Skip to content
Field Atlas

Atlas / Mathematics

Domain · 9 threads · 49 fields

Mathematics

How space split into many geometries and whole numbers into theories of their own, how calculus was made rigorous and mathematics met its own limits, and how solving equations became the study of symmetry. How counting puzzles became the mathematics of networks and equations of motion led to chaos, how reasoning from data and calculating by machine became mathematics of their own, and what forced each split.

Thread 01 · 9 fields

The Geometry Thread

From Euclid's axioms to the shape of three-dimensional space. One awkward postulate split geometry in two. The two halves met again in Riemann's lecture of 1854. What grew from that meeting eventually settled Poincaré's question about the shape of space, and it still runs into fog in dimension four. A second branch runs through the painter's perspective to the geometry of polynomial equations, and on to Fermat's Last Theorem.

Scroll the map sideways →

Field tree: Euclidean Geometry, Projective Geometry, Differential Geometry of Surfaces, Non-Euclidean Geometry, Algebraic Topology, Riemannian Geometry, Algebraic Geometry, Geometric Topology, Geometric Analysis. Each field links to its page.Euclidean GeometryEMERGED C. 300 BCEc. 300 BCE · REFORMULATION: The Elements puts geometry in axiomatic orderC. 300 BCE REFORMULATIONc. 150 – 1733 · CRISIS: Centuries of failed proofs of the fifth postulateC. 150 – 1733 CRISIS1637 · REFORMULATION · contested: Coordinates turn geometry into algebra1637 REFORMULATION1837 · DISPROOF: Straightedge and compass cannot double the cube or trisect the angle1837 DISPROOF1899 · REFORMULATION: Hilbert rebuilds the foundations1899 REFORMULATIONContinues into Calculus, drawn on The Analysis Thread↘ INTO CALCULUS · THE ANALYSIS THREADContinues into Elementary Number Theory, drawn on The Number Theory Thread↘ INTO ELEMENTARY NUMBER THEORY · THE NUMBER THEORY THREAD1 unresolved problemProjective GeometryEMERGED 1630S – 1820Sc. 1415 – 1435 · REFORMULATION · contested: Painters find the geometry of perspectiveC. 1415 – 1435 REFORMULATION1639 – 1640 · REFORMULATION: Desargues treats conics by projection and adds points at infinity1639 – 1640 REFORMULATION1822 · REFORMULATION: Poncelet founds projective geometry as a subject1822 REFORMULATION1825 – 1827 · REFORMULATION · contested: The principle of duality1825 – 1827 REFORMULATION1859 – 1871 · REFORMULATION: Cayley and Klein put distance inside projective geometry1859 – 1871 REFORMULATION1 unresolved problemDifferential Geometry of SurfacesEMERGED 18TH CENTURY – 1820S1760 (published 1767) · PROOF: Euler finds the two principal curvatures of a surface1760 (PUBLISHED 1767) PROOF1827 · PROOF: The Theorema Egregium — curvature is intrinsic1827 PROOF1847 – 1851 · PROOF: The Frenet–Serret formulas describe every space curve1847 – 1851 PROOF1848 · PROOF: The Gauss–Bonnet theorem ties curvature to angles1848 PROOF1901 · PROOF: The full hyperbolic plane does not fit in ordinary space1901 PROOFBranches from Calculus, drawn on The Analysis Thread↖ FROM CALCULUS · THE ANALYSIS THREAD1 unresolved problemNon-Euclidean GeometryEMERGED 1820S – 1830S1829 – 1832 · REFORMULATION · contested: Hyperbolic geometry is published as a geometry in its own right1829 – 1832 REFORMULATION1868 · PROOF: Beltrami shows hyperbolic geometry is as consistent as Euclid's1868 PROOF1872 · REFORMULATION: Klein's Erlangen Program redefines what a geometry is1872 REFORMULATION1882 · REFORMULATION: Poincaré finds hyperbolic geometry inside complex analysis1882 REFORMULATIONContinues into Group Theory, drawn on The Algebra Thread↘ INTO GROUP THEORY · THE ALGEBRA THREADAlgebraic TopologyEMERGED 1895 (ROOTS FROM 1750)1750 – 1758 · PROOF · contested: Euler's polyhedron formula1750 – 1758 PROOF1813 · CRISIS: Polyhedra with tunnels break Euler's formula1813 CRISIS1895 · REFORMULATION: Poincaré's Analysis Situs1895 REFORMULATION1911 – 1912 · PROOF: Brouwer's fixed-point and degree theorems1911 – 1912 PROOFc. 1925 · REFORMULATION: Homology becomes a group, not a countC. 1925 REFORMULATION1931 · PROOF: The Hopf fibration wraps a 3-sphere around a 2-sphere1931 PROOF1945 – 1952 · REFORMULATION: Homology is given axioms1945 – 1952 REFORMULATION1 unresolved problemRiemannian GeometryEMERGED 18541854 (published 1868) · REFORMULATION: Riemann's lecture "On the hypotheses which lie at the foundations of geometry"1854 (PUBLISHED 1868) REFORMULATION1900 · REFORMULATION: The absolute differential calculus1900 REFORMULATION1917 · REFORMULATION: Levi-Civita introduces parallel transport1917 REFORMULATION1941 · PROOF: Positive curvature forces a space to close up1941 PROOF1956 · PROOF: Nash proves every Riemannian manifold fits in some Euclidean space1956 PROOF1982 · PROOF: Hamilton introduces the Ricci flow1982 PROOF1 unresolved problemAlgebraic GeometryEMERGED 1850S – 1950S1779 · PROOF · contested: Bézout counts the intersections of curves1779 PROOF1857 · REFORMULATION: Riemann turns algebraic curves into surfaces1857 REFORMULATIONc. 1890 – 1946 · CRISIS · contested: The Italian school's intuition outruns its proofsC. 1890 – 1946 CRISIS1949 · CONJECTURE: The Weil conjectures1949 CONJECTURE1960 – 1967 · REFORMULATION: Grothendieck rebuilds the field on schemes1960 – 1967 REFORMULATION1974 · PROOF: Deligne proves the last Weil conjecture1974 PROOF1994 – 1995 · PROOF: Wiles proves Fermat's Last Theorem through elliptic curves1994 – 1995 PROOFContinues into Arithmetic Geometry, drawn on The Number Theory Thread↘ INTO ARITHMETIC GEOMETRY · THE NUMBER THEORY THREAD3 unresolved problemsGeometric TopologyEMERGED 1904 – 1980S1904 · CONJECTURE: Poincaré asks whether simple connectivity characterises the 3-sphere1904 CONJECTURE1956 · DISPROOF: Milnor finds exotic 7-spheres1956 DISPROOF1961 · PROOF: The Poincaré conjecture falls in dimensions five and up1961 PROOF1982 · CONJECTURE: Thurston's geometrization conjecture1982 CONJECTURE1982 · PROOF: Freedman settles the topological 4-dimensional case1982 PROOF2002 – 2003 · PROOF · contested: Perelman proves geometrization with Ricci flow2002 – 2003 PROOF2012 · PROOF: Agol proves the virtual Haken conjecture2012 PROOF2 unresolved problemsGeometric AnalysisEMERGED 1930S – 1970S1930 – 1931 · PROOF · contested: The Plateau problem is solved1930 – 1931 PROOF1976 – 1978 · PROOF: Yau proves the Calabi conjecture1976 – 1978 PROOF1979 · PROOF: The positive mass theorem1979 PROOF1981 · PROOF: Sacks and Uhlenbeck tame the bubbles1981 PROOF2017 – 2018 · PROOF: Infinitely many minimal surfaces in every closed 3-manifold2017 – 2018 PROOF2 unresolved problems
Charted: settled results
Contested: sources disagree on priority or causation
Unmapped: open problems

