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St Petersburg Mathematical Society is founded.
Heawood publishes Map colour theorems in which he points out the error in Kempe's proof of the Four Colour Theorem. He proves that five colours suffice. (See this History Topic.)
Fedorov and Schönflies independently classify crystallographic space groups showing that there are 230 of them.
Poincaré publishes the first of three volumes of Les Méthodes nouvelles de la mécanique céleste (New Methods in Celestial Mechanics). He aims to completely characterise all motions of mechanical systems, invoking an analogy with fluid flow. He also shows that series expansions previously used in studying the three-body problem, for example by Delaunay, were convergent, but not in general uniformly convergent. This puts in doubt the stability proofs of the solar system given by Lagrange and Laplace.
Pearson publishes the first in a series of 18 papers, written over the next 18 years, which introduce a number of fundamental concepts to the study of statistics. These papers contain contributions to regression analysis, the correlation coefficient and includes the chi-square test of statistical significance.
Poincaré begins work on algebraic topology.
Borel introduces "Borel measure".
Cartan, in his doctoral dissertation, classifies all finite dimensional simple Lie algebras over the complex numbers.
Poincaré publishes Analysis situs his first work on topology which gives an early systematic treatment of the topic. He is the originator of algebraic topology publishing six papers on the topic. He introduces fundamental groups.
Cantor publishes the first of two major surveys on transfinite arithmetic.
Heinrich Weber publishes his famous text Lehrbuch der Algebra (Lectures on Algebra).
The prime number theorem is proved independently by Hadamard and de la Vallée-Poussin. This theorem gives an estimate of the number of primes there are up to a given number, showing that the number of primes less than n tends to infinity as n/log n.
Cesàro publishes Lezione di geometria intrinseca in which he formulates intrinsic geometry.
Frobenius introduces group characters.
Hensel invents the p-adic numbers.
Burali-Forti is the first to discover of a set theory paradox.
Burnside publishes The Theory of Groups of Finite Order.
Frobenius begins the study of the representation theory of groups.
Frobenius introduces the notion of induced representations and the "Frobenius Reciprocity Theorem".
Hadamard's work on geodesics on surfaces of negative curvature lays the foundations of symbolic dynamics.
Hilbert publishes Grundlagen der Geometrie (Foundations of Geometry) putting geometry in a formal axiomatic setting.
Lyapunov devises methods which provide ways of determining the stability of sets of ordinary differential equations.
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.
Goursat begins publication of Cours d'analyse mathematique which introduces many new analysis concepts.
Fredholm develops his theory of integral equations in Sur une nouvelle méthode pour la résolution du problème de Dirichlet.
Fejér publishes a fundamental summation theorem for Fourier series.
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.
List of mathematicians alive in 1900.
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JOC/EFR August 2001
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