List of important publications in mathematics
This article is written like a personal reflection, personal essay, or argumentative essay that states a Wikipedia editor's personal feelings or presents an original argument about a topic. (August 2020) |
This is a list of important publications in mathematics, organized by field.
Some reasons a particular publication might be regarded as important:
- Topic creator – A publication that created a new topic
- Breakthrough – A publication that changed scientific knowledge significantly
- Influence – A publication which has significantly influenced the world or has had a massive impact on the teaching of mathematics.
Among published compilations of important publications in mathematics are Landmark writings in Western mathematics 1640–1940 by Ivor Grattan-Guinness[2] and A Source Book in Mathematics by David Eugene Smith.[3]
Algebra
Theory of equations
Sulba Sutra
- Baudhayana(8th century BCE)
Believed to have been written around the 8th century BCE, this is one of the oldest mathematical texts. It laid the foundations of Indian mathematics and was influential in South Asia. Though this was primarily a geometrical text, it also contained some important algebraic developments, including the list of Pythagorean triples discovered algebraically, geometric solutions of linear equations, the use of quadratic equations and square root of 2.
The Nine Chapters on the Mathematical Art
- The Nine Chapters on the Mathematical Art from the 10th–2nd century BCE.
Contains the earliest description of Gaussian elimination for solving system of linear equations, it also contains method for finding square root and cubic root.
Haidao Suanjing
- Liu Hui (220-280 CE)
Contains the application of right angle triangles for survey of depth or height of distant objects.
Sunzi Suanjing
- Sunzi (5th century CE)
Contains the earliest description of Chinese remainder theorem.
Aryabhatiya
- Aryabhata (499 CE)
The text contains 33 verses covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations. It also gave the modern standard algorithm for solving first-order diophantine equations.
Jigu Suanjing
Jigu Suanjing (626 CE)
This book by Tang dynasty mathematician Wang Xiaotong contains the world's earliest third order equation.[citation needed]
Brāhmasphuṭasiddhānta
- Brahmagupta (628 CE)
Contained rules for manipulating both negative and positive numbers, rules for dealing the number zero, a method for computing square roots, and general methods of solving linear and some quadratic equations, solution to Pell's equation. [4] [5] [6] [7]
Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa'l-muqābala
- Muhammad ibn Mūsā al-Khwārizmī(820 CE)
The first book on the systematic
Līlāvatī, Siddhānta Shiromani and Bijaganita
One of the major treatises on mathematics by Bhāskara II provides the solution for indeterminate equations of 1st and 2nd order.
Yigu yanduan
- Liu Yi (12th century)
Contains the earliest invention of 4th order polynomial equation.[citation needed]
Mathematical Treatise in Nine Sections
- Qin Jiushao (1247)
This 13th-century book contains the earliest complete solution of 19th-century
Ceyuan haijing
- Li Zhi(1248)
Contains the application of high order polynomial equation in solving complex geometry problems.
Jade Mirror of the Four Unknowns
- Zhu Shijie (1303)
Contains the method of establishing system of high order polynomial equations of up to four unknowns.
Ars Magna
- Gerolamo Cardano (1545)
Otherwise known as The Great Art, provided the first published methods for solving
Vollständige Anleitung zur Algebra
- Leonhard Euler (1770)
Also known as Elements of Algebra, Euler's textbook on elementary algebra is one of the first to set out algebra in the modern form we would recognize today. The first volume deals with determinate equations, while the second part deals with Diophantine equations. The last section contains a proof of Fermat's Last Theorem for the case n = 3, making some valid assumptions regarding that Euler did not prove.[11]
Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse
- Carl Friedrich Gauss (1799)
Gauss' doctoral dissertation,[12] which contained a widely accepted (at the time) but incomplete proof[13] of the fundamental theorem of algebra.
Abstract algebra
Group theory
Réflexions sur la résolution algébrique des équations
- Joseph Louis Lagrange(1770)
The title means "Reflections on the algebraic solutions of equations". Made the prescient observation that the roots of the
Articles Publiés par Galois dans les Annales de Mathématiques
- Journal de Mathematiques pures et Appliquées, II (1846)
Posthumous publication of the mathematical manuscripts of Évariste Galois by Joseph Liouville. Included are Galois' papers Mémoire sur les conditions de résolubilité des équations par radicaux and Des équations primitives qui sont solubles par radicaux.