Fields in this thread

  1. c. 300 BCEEuclidean GeometryWhat can be deduced about space from a handful of assumptions no one thinks to doubt?Root of the thread
  2. 1630s – 1820sProjective GeometryWhich properties of a figure survive projection, the way a painter projects a scene onto a canvas?Branched from Euclidean Geometry
  3. 18th century – 1820sDifferential Geometry of SurfacesHow do you measure the bending of a curve or surface with calculus, and how much of that bending can be detected from inside the surface?Branched from Euclidean Geometry + Calculus (The Analysis Thread)
  4. 1820s – 1830sNon-Euclidean GeometryWhat does geometry look like if the parallel postulate is simply false?Branched from Euclidean Geometry
  5. 1895 (roots from 1750)Algebraic TopologyWhich features of a shape survive any amount of stretching, and how can algebra detect them?Branched from Euclidean Geometry
  6. 1854Riemannian GeometryWhat is geometry when a space is known only from the inside, in any number of dimensions?Branched from Differential Geometry of Surfaces + Non-Euclidean Geometry
  7. 1850s – 1950sAlgebraic GeometryWhat shapes are defined by polynomial equations, and what does their geometry reveal about the equations' solutions?Branched from Projective Geometry + Algebraic Topology
  8. 1904 – 1980sGeometric TopologyCan every three-dimensional space be classified, and does geometry decide its shape?Branched from Algebraic Topology + Riemannian Geometry + Non-Euclidean Geometry
  9. 1930s – 1970sGeometric AnalysisWhat can the solutions of differential equations reveal about the shape of a curved space?Branched from Riemannian Geometry + Differential Geometry of Surfaces

Thread 02 · 5 fields

The Number Theory Thread

From Euclid's proof that the primes never end to the arithmetic that secures the internet. The whole numbers look like the simplest objects in mathematics, yet questions a child could ask about them have taken centuries. The effort to prove Fermat's Last Theorem split number theory in two: an analytic branch that counts primes with calculus, and an algebraic branch that builds new number systems when unique factorisation fails. Both rejoined geometry in the proof of Fermat's theorem, and their oldest problems now guard every encrypted connection. The fog here is some of the densest in mathematics: the Riemann hypothesis, the twin primes, abc.

Scroll the map sideways →

Field tree: Elementary Number Theory, Algebraic Number Theory, Analytic Number Theory, Arithmetic Geometry, Public-Key Cryptography. Each field links to its page.Elementary Number TheoryEMERGED C. 300 BCE – 1801c. 300 BCE · PROOF: Euclid proves the primes never endC. 300 BCE PROOFc. 250 CE · REFORMULATION: Diophantus' ArithmeticaC. 250 CE REFORMULATIONc. 1637 · CONJECTURE: Fermat's note in the marginC. 1637 CONJECTURE1732 · DISPROOF: Euler factors Fermat's "always prime" number1732 DISPROOF1801 · REFORMULATION · contested: Gauss's Disquisitiones Arithmeticae1801 REFORMULATIONBranches from Euclidean Geometry, drawn on The Geometry Thread↖ FROM EUCLIDEAN GEOMETRY · THE GEOMETRY THREAD2 unresolved problemsAlgebraic Number TheoryEMERGED 1832 – 18711832 · REFORMULATION: Gauss introduces the Gaussian integers1832 REFORMULATION1847 · CRISIS: Lamé's proof of Fermat's Last Theorem collapses1847 CRISIS1844 – 1850 · REFORMULATION: Kummer's ideal numbers1844 – 1850 REFORMULATION1871 · REFORMULATION: Dedekind replaces ideal numbers with ideals1871 REFORMULATION1920 – 1927 · PROOF: Class field theory is completed1920 – 1927 PROOF1952 – 1967 · PROOF · contested: Gauss's class number one problem is solved1952 – 1967 PROOFContinues into Abstract Algebra, drawn on The Algebra Thread↘ INTO ABSTRACT ALGEBRA · THE ALGEBRA THREAD2 unresolved problemsAnalytic Number TheoryEMERGED 1737 – 18961737 · PROOF: Euler links the primes to an infinite series1737 PROOF1792 – 1798 · CONJECTURE · contested: The prime number theorem is conjectured1792 – 1798 CONJECTURE1837 · PROOF: Dirichlet's primes in arithmetic progressions1837 PROOF1859 · CONJECTURE: Riemann's paper on the number of primes1859 CONJECTURE1896 · PROOF: The prime number theorem is proved1896 PROOF1948 – 1949 · PROOF · contested: An "elementary" proof of the prime number theorem1948 – 1949 PROOF2013 · PROOF: Bounded gaps between primes2013 PROOFBranches from Complex Analysis, drawn on The Analysis Thread↖ FROM COMPLEX ANALYSIS · THE ANALYSIS THREAD2 unresolved problemsArithmetic GeometryEMERGED 1922 – 19831922 · PROOF: Mordell's theorem and conjecture1922 PROOF1955 – 1967 · CONJECTURE · contested: The modularity conjecture1955 – 1967 CONJECTURE1983 · PROOF: Faltings proves the Mordell conjecture1983 PROOF1985 – 1990 · PROOF: Fermat's Last Theorem is reduced to modularity1985 – 1990 PROOF1999 – 2001 · PROOF: Every elliptic curve over the rationals is modular1999 – 2001 PROOFBranches from Algebraic Geometry, drawn on The Geometry Thread↖ FROM ALGEBRAIC GEOMETRY · THE GEOMETRY THREAD1 unresolved problemPublic-Key CryptographyEMERGED 1976 – 19851976 · REFORMULATION · contested: Diffie and Hellman: New Directions in Cryptography1976 REFORMULATION1977 · REFORMULATION · contested: RSA turns Fermat's little theorem into a lock1977 REFORMULATION1985 · REFORMULATION: Elliptic-curve cryptography1985 REFORMULATION1994 · CRISIS: Shor's algorithm threatens it all1994 CRISIS2002 · PROOF: PRIMES is in P2002 PROOF2016 – 2024 · REFORMULATION: Post-quantum standards are chosen2016 – 2024 REFORMULATIONBranches from Computational Complexity, drawn on The Foundations Thread↖ FROM COMPUTATIONAL COMPLEXITY · THE FOUNDATIONS THREAD2 unresolved problems
Charted: settled results
Contested: sources disagree on priority or causation
Unmapped: open problems

