dissertation is important in the theory of Diophantine approximations.
*Fundamenta Mathematica*is founded by Sierpinski and Mazurkiewicz.

#### 1921

- Keynes publishes his
*Treatise on Probability*in which he argues that probability is a logical relation and so it is objective. A statement involving probability relations has a truth-value independent of people's opinions. This is to have a profound effect on statistics as well as economics. - Fisher introduces the concept of likelihood into statistics.
- Borel publishes the first in a series of papers on game theory and becomes the first to define games of strategy.
- Emmy Noether publishes
*Idealtheorie in Ringbereichen*which is of fundamental importance in the development of modern abstract algebra.

#### 1922

- Richardson publishes
*Weather Prediction by Numerical Process*. He is the first to apply mathematics, in particular the method of finite differences, to predicting the weather. The calculations are prohibitive by hand calculation and only the development of computers will make his idea a reality. - Banach is awarded his habilitation for a thesis on measure theory. He begins his work on a development of normed vector spaces.
- Fraenkel attempts to put set theory into an axiomatic setting.
- Chebotaryov proves the density theorem on primes in an arithmetic progression.
- Fejér and Riesz publish an important work on conformal mappings.
- Kolmogorov constructs a summable function which diverges almost everywhere.

#### 1923

- Study publishes important work on real and complex algebras of low dimension.

#### 1924

- Alexander introduces the now famous "Alexander horned sphere".

#### 1925

- Fisher publishes
*Statistical Methods for Research Workers*. He gives experimental and statistical methods which can be used in biology. - Whitehead publishes
*Science and the Modern World*. It results from a series of lectures given in the United States and serves as an introduction to his later metaphysics. He considers the growth, success, and impact of "scientific materialism" which is the notion that nature is merely matter and energy. - Besicovitch solves "Kakeya's problem" on minimising areas.
- Krull proves the "Krull-Schmidt theorem" for decomposing abelian groups of operators.

#### 1926

- Reidemeister publishes an important book on knot theory:
*Knoten und gruppen*. - Artin and Schreier publish a paper on ordering formally real fields and real closed fields.
- Banach and Tarski publish the "Banach-Tarski paradox" in a joint paper in
*Fundamenta Mathematicae*:*Sur la decomposition des ensembles de points en parties respectivement congruentes*.

#### 1927

- Emmy Noether, Helmut Hasse and Richard Brauer work on non-commutative algebras.
- Artin publishes his reciprocity law in
*Beweis des allgemeinen Reziprozitätsgesetzes*.

#### 1928

- Von Mises publishes
*Probability, Statistics and Truth*. - Von Neumann proves the minimax theorem in game theory.
- Hopf introduces homology groups.

#### 1929

- Gelfond makes his Conjecture about the linear independence of algebraic numbers over the rational numbers.