Timeline of Mathematics - 20th Century

20th Century

• 1901 – Élie Cartan develops the exterior derivative,
• 1903 – Carle David Tolme Runge presents a fast Fourier Transform algorithm,
• 1903 – Edmund Georg Hermann Landau gives considerably simpler proof of the prime number theorem.
• 1908 – Ernst Zermelo axiomizes set theory, thus avoiding Cantor's contradictions,
• 1908 – Josip Plemelj solves the Riemann problem about the existence of a differential equation with a given monodromic group and uses Sokhotsky – Plemelj formulae,
• 1912 – Luitzen Egbertus Jan Brouwer presents the Brouwer fixed-point theorem,
• 1912 – Josip Plemelj publishes simplified proof for the Fermat's Last Theorem for exponent n = 5,
• 1919 – Viggo Brun defines Brun's constant B₂ for twin primes,
• 1928 – John von Neumann begins devising the principles of game theory and proves the minimax theorem,
• 1930 – Casimir Kuratowski shows that the three-cottage problem has no solution,
• 1931 – Kurt Gödel proves his incompleteness theorem which shows that every axiomatic system for mathematics is either incomplete or inconsistent,
• 1931 – Georges de Rham develops theorems in cohomology and characteristic classes,
• 1933 – Karol Borsuk and Stanislaw Ulam present the Borsuk-Ulam antipodal-point theorem,
• 1933 – Andrey Nikolaevich Kolmogorov publishes his book Basic notions of the calculus of probability (Grundbegriffe der Wahrscheinlichkeitsrechnung) which contains an axiomatization of probability based on measure theory,
• 1940 – Kurt Gödel shows that neither the continuum hypothesis nor the axiom of choice can be disproven from the standard axioms of set theory,
• 1942 – G.C. Danielson and Cornelius Lanczos develop a Fast Fourier Transform algorithm,
• 1943 – Kenneth Levenberg proposes a method for nonlinear least squares fitting,
• 1945 – Stephen Cole Kleene introduces realizability,
• 1945 - Saunders Mac Lane and Samuel Eilenberg start category theory
• 1945 - Norman Steenrod and Samuel Eilenberg give the Eilenberg–Steenrod axioms for (co-)homology
• 1948 – John von Neumann mathematically studies self-reproducing machines,
• 1949 – John von Neumann computes π to 2,037 decimal places using ENIAC,
• 1950 – Stanislaw Ulam and John von Neumann present cellular automata dynamical systems,
• 1953 – Nicholas Metropolis introduces the idea of thermodynamic simulated annealing algorithms,
• 1955 – H. S. M. Coxeter et al. publish the complete list of uniform polyhedron,
• 1955 – Enrico Fermi, John Pasta, and Stanislaw Ulam numerically study a nonlinear spring model of heat conduction and discover solitary wave type behavior,
• 1956 – Noam Chomsky describes an hierarchy of formal languages,
• 1958 - Alexander Grothendieck's proof of the Grothendieck-Riemann-Roch theorem is published
• 1960 – C. A. R. Hoare invents the quicksort algorithm,
• 1960 – Irving S. Reed and Gustave Solomon present the Reed-Solomon error-correcting code,
• 1961 – Daniel Shanks and John Wrench compute π to 100,000 decimal places using an inverse-tangent identity and an IBM-7090 computer,
• 1962 – Donald Marquardt proposes the Levenberg–Marquardt nonlinear least squares fitting algorithm,
• 1963 – Paul Cohen uses his technique of forcing to show that neither the continuum hypothesis nor the axiom of choice can be proven from the standard axioms of set theory,
• 1963 – Martin Kruskal and Norman Zabusky analytically study the Fermi-Pasta-Ulam heat conduction problem in the continuum limit and find that the KdV equation governs this system,
• 1963 – meteorologist and mathematician Edward Norton Lorenz published solutions for a simplified mathematical model of atmospheric turbulence – generally known as chaotic behaviour and strange attractors or Lorenz Attractor – also the Butterfly Effect,
• 1965 – Iranian mathematician Lotfi Asker Zadeh founded fuzzy set theory as an extension of the classical notion of set and he founded the field of Fuzzy Mathematics,
• 1965 – Martin Kruskal and Norman Zabusky numerically study colliding solitary waves in plasmas and find that they do not disperse after collisions,
• 1965 – James Cooley and John Tukey present an influential Fast Fourier Transform algorithm,
• 1966 – E.J. Putzer presents two methods for computing the exponential of a matrix in terms of a polynomial in that matrix,
• 1966 – Abraham Robinson presents Non-standard analysis.
• 1967 – Robert Langlands formulates the influential Langlands program of conjectures relating number theory and representation theory,
• 1968 – Michael Atiyah and Isadore Singer prove the Atiyah-Singer index theorem about the index of elliptic operators,
• 1973 – Lotfi Zadeh founded the field of fuzzy logic,
• 1975 – Benoît Mandelbrot publishes Les objets fractals, forme, hasard et dimension,
• 1976 – Kenneth Appel and Wolfgang Haken use a computer to prove the Four color theorem,
• 1981 – Richard Feynman gives an influential talk "Simulating Physics with Computers" (in 1980 Yuri Manin proposed the same idea about quantum computations in "Computable and Uncomputable" (in Russian)),
• 1983 – Gerd Faltings proves the Mordell conjecture and thereby shows that there are only finitely many whole number solutions for each exponent of Fermat's Last Theorem,
• 1983 – the classification of finite simple groups, a collaborative work involving some hundred mathematicians and spanning thirty years, is completed,
• 1985 – Louis de Branges de Bourcia proves the Bieberbach conjecture,
• 1987 – Yasumasa Kanada, David Bailey, Jonathan Borwein, and Peter Borwein use iterative modular equation approximations to elliptic integrals and a NEC SX-2 supercomputer to compute π to 134 million decimal places,
• 1991 – Alain Connes and John W. Lott develop non-commutative geometry,
• 1992 – David Deutsch and Richard Jozsa develop the Deutsch–Jozsa algorithm, one of the first examples of a quantum algorithm that is exponentially faster than any possible deterministic classical algorithm.
• 1994 – Andrew Wiles proves part of the Taniyama–Shimura conjecture and thereby proves Fermat's Last Theorem,
• 1994 – Peter Shor formulates Shor's algorithm, a quantum algorithm for integer factorization,
• 1998 – Thomas Callister Hales (almost certainly) proves the Kepler conjecture,
• 1999 – the full Taniyama–Shimura conjecture is proved,
• 2000 – the Clay Mathematics Institute proposes the seven Millennium Prize Problems of unsolved important classic mathematical questions.