Fields in this thread

  1. c. 300 BCE – 1801Elementary Number TheoryWhat can be proved about the whole numbers and how they divide one another?Branched from Euclidean Geometry (The Geometry Thread)
  2. 1832 – 1871Algebraic Number TheoryWhat happens to primes and factorisation when the whole numbers are extended to larger number systems?Branched from Elementary Number Theory
  3. 1737 – 1896Analytic Number TheoryHow are the primes distributed, and why can calculus answer questions about whole numbers?Branched from Elementary Number Theory + Complex Analysis (The Analysis Thread)
  4. 1922 – 1983Arithmetic GeometryWhat does the shape of an equation's solution set reveal about its solutions in whole or rational numbers?Branched from Algebraic Number Theory + Analytic Number Theory + Algebraic Geometry (The Geometry Thread)
  5. 1976 – 1985Public-Key CryptographyHow can two strangers communicate in secret without ever having shared a secret key?Branched from Elementary Number Theory + Arithmetic Geometry + Computational Complexity (The Foundations Thread)

Thread 03 · 5 fields

The Analysis Thread

From Archimedes' curved areas to the laws of chance. Calculus worked brilliantly from the day Newton and Leibniz invented it, but nobody could say what its infinitely small quantities were. Fourier's claim that any function is a sum of waves pushed intuition past breaking point, and repairing the foundations produced rigorous limits, a precise definition of the real numbers, and a theory of measure strong enough to put probability on firm ground. The fog here sits where waves concentrate: problems like Kakeya's needle, and the critical point of random networks.

Scroll the map sideways →

Field tree: Calculus, Fourier Analysis, Complex Analysis, Real Analysis, Probability Theory. Each field links to its page.CalculusEMERGED 1665 – 1700c. 250 BCE · PROOF: Archimedes computes curved areas by exhaustionC. 250 BCE PROOF1665 – 1684 · REFORMULATION · contested: Newton and Leibniz invent the calculus1665 – 1684 REFORMULATION1734 · CRISIS: Berkeley's "ghosts of departed quantities"1734 CRISIS1748 · REFORMULATION: Euler makes the function central1748 REFORMULATIONBranches from Euclidean Geometry, drawn on The Geometry Thread↖ FROM EUCLIDEAN GEOMETRY · THE GEOMETRY THREADContinues into Differential Equations, drawn on The Dynamics Thread↘ INTO DIFFERENTIAL EQUATIONS · THE DYNAMICS THREADContinues into Differential Geometry of Surfaces, drawn on The Geometry Thread↘ INTO DIFFERENTIAL GEOMETRY OF SURFACES · THE GEOMETRY THREADContinues into Numerical Analysis, drawn on The Computation Thread↘ INTO NUMERICAL ANALYSIS · THE COMPUTATION THREADFourier AnalysisEMERGED 1807 – 18291747 – 1759 · CRISIS: The vibrating string controversy1747 – 1759 CRISIS1807 – 1822 · REFORMULATION · contested: Fourier's theory of heat1807 – 1822 REFORMULATION1829 · PROOF: Dirichlet proves when Fourier series converge1829 PROOF1965 · REFORMULATION · contested: The fast Fourier transform1965 REFORMULATION1966 · PROOF: Carleson's theorem1966 PROOF1 unresolved problemComplex AnalysisEMERGED 1799 – 18511545 – 1572 · REFORMULATION: Square roots of negative numbers appear in the cubic formula1545 – 1572 REFORMULATION1799 – 1831 · REFORMULATION · contested: Complex numbers become points in a plane1799 – 1831 REFORMULATION1814 – 1831 · PROOF: Cauchy's integral theorem and formula1814 – 1831 PROOF1851 · REFORMULATION: Riemann's geometric theory of complex functions1851 REFORMULATION1870 · CRISIS: Weierstrass undermines the Dirichlet principle1870 CRISISBranches from Theory of Equations, drawn on The Algebra Thread↖ FROM THEORY OF EQUATIONS · THE ALGEBRA THREADContinues into Analytic Number Theory, drawn on The Number Theory Thread↘ INTO ANALYTIC NUMBER THEORY · THE NUMBER THEORY THREADContinues into Complex Dynamics, drawn on The Dynamics Thread↘ INTO COMPLEX DYNAMICS · THE DYNAMICS THREADReal AnalysisEMERGED 1821 – 19021821 · REFORMULATION · contested: Cauchy's Cours d'analyse1821 REFORMULATION1854 (published 1868) · REFORMULATION: Riemann defines the integral1854 (PUBLISHED 1868) REFORMULATION1872 · DISPROOF: A continuous curve with no slope anywhere1872 DISPROOF1872 – 1874 · REFORMULATION: The real numbers are constructed1872 – 1874 REFORMULATION1902 · REFORMULATION: Lebesgue's integral1902 REFORMULATIONContinues into Continuous Optimisation, drawn on The Computation Thread↘ INTO CONTINUOUS OPTIMISATION · THE COMPUTATION THREADContinues into Ergodic Theory, drawn on The Dynamics Thread↘ INTO ERGODIC THEORY · THE DYNAMICS THREADContinues into Set Theory, drawn on The Foundations Thread↘ INTO SET THEORY · THE FOUNDATIONS THREADContinues into Stochastic Processes, drawn on The Statistics Thread↘ INTO STOCHASTIC PROCESSES · THE STATISTICS THREADProbability TheoryEMERGED 1654 – 19331654 · REFORMULATION: Pascal and Fermat solve the problem of points1654 REFORMULATION1713 · PROOF: Bernoulli's law of large numbers1713 PROOF1733 – 1810 · PROOF: The central limit theorem1733 – 1810 PROOF1763 – 1774 · PROOF · contested: Bayes' theorem and inverse probability1763 – 1774 PROOF1933 · REFORMULATION: Kolmogorov's axioms1933 REFORMULATIONContinues into Bayesian Statistics, drawn on The Statistics Thread↘ INTO BAYESIAN STATISTICS · THE STATISTICS THREADContinues into Information Theory, drawn on The Statistics Thread↘ INTO INFORMATION THEORY · THE STATISTICS THREADContinues into Monte Carlo Methods, drawn on The Computation Thread↘ INTO MONTE CARLO METHODS · THE COMPUTATION THREADContinues into Probabilistic Combinatorics, drawn on The Combinatorics Thread↘ INTO PROBABILISTIC COMBINATORICS · THE COMBINATORICS THREADContinues into Statistical Inference, drawn on The Statistics Thread↘ INTO STATISTICAL INFERENCE · THE STATISTICS THREADContinues into Stochastic Processes, drawn on The Statistics Thread↘ INTO STOCHASTIC PROCESSES · THE STATISTICS THREAD1 unresolved problem
Charted: settled results
Contested: sources disagree on priority or causation
Unmapped: open problems