Traité des substitutions et des équations algébriques
- Camille Jordan (1870)
Online version: Online version
Traité des substitutions et des équations algébriques (Treatise on Substitutions and Algebraic Equations). The first book on group theory, giving a then-comprehensive study of permutation groups and Galois theory. In this book, Jordan introduced the notion of a
Theorie der Transformationsgruppen
- Sophus Lie, Friedrich Engel (1888–1893).
Publication data: 3 volumes, B.G. Teubner, Verlagsgesellschaft, mbH, Leipzig, 1888–1893. Volume 1, Volume 2, Volume 3.
The first comprehensive work on
Solvability of groups of odd order
- John Thompson(1960)
Description: Gave a complete proof of the solvability of finite groups of odd order, establishing the long-standing Burnside conjecture that all finite non-abelian simple groups are of even order. Many of the original techniques used in this paper were used in the eventual classification of finite simple groups.
Homological algebra
Homological Algebra
- Henri Cartan and Samuel Eilenberg (1956)
Provided the first fully worked out treatment of abstract homological algebra, unifying previously disparate presentations of homology and cohomology for associative algebras, Lie algebras, and groups into a single theory.
"Sur Quelques Points d'Algèbre Homologique"
- Alexander Grothendieck (1957)
Often referred to as the "Tôhoku paper", it revolutionized homological algebra by introducing abelian categories and providing a general framework for Cartan and Eilenberg's notion of derived functors.
Algebraic geometry
Theorie der Abelschen Functionen
- Bernhard Riemann (1857)
Publication data: Journal für die Reine und Angewandte Mathematik
Developed the concept of Riemann surfaces and their topological properties beyond Riemann's 1851 thesis work, proved an index theorem for the genus (the original formulation of the Riemann–Hurwitz formula), proved the Riemann inequality for the dimension of the space of meromorphic functions with prescribed poles (the original formulation of the Riemann–Roch theorem), discussed birational transformations of a given curve and the dimension of the corresponding moduli space of inequivalent curves of a given genus, and solved more general inversion problems than those investigated by Abel and Jacobi. André Weil once wrote that this paper "is one of the greatest pieces of mathematics that has ever been written; there is not a single word in it that is not of consequence."[16]
Faisceaux Algébriques Cohérents
Publication data: Annals of Mathematics, 1955
FAC, as it is usually called, was foundational for the use of
Géométrie Algébrique et Géométrie Analytique
- Jean-Pierre Serre (1956)
In
Le théorème de Riemann–Roch, d'après A. Grothendieck
- Armand Borel, Jean-Pierre Serre (1958)
Borel and Serre's exposition of Grothendieck's version of the
Éléments de géométrie algébrique
- Alexander Grothendieck (1960–1967)
Written with the assistance of
Séminaire de géométrie algébrique
- Alexander Grothendieck et al.
These seminar notes on Grothendieck's reworking of the foundations of algebraic geometry report on work done at
Number theory
Brāhmasphuṭasiddhānta
- Brahmagupta (628)
Brahmagupta's Brāhmasphuṭasiddhānta is the first book that mentions zero as a number, hence Brahmagupta is considered the first to formulate the concept of zero. The current system of the four fundamental operations (addition, subtraction, multiplication and division) based on the Hindu-Arabic number system also first appeared in Brahmasphutasiddhanta. It was also one of the first texts to provide concrete ideas on positive and negative numbers.
