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1900

Hilbert poses 23 problems at the Second International Congress of Mathematicians in Paris as a challenge for the 20th century. The problems include the continuum hypothesis, the well ordering of the real numbers, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of "Dirichlet's principle" and many more. Many of the problems were solved during the 20th century, and each time one of the problems was solved it was a major event for mathematics.

1900

Goursat begins publication of *Cours d'analyse mathematique* which introduces many new analysis concepts.

1900

Fredholm develops his theory of integral equations in *Sur une nouvelle méthode pour la résolution du problème de Dirichlet*.

1900

Fejér publishes a fundamental summation theorem for Fourier series.

1900

Levi-Civita and Ricci-Curbastro publish *Méthodes de calcul differential absolu et leures applications* in which they set up the theory of tensors in the form that will be used in the general theory of relativity 15 years later.

1901

Russell discovers "Russell's paradox" which illustrates in a simple fashion the problems inherent in naive set theory.

1901

Planck proposes quantum theory. (See this History Topic.)

1901

The Runge-Kutta method for numerically solving ordinary differential equations is proposed.

1901

Lebesgue formulates the theory of measure.

1901

Dickson publishes *Linear groups with an exposition of the Galois field theory*.

1902

Lebesgue gives the definition of the "Lebesgue integral".

1902

Beppo Levi states the axiom of choice for the first time.

1902

Gibbs publishes *Elementary Principles of Statistical Mechanics* which is a beautiful account putting the foundations of statistical mechanics on a firm foundation.

1903

Castelnuovo publishes *Geometria analitica e proiettiva* his most important work in algebraic geometry.

1904

Zermelo uses the axiom of choice to prove that every set can be well ordered.

1904

Lorentz introduces the "Lorentz transformations". (See this History Topic.)

1904

Poincaré proposes the Poincaré Conjecture, namely that any closed 3-dimensional manifold which is homotopy equivalent to the 3-sphere must be the 3-sphere.

1904

Poincaré gives a lecture in which he proposes a theory of relativity to explain the "Michelson and Morley experiment". (See this History Topic.)

1905

Einstein publishes the special theory of relativity. (See this History Topic.)

1905

Lasker proves the decomposition theorem for ideals into primary ideals in a polynomial ring.

1906

Fréchet, in his dissertation, investigated functionals on a metric space and formulated the abstract notion of compactness.

1906

Markov studies random processes that are subsequently known as "Markov chains".

1906

Bateman applies Laplace transforms to integral equations.

1906

Koch publishes *Une methode geometrique elementaire pour l'etude de certaines questions de la theorie des courbes plane* which contains the "Koch curve". It is a continuous curve which is of infinite length and nowhere differentiable.

1907

Fréchet discovers an integral representation theorem for functionals on the space of "quadratic Lebesgue integrable functions". A similar result was discovered independently by Riesz.

1907

Einstein publishes his principle of equivalence, in which says that gravitational acceleration is indistinguishable from acceleration caused by mechanical forces. It is a key ingredient of general relativity. (See this History Topic.)

1907

Heegaard and Dehn publish *Analysis Situs* which marks the beginnings of combinatorial topology.

1907

Brouwer's doctoral thesis on the foundations of mathematics attacked the logical foundations of mathematics and marks the beginning of the Intuitionist School.

1907

Dehn formulates the word problem and the isomorphism problem for group presentations.

1907

Riesz proves the theorem now called the "Riesz-Fischer theorem" concerning Fourier analysis on Hilbert space.

1908

Gosset introduces "Student's *t*-test" to handle small samples.

1908

Hardy and Weinberg present a law describing how the proportions of dominant and recessive genetic traits would be propagated in a large population. This establishes the mathematical basis for population genetics.

1908

Zermelo publishes *Untersuchungen über die Grundlagen der Mengenlehre* (*Investigations on the Foundations of Set Theory*). He bases set theory on seven axioms : Axiom of extensionality, Axiom of elementary sets, Axiom of separation, Power set axiom, Union axiom, Axiom of choice and Axiom of infinity. This aims to overcome the difficulties with set theory encountered by Cantor.

1908

Poincaré publishes *Science et méthode* (*Science and Method*), perhaps his most famous popular work.

1909

Carmichael investigates pseudoprimes.

1909

Edmund Landau gives the first systematic presentation of analytic number theory.

1910

Russell and Whitehead publish the first volume of *Principia Mathematica*. They attempt to put the whole of mathematics on a logical foundation. They were able to provide detailed derivations of many major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory. The third and final volume will appear three years later, while a fourth volume on geometry was planned but never completed.

1910

Steinitz gives the first abstract definition of a field in *Algebraische Theorie der Körper*.

JOC/EFR May 2015
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