Fields in this thread

  1. 1665 – 1700CalculusHow can quantities that change continuously be measured, and infinitely many infinitely small pieces be added up?Branched from Euclidean Geometry (The Geometry Thread)
  2. 1807 – 1829Fourier AnalysisCan every signal be built from simple waves, and what does that decomposition reveal?Branched from Calculus
  3. 1799 – 1851Complex AnalysisWhat happens to calculus when numbers are allowed to be complex?Branched from Calculus + Theory of Equations (The Algebra Thread)
  4. 1821 – 1902Real AnalysisWhat exactly are limits, continuity and the real numbers, and why did calculus need them defined?Branched from Calculus + Fourier Analysis
  5. 1654 – 1933Probability TheoryHow can chance be measured, and what laws does randomness obey in the long run?Branched from Calculus + Real Analysis

Thread 04 · 5 fields

The Foundations Thread

From Aristotle's syllogisms to the limits of computation. Around 1900 mathematicians tried to rest all of mathematics on one secure foundation: logic made exact, and sets as the universal building material. The attempt produced paradoxes, a bitter feud, and in 1931 Gödel's proof that no such foundation can ever be complete. Out of that wreckage came the theory of computation, which is the blueprint of every computer, and the deepest open question in computer science. The fog here is P versus NP, and the continuum hypothesis, a question the standard axioms cannot answer at all.

Scroll the map sideways →

Field tree: Mathematical Logic, Set Theory, Metamathematics, Computability Theory, Computational Complexity. Each field links to its page.Mathematical LogicEMERGED 1847 – 1930c. 350 BCE · REFORMULATION: Aristotle's syllogisticC. 350 BCE REFORMULATION1847 – 1854 · REFORMULATION: Boole turns logic into algebra1847 – 1854 REFORMULATION1879 · REFORMULATION · contested: Frege's Begriffsschrift and the quantifier1879 REFORMULATION1910 – 1913 · REFORMULATION: Principia Mathematica1910 – 1913 REFORMULATION1929 – 1930 · PROOF: Gödel's completeness theorem1929 – 1930 PROOF1 unresolved problemSet TheoryEMERGED 1874 – 19081878 · CONJECTURE: Cantor's continuum hypothesis1878 CONJECTURE1891 · PROOF: The diagonal argument1891 PROOF1901 – 1903 · CRISIS · contested: Russell's paradox1901 – 1903 CRISIS1904 – 1908 · REFORMULATION · contested: Zermelo axiomatises set theory1904 – 1908 REFORMULATION1938 – 1963 · PROOF: The continuum hypothesis is independent1938 – 1963 PROOFBranches from Real Analysis, drawn on The Analysis Thread↖ FROM REAL ANALYSIS · THE ANALYSIS THREAD1 unresolved problemMetamathematicsEMERGED 1900 – 19361900 – 1928 · CONJECTURE: Hilbert's programme1900 – 1928 CONJECTURE1918 – 1928 · CRISIS: The foundations dispute1918 – 1928 CRISIS1931 · DISPROOF: Gödel's incompleteness theorems1931 DISPROOF1936 · PROOF: Gentzen proves arithmetic consistent1936 PROOF1977 · PROOF: A natural statement arithmetic cannot prove1977 PROOF2005 – 2022 · REFORMULATION: Machines check major theorems2005 – 2022 REFORMULATIONComputability TheoryEMERGED 1931 – 19361936 · DISPROOF · contested: The decision problem has no solution1936 DISPROOF1936 – 1945 · REFORMULATION · contested: From universal machine to stored-program computer1936 – 1945 REFORMULATION1953 · PROOF: Rice's theorem1953 PROOF1950 – 1970 · DISPROOF: Hilbert's tenth problem is unsolvable1950 – 1970 DISPROOF2024 · PROOF: The fifth Busy Beaver number is determined2024 PROOF1 unresolved problemComputational ComplexityEMERGED 1965 – 19721965 · REFORMULATION: Polynomial time as the meaning of "efficient"1965 REFORMULATION1971 – 1973 · PROOF · contested: The Cook–Levin theorem1971 – 1973 PROOF1972 · PROOF: Karp's 21 NP-complete problems1972 PROOF1975 – 2009 · CRISIS: The barriers to proving P ≠ NP1975 – 2009 CRISIS1992 – 1998 · PROOF: The PCP theorem1992 – 1998 PROOFContinues into Combinatorial Optimisation, drawn on The Combinatorics Thread↘ INTO COMBINATORIAL OPTIMISATION · THE COMBINATORICS THREADContinues into Public-Key Cryptography, drawn on The Number Theory Thread↘ INTO PUBLIC-KEY CRYPTOGRAPHY · THE NUMBER THEORY THREADContinues into Statistical Learning Theory, drawn on The Statistics Thread↘ INTO STATISTICAL LEARNING THEORY · THE STATISTICS THREAD1 unresolved problem
Charted: settled results
Contested: sources disagree on priority or causation
Unmapped: open problems

Fields in this thread

  1. 1847 – 1930Mathematical LogicWhat makes an argument valid, and can reasoning itself be turned into calculation?Root of the thread
  2. 1874 – 1908Set TheoryWhat is infinity, and can all of mathematics be built out of collections?Branched from Mathematical Logic + Real Analysis (The Analysis Thread)
  3. 1900 – 1936MetamathematicsWhat can mathematics prove about itself, about its own consistency, completeness and limits?Branched from Mathematical Logic + Set Theory
  4. 1931 – 1936Computability TheoryWhat can be computed at all, by any mechanical procedure?Branched from Metamathematics
  5. 1965 – 1972Computational ComplexityWhich problems can be solved efficiently, and why do some seem to need astronomical time?Branched from Computability Theory

Thread 05 · 5 fields

The Algebra Thread

From Babylonian recipes for finding an unknown to the mathematics of symmetry. For three thousand years algebra meant solving equations, and each advance, from the cubic formula to the proof that the quintic has none, came from a harder question about the roots. Galois answered the last of those questions with symmetry, and symmetry became the subject: groups, rings and fields, studied for their own sake after Emmy Noether, and represented as matrices that turned out to describe atoms and particles. The fog here includes the inverse Galois problem and the Jacobian conjecture.

Scroll the map sideways →

Field tree: Theory of Equations, Galois Theory, Group Theory, Abstract Algebra, Representation Theory. Each field links to its page.Theory of EquationsEMERGED C. 820 – 1799c. 1800 BCE · REFORMULATION: Babylonian scribes solve quadratic problemsC. 1800 BCE REFORMULATIONc. 820 · REFORMULATION: Al-Khwārizmī's book of al-jabrC. 820 REFORMULATION1515 – 1545 · PROOF · contested: The cubic and quartic are solved1515 – 1545 PROOF1591 · REFORMULATION: Viète puts letters for the known quantities1591 REFORMULATION1799 · PROOF · contested: The fundamental theorem of algebra1799 PROOFContinues into Complex Analysis, drawn on The Analysis Thread↘ INTO COMPLEX ANALYSIS · THE ANALYSIS THREAD1 unresolved problemGalois TheoryEMERGED 1770 – 18461770 – 1771 · REFORMULATION: Lagrange asks why the old formulas work1770 – 1771 REFORMULATION1799 – 1824 · DISPROOF · contested: No general formula for the quintic1799 – 1824 DISPROOF1830 – 1832 · REFORMULATION: Galois's theory of equations1830 – 1832 REFORMULATION1870 · REFORMULATION: Jordan's treatise on substitutions1870 REFORMULATION1942 · REFORMULATION: Artin's modern Galois theory1942 REFORMULATION1 unresolved problemGroup TheoryEMERGED 1832 – 18821854 · REFORMULATION: Cayley defines the abstract group1854 REFORMULATION1872 · PROOF: The Sylow theorems1872 PROOF1963 · PROOF: The odd order theorem1963 PROOF1968 · DISPROOF: The Burnside problem has a negative answer1968 DISPROOF1955 – 2004 · PROOF · contested: The classification of finite simple groups1955 – 2004 PROOFBranches from Non-Euclidean Geometry, drawn on The Geometry Thread↖ FROM NON-EUCLIDEAN GEOMETRY · THE GEOMETRY THREAD1 unresolved problemAbstract AlgebraEMERGED 1843 – 19311843 · REFORMULATION: Hamilton's quaternions1843 REFORMULATION1890 · PROOF: Hilbert's basis theorem1890 PROOF1910 · REFORMULATION: Steinitz's abstract theory of fields1910 REFORMULATION1921 · REFORMULATION: Emmy Noether's theory of ideals1921 REFORMULATION1930 – 1931 · REFORMULATION: Van der Waerden's Moderne Algebra1930 – 1931 REFORMULATIONBranches from Algebraic Number Theory, drawn on The Number Theory Thread↖ FROM ALGEBRAIC NUMBER THEORY · THE NUMBER THEORY THREAD1 unresolved problemRepresentation TheoryEMERGED 1873 – 19311873 – 1893 · REFORMULATION: Sophus Lie's continuous groups1873 – 1893 REFORMULATION1888 – 1894 · PROOF · contested: The simple Lie algebras are classified1888 – 1894 PROOF1896 · REFORMULATION: Frobenius invents group characters1896 REFORMULATION1925 – 1931 · REFORMULATION: Representation theory becomes the language of quantum mechanics1925 – 1931 REFORMULATION1979 – 1992 · PROOF: Monstrous moonshine1979 – 1992 PROOF
Charted: settled results
Contested: sources disagree on priority or causation
Unmapped: open problems

Fields in this thread

  1. c. 820 – 1799Theory of EquationsHow can an unknown quantity be found from an equation, and is there always a formula for it?Root of the thread
  2. 1770 – 1846Galois TheoryWhy can some polynomial equations be solved by a formula and others not?Branched from Theory of Equations
  3. 1832 – 1882Group TheoryWhat is symmetry, and what are all the ways a structure can be transformed into itself?Branched from Galois Theory + Non-Euclidean Geometry (The Geometry Thread)
  4. 1843 – 1931Abstract AlgebraWhat do all number-like systems have in common, and can algebra be done without numbers at all?Branched from Group Theory + Algebraic Number Theory (The Number Theory Thread)
  5. 1873 – 1931Representation TheoryHow can abstract symmetries be made concrete as matrices, and what does that reveal?Branched from Group Theory + Abstract Algebra

Thread 06 · 5 fields

The Combinatorics Thread

From counting arrangements to the structure of networks. For most of its history combinatorics was a collection of puzzles: how many ways to choose, whether a walk can cross every bridge once, how many colours a map needs. In the twentieth century the puzzles became a subject. Ramsey showed that complete disorder is impossible, Erdős showed that randomness proves what explicit construction cannot, and the need to route, schedule and match at scale turned graphs into the mathematics of computing. The fog here is close to the surface: nobody knows the smallest party of guests guaranteed to contain five mutual friends or five mutual strangers.

Scroll the map sideways →

Field tree: Enumerative Combinatorics, Graph Theory, Extremal Combinatorics, Combinatorial Optimisation, Probabilistic Combinatorics. Each field links to its page.Enumerative CombinatoricsEMERGED C. 200 BCE – 19641654 · PROOF · contested: Pascal's arithmetical triangle1654 PROOF1740 – 1748 · REFORMULATION: Euler turns counting into algebra1740 – 1748 REFORMULATION1918 · PROOF: A formula for the partition numbers1918 PROOF1937 · REFORMULATION · contested: Pólya's enumeration theorem1937 REFORMULATION1964 · REFORMULATION: Rota's foundations of combinatorial theory1964 REFORMULATION1 unresolved problemGraph TheoryEMERGED 1736 – 19361736 · PROOF: The bridges of Königsberg1736 PROOF1847 · PROOF: Kirchhoff's circuits and spanning trees1847 PROOF1852 · CONJECTURE: Four colours suffice?1852 CONJECTURE1879 – 1890 · DISPROOF: Kempe's proof collapses1879 – 1890 DISPROOF1936 · REFORMULATION: The first book on graph theory1936 REFORMULATION1976 · PROOF · contested: The four colour theorem, by computer1976 PROOF1983 – 2004 · PROOF: The graph minor theorem1983 – 2004 PROOF1 unresolved problemExtremal CombinatoricsEMERGED 1916 – 19751916 · PROOF: Schur's theorem1916 PROOF1927 · PROOF: Van der Waerden's theorem1927 PROOF1930 · PROOF: Ramsey's theorem1930 PROOF1935 · PROOF: The happy ending problem1935 PROOF1941 · PROOF: Turán's theorem1941 PROOF1975 · PROOF: Szemerédi's theorem1975 PROOF2023 · PROOF: An exponential improvement for Ramsey numbers2023 PROOF2 unresolved problemsCombinatorial OptimisationEMERGED 1939 – 19791939 – 1947 · REFORMULATION · contested: Linear programming and the simplex method1939 – 1947 REFORMULATION1956 · PROOF: The max-flow min-cut theorem1956 PROOF1959 · PROOF: Dijkstra's shortest-path algorithm1959 PROOF1965 · PROOF: Edmonds's matching algorithm1965 PROOF1976 – 2020 · PROOF · contested: Half again as long, and then slightly less1976 – 2020 PROOF1979 · PROOF: Linear programming in polynomial time1979 PROOFBranches from Computational Complexity, drawn on The Foundations Thread↖ FROM COMPUTATIONAL COMPLEXITY · THE FOUNDATIONS THREAD1 unresolved problemProbabilistic CombinatoricsEMERGED 1947 – 19751947 · PROOF: A random colouring beats every explicit one1947 PROOF1959 · DISPROOF: Networks with no short cycles that need many colours1959 DISPROOF1959 – 1960 · REFORMULATION: The evolution of random graphs1959 – 1960 REFORMULATION1975 · PROOF: The Lovász local lemma1975 PROOF2022 · PROOF: The Kahn–Kalai conjecture2022 PROOFBranches from Probability Theory, drawn on The Analysis Thread↖ FROM PROBABILITY THEORY · THE ANALYSIS THREAD1 unresolved problem
Charted: settled results
Contested: sources disagree on priority or causation
Unmapped: open problems

Fields in this thread

  1. c. 200 BCE – 1964Enumerative CombinatoricsHow many ways can something be arranged, and can the answer be found without listing them all?One of the thread's roots
  2. 1736 – 1936Graph TheoryWhat can be said about a network from the pattern of its connections alone?One of the thread's roots
  3. 1916 – 1975Extremal CombinatoricsHow large can a structure grow before some pattern is forced to appear inside it?Branched from Graph Theory + Enumerative Combinatorics
  4. 1939 – 1979Combinatorial OptimisationAmong astronomically many possible arrangements, how can the best one be found without trying them all?Branched from Graph Theory + Computational Complexity (The Foundations Thread)
  5. 1947 – 1975Probabilistic CombinatoricsCan chance prove that something exists, and what does a typical large network look like?Branched from Extremal Combinatorics + Probability Theory (The Analysis Thread)

Thread 07 · 5 fields

The Dynamics Thread

From Newton's laws of motion to the limits of prediction. Differential equations promised that the future follows from the present, and for two centuries mathematicians tried to solve them. Most cannot be solved by formula, so Poincaré learned to describe their solutions without solving them, and in doing so found the first hint of chaos. Computers later showed chaos everywhere: deterministic systems whose long-term behaviour is unpredictable in practice, yet obeys laws of its own, from the statistics of ergodic theory to the infinite detail of the Mandelbrot set. The fog here includes Hilbert's sixteenth problem and whether the Mandelbrot set is locally connected.

Scroll the map sideways →

Field tree: Differential Equations, Dynamical Systems, Chaos Theory, Ergodic Theory, Complex Dynamics. Each field links to its page.Differential EquationsEMERGED 1671 – 18901696 – 1697 · PROOF: The brachistochrone challenge1696 – 1697 PROOF1743 – 1768 · REFORMULATION: Euler systematises the subject1743 – 1768 REFORMULATION1824 – 1890 · PROOF: Solutions exist and are unique1824 – 1890 PROOF1841 · DISPROOF: Most equations can't be solved by formula1841 DISPROOF1874 – 1875 · PROOF: The Cauchy–Kovalevskaya theorem1874 – 1875 PROOFBranches from Calculus, drawn on The Analysis Thread↖ FROM CALCULUS · THE ANALYSIS THREADContinues into Numerical Methods for PDEs, drawn on The Computation Thread↘ INTO NUMERICAL METHODS FOR PDES · THE COMPUTATION THREAD1 unresolved problemDynamical SystemsEMERGED 1881 – 19671881 – 1886 · REFORMULATION: Poincaré's qualitative theory1881 – 1886 REFORMULATION1889 – 1890 · CRISIS: The prize memoir's error1889 – 1890 CRISIS1892 · PROOF: Lyapunov's theory of stability1892 PROOF1913 · PROOF: Poincaré's last geometric theorem1913 PROOF1954 – 1963 · PROOF: The KAM theorem1954 – 1963 PROOF1960 – 1967 · REFORMULATION: Smale's horseshoe1960 – 1967 REFORMULATION1 unresolved problemChaos TheoryEMERGED 1961 – 19781945 · PROOF: Wild solutions in a radio equation1945 PROOF1961 – 1963 · REFORMULATION: Lorenz's weather model1961 – 1963 REFORMULATION1964 – 1975 · PROOF · contested: Period three implies chaos1964 – 1975 PROOF1975 – 1982 · CONJECTURE: Feigenbaum's universal constant1975 – 1982 CONJECTURE1976 · REFORMULATION: May's simple models with complicated dynamics1976 REFORMULATION2002 · PROOF: The Lorenz attractor exists2002 PROOF1 unresolved problemErgodic TheoryEMERGED 1871 – 19591871 · CONJECTURE: Boltzmann's ergodic hypothesis1871 CONJECTURE1890 · PROOF: The recurrence theorem1890 PROOF1931 – 1932 · PROOF · contested: The ergodic theorems1931 – 1932 PROOF1958 – 1959 · REFORMULATION: Entropy for dynamical systems1958 – 1959 REFORMULATION1970 · PROOF: Ornstein's isomorphism theorem1970 PROOF1977 · PROOF: Ergodic theory proves Szemerédi's theorem1977 PROOFBranches from Real Analysis, drawn on The Analysis Thread↖ FROM REAL ANALYSIS · THE ANALYSIS THREAD1 unresolved problemComplex DynamicsEMERGED 1918 – 19851918 – 1920 · REFORMULATION · contested: Fatou and Julia found the theory of iteration1918 – 1920 REFORMULATION1978 – 1980 · REFORMULATION · contested: The Mandelbrot set is drawn1978 – 1980 REFORMULATION1982 · PROOF: The Mandelbrot set is connected1982 PROOF1985 · PROOF: No wandering domains1985 PROOF1998 · PROOF: The boundary has dimension two1998 PROOFBranches from Complex Analysis, drawn on The Analysis Thread↖ FROM COMPLEX ANALYSIS · THE ANALYSIS THREAD1 unresolved problem
Charted: settled results
Contested: sources disagree on priority or causation
Unmapped: open problems

Fields in this thread

  1. 1671 – 1890Differential EquationsIf we know how something is changing at every instant, can we work out where it will be?Branched from Calculus (The Analysis Thread)
  2. 1881 – 1967Dynamical SystemsWhat does a system do in the long run, when its equations can't be solved?Branched from Differential Equations
  3. 1961 – 1978Chaos TheoryHow can a system that follows exact rules be impossible to predict, and what order is hidden in its disorder?Branched from Dynamical Systems
  4. 1871 – 1959Ergodic TheoryWhen does the long-run behaviour of one trajectory match the average over all possible states?Branched from Dynamical Systems + Real Analysis (The Analysis Thread)
  5. 1918 – 1985Complex DynamicsWhat happens when a simple formula on the complex numbers is applied over and over again?Branched from Dynamical Systems + Complex Analysis (The Analysis Thread)

Thread 08 · 5 fields

The Statistics Thread

From combining the observations of astronomers to machines that learn from examples. Probability predicts data from a known chance mechanism. Statistics runs the argument backwards, from data to the mechanism, and for two centuries it was argued over as much as it was used. Least squares began in a priority dispute, Fisher and Neyman feuded over what a test means, and Bayesian reasoning was nearly banished before computers brought it back. Along the way Shannon measured information itself, Markov and Wiener gave laws to quantities that wander at random, and Vapnik and Valiant asked when a rule learned from examples can be trusted. The fog here is close to daily life: how to make published findings reliable, and why giant neural networks generalise when the theory says they should not.

Scroll the map sideways →

Field tree: Statistical Inference, Information Theory, Stochastic Processes, Bayesian Statistics, Statistical Learning Theory. Each field links to its page.Statistical InferenceEMERGED 1805 – 19351805 – 1809 · REFORMULATION · contested: The method of least squares1805 – 1809 REFORMULATION1886 · REFORMULATION: Galton's regression to the mean1886 REFORMULATION1900 · PROOF: Pearson's chi-squared test1900 PROOF1908 · PROOF: Student's t-distribution1908 PROOF1922 – 1935 · REFORMULATION: Fisher rebuilds statistics1922 – 1935 REFORMULATION1933 · REFORMULATION · contested: The Neyman–Pearson theory of testing1933 REFORMULATION2005 – 2015 · CRISIS · contested: The replication crisis2005 – 2015 CRISISBranches from Probability Theory, drawn on The Analysis Thread↖ FROM PROBABILITY THEORY · THE ANALYSIS THREAD1 unresolved problemInformation TheoryEMERGED 1924 – 19521924 – 1928 · REFORMULATION: Nyquist and Hartley measure the telegraph1924 – 1928 REFORMULATION1948 · PROOF: A mathematical theory of communication1948 PROOF1950 · PROOF: Hamming's error-correcting codes1950 PROOF1952 · PROOF: Huffman coding1952 PROOF1964 – 1969 · REFORMULATION · contested: The information in a single object1964 – 1969 REFORMULATION1993 – 1996 · REFORMULATION: Codes that reach Shannon's limit1993 – 1996 REFORMULATIONBranches from Probability Theory, drawn on The Analysis Thread↖ FROM PROBABILITY THEORY · THE ANALYSIS THREAD1 unresolved problemStochastic ProcessesEMERGED 1900 – 19441900 · REFORMULATION: Bachelier's theory of speculation1900 REFORMULATION1906 – 1913 · PROOF: Markov chains1906 – 1913 PROOF1923 · PROOF: Wiener constructs Brownian motion1923 PROOF1931 · PROOF: Kolmogorov's equations1931 PROOF1944 · REFORMULATION · contested: Itô's stochastic calculus1944 REFORMULATION1973 · REFORMULATION: The Black–Scholes formula1973 REFORMULATION1998 · REFORMULATION: The web as a Markov chain1998 REFORMULATIONBranches from Probability Theory, drawn on The Analysis Thread↖ FROM PROBABILITY THEORY · THE ANALYSIS THREADBranches from Real Analysis, drawn on The Analysis Thread↖ FROM REAL ANALYSIS · THE ANALYSIS THREAD1 unresolved problemBayesian StatisticsEMERGED 1774 – 19901774 – 1814 · PROOF: Laplace's rule of succession1774 – 1814 PROOF1922 – 1925 · CRISIS: The eclipse of inverse probability1922 – 1925 CRISIS1926 – 1954 · REFORMULATION: Probability as degree of belief1926 – 1954 REFORMULATION1939 · REFORMULATION: Jeffreys's Theory of Probability1939 REFORMULATION1940 – 1941 · REFORMULATION: Bayes breaks the Enigma1940 – 1941 REFORMULATION1990 · REFORMULATION: The MCMC revolution1990 REFORMULATIONBranches from Probability Theory, drawn on The Analysis Thread↖ FROM PROBABILITY THEORY · THE ANALYSIS THREAD1 unresolved problemStatistical Learning TheoryEMERGED 1958 – 19951958 · REFORMULATION: The perceptron1958 REFORMULATION1969 · DISPROOF · contested: The limits of perceptrons1969 DISPROOF1971 · PROOF: Vapnik–Chervonenkis theory1971 PROOF1984 · REFORMULATION: A theory of the learnable1984 REFORMULATION1986 · REFORMULATION · contested: Backpropagation1986 REFORMULATION1992 – 1995 · PROOF: Support vector machines1992 – 1995 PROOF2016 – 2019 · CRISIS: Deep networks break the rules2016 – 2019 CRISISBranches from Computational Complexity, drawn on The Foundations Thread↖ FROM COMPUTATIONAL COMPLEXITY · THE FOUNDATIONS THREAD1 unresolved problem
Charted: settled results
Contested: sources disagree on priority or causation
Unmapped: open problems

Fields in this thread

  1. 1805 – 1935Statistical InferenceHow can conclusions about a whole population be drawn from a limited sample, with a known risk of being wrong?Branched from Probability Theory (The Analysis Thread)
  2. 1924 – 1952Information TheoryHow much information does a message contain, and how fast can it be sent reliably through a noisy channel?Branched from Probability Theory (The Analysis Thread)
  3. 1900 – 1944Stochastic ProcessesWhat laws govern quantities that change randomly over time?Branched from Probability Theory (The Analysis Thread) + Real Analysis (The Analysis Thread)
  4. 1774 – 1990Bayesian StatisticsHow should a degree of belief be updated as evidence arrives, and can probability measure belief at all?Branched from Probability Theory (The Analysis Thread) + Statistical Inference
  5. 1958 – 1995Statistical Learning TheoryWhen can a rule learned from examples be trusted on cases it has never seen?Branched from Statistical Inference + Information Theory + Computational Complexity (The Foundations Thread)

Thread 09 · 5 fields

The Computation Thread

From Newton's method to the training of neural networks. For centuries, numbers were computed by hand, by people who followed rules and made mistakes, and the question was only how to get an answer at all. Electronic computers answered that and raised a harder question: can an answer produced by billions of rounded operations be trusted? Turing and Wilkinson showed how to tell a bad method from a bad problem. Richardson's failed weather forecast became a daily routine once the grid was made to keep up with the physics, games of chance on the ENIAC became Monte Carlo methods, and Cauchy's idea of walking downhill now trains artificial intelligence. The fog here is how fast two matrices can be multiplied, and why gradient descent trains deep networks as well as it does.

Scroll the map sideways →

Field tree: Numerical Analysis, Numerical Methods for PDEs, Continuous Optimisation, Monte Carlo Methods, Numerical Linear Algebra. Each field links to its page.Numerical AnalysisEMERGED 1669 – 19631669 – 1690 · REFORMULATION · contested: Newton's method for roots1669 – 1690 REFORMULATION1822 – 1843 · REFORMULATION: Calculation by machine1822 – 1843 REFORMULATION1901 · DISPROOF: More points can make things worse1901 DISPROOF1960 – 1963 · REFORMULATION: Backward error analysis1960 – 1963 REFORMULATION1985 · REFORMULATION: The IEEE 754 standard1985 REFORMULATION1991 – 1996 · CRISIS: Rounding error kills1991 – 1996 CRISISBranches from Calculus, drawn on The Analysis Thread↖ FROM CALCULUS · THE ANALYSIS THREAD1 unresolved problemNumerical Methods for PDEsEMERGED 1922 – 19771922 · CRISIS: Richardson's forecast by hand1922 CRISIS1928 · PROOF: The Courant–Friedrichs–Lewy condition1928 PROOF1950 · REFORMULATION: The first computer forecast1950 REFORMULATION1943 – 1960 · REFORMULATION · contested: The finite element method1943 – 1960 REFORMULATION1964 – 1977 · REFORMULATION: Multigrid1964 – 1977 REFORMULATION2005 – 2006 · REFORMULATION: Two black holes merge on a computer2005 – 2006 REFORMULATIONBranches from Differential Equations, drawn on The Dynamics Thread↖ FROM DIFFERENTIAL EQUATIONS · THE DYNAMICS THREAD1 unresolved problemContinuous OptimisationEMERGED 1847 – 19701788 · REFORMULATION: Lagrange multipliers1788 REFORMULATION1847 · REFORMULATION: Gradient descent1847 REFORMULATION1939 – 1951 · PROOF · contested: The Karush–Kuhn–Tucker conditions1939 – 1951 PROOF1951 · REFORMULATION: Stochastic approximation1951 REFORMULATION1959 – 1970 · REFORMULATION: Quasi-Newton methods1959 – 1970 REFORMULATION1984 – 1994 · PROOF: Interior-point methods for convex problems1984 – 1994 PROOFBranches from Real Analysis, drawn on The Analysis Thread↖ FROM REAL ANALYSIS · THE ANALYSIS THREAD1 unresolved problemMonte Carlo MethodsEMERGED 1946 – 19531777 · PROOF: Buffon's needle1777 PROOF1946 – 1949 · REFORMULATION: The Monte Carlo method at Los Alamos1946 – 1949 REFORMULATION1953 · REFORMULATION · contested: The Metropolis algorithm1953 REFORMULATION1960 – 1967 · REFORMULATION: Quasi-random points1960 – 1967 REFORMULATION1968 · CRISIS: Random numbers fall mainly in the planes1968 CRISIS1970 – 1990 · REFORMULATION: Markov chains enter statistics1970 – 1990 REFORMULATIONBranches from Probability Theory, drawn on The Analysis Thread↖ FROM PROBABILITY THEORY · THE ANALYSIS THREAD1 unresolved problemNumerical Linear AlgebraEMERGED 1947 – 1969c. 1st century CE · REFORMULATION: Elimination in the Nine ChaptersC. 1ST CENTURY CE REFORMULATION1947 · PROOF: Rounding error in matrix inversion1947 PROOF1948 · REFORMULATION: Turing's condition number1948 REFORMULATION1952 · REFORMULATION: The conjugate gradient method1952 REFORMULATION1959 – 1961 · REFORMULATION · contested: The QR algorithm1959 – 1961 REFORMULATION1969 · DISPROOF: Elimination is not optimal1969 DISPROOF1 unresolved problem
Charted: settled results
Contested: sources disagree on priority or causation
Unmapped: open problems

Fields in this thread

  1. 1669 – 1963Numerical AnalysisHow can a finite machine, doing finitely many rounded operations, give answers we can trust?Branched from Calculus (The Analysis Thread)
  2. 1922 – 1977Numerical Methods for PDEsHow do you turn the continuous equations of physics into a finite computation whose answer converges to the truth?Branched from Numerical Analysis + Differential Equations (The Dynamics Thread)
  3. 1847 – 1970Continuous OptimisationHow do you find the lowest point of a function of thousands or billions of variables?Branched from Numerical Analysis + Real Analysis (The Analysis Thread)
  4. 1946 – 1953Monte Carlo MethodsHow can randomness compute answers that careful deterministic methods cannot reach?Branched from Numerical Analysis + Probability Theory (The Analysis Thread)
  5. 1947 – 1969Numerical Linear AlgebraHow do you solve a million linear equations on a machine that rounds every operation, and know the answer is right?Branched from Numerical Analysis