De fractionibus continuis dissertatio
- Leonhard Euler (1744)
First presented in 1737, this paper
Recherches d'Arithmétique
- Joseph Louis Lagrange(1775)
Developed a general theory of binary quadratic forms to handle the general problem of when an integer is representable by the form . This included a reduction theory for binary quadratic forms, where he proved that every form is equivalent to a certain canonically chosen reduced form.[21][22]
Disquisitiones Arithmeticae
- Carl Friedrich Gauss (1801)
The
"Beweis des Satzes, daß jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält"
Pioneering paper in
"Über die Anzahl der Primzahlen unter einer gegebenen Grösse"
- Bernhard Riemann (1859)
"Über die Anzahl der Primzahlen unter einer gegebenen Grösse" (or "On the Number of Primes Less Than a Given Magnitude") is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monthly Reports of the Berlin Academy. Although it is the only paper he ever published on number theory, it contains ideas which influenced dozens of researchers during the late 19th century and up to the present day. The paper consists primarily of definitions, heuristic arguments, sketches of proofs, and the application of powerful analytic methods; all of these have become essential concepts and tools of modern
Vorlesungen über Zahlentheorie
Vorlesungen über Zahlentheorie (Lectures on Number Theory) is a textbook of number theory written by German mathematicians P. G. Lejeune Dirichlet and R. Dedekind, and published in 1863. The Vorlesungen can be seen as a watershed between the classical number theory of
Zahlbericht
- David Hilbert (1897)
Unified and made accessible many of the developments in algebraic number theory made during the nineteenth century. Although criticized by André Weil (who stated "more than half of his famous Zahlbericht is little more than an account of Kummer's number-theoretical work, with inessential improvements")[27] and Emmy Noether,[28] it was highly influential for many years following its publication.
Fourier Analysis in Number Fields and Hecke's Zeta-Functions
- John Tate (1950)
Generally referred to simply as
"Automorphic Forms on GL(2)"
- Hervé Jacquet and Robert Langlands (1970)
This publication offers evidence towards Langlands' conjectures by reworking and expanding the classical theory of modular forms and their L-functions through the introduction of representation theory.
"La conjecture de Weil. I."
- Pierre Deligne (1974)
Proved the Riemann hypothesis for varieties over finite fields, settling the last of the open Weil conjectures.
"Endlichkeitssätze für abelsche Varietäten über Zahlkörpern"
- Gerd Faltings (1983)
Faltings proves a collection of important results in this paper, the most famous of which is the first proof of the
"Modular Elliptic Curves and Fermat's Last Theorem"
- Andrew Wiles (1995)
This article proceeds to prove a special case of the
The geometry and cohomology of some simple Shimura varieties
- Michael Harris and Richard Taylor (2001)
Harris and Taylor provide the first proof of the
"Le lemme fondamental pour les algèbres de Lie"
- Ngô Bảo Châu (2008)
Ngô Bảo Châu proved a long-standing unsolved problem in the classical Langlands program, using methods from the Geometric Langlands program.
"Perfectoid space"
- Peter Scholze (2012)
Peter Scholze introduced Perfectoid space.
Analysis
Introductio in analysin infinitorum
- Leonhard Euler (1748)
The eminent historian of mathematics
Yuktibhāṣā
- Jyeshtadeva(1501)
Written in India in 1530, [35]
Calculus
Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illi calculi genus
- Gottfried Leibniz(1684)
Leibniz's first publication on differential calculus, containing the now familiar notation for differentials as well as rules for computing the derivatives of powers, products and quotients.
Philosophiae Naturalis Principia Mathematica
- Isaac Newton (1687)
The Philosophiae Naturalis Principia Mathematica (
Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum
- Leonhard Euler (1755)
Published in two books,
Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe
- Bernhard Riemann (1867)
Written in 1853, Riemann's work on trigonometric series was published posthumously. In it, he extended Cauchy's definition of the integral to that of the Riemann integral, allowing some functions with dense subsets of discontinuities on an interval to be integrated (which he demonstrated by an example).[42] He also stated the Riemann series theorem,[42] proved the Riemann–Lebesgue lemma for the case of bounded Riemann integrable functions,[43] and developed the Riemann localization principle.[44]
Intégrale, longueur, aire
- Henri Lebesgue (1901)
Lebesgue's
Complex analysis
Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse
- Bernhard Riemann (1851)
Riemann's doctoral dissertation introduced the notion of a Riemann surface, conformal mapping, simple connectivity, the Riemann sphere, the Laurent series expansion for functions having poles and branch points, and the Riemann mapping theorem.
Functional analysis
Théorie des opérations linéaires
- Stefan Banach (1932; originally published 1931 in Polish under the title Teorja operacyj.)
- Zbl 0005.20901. Archived from the original(PDF) on 11 January 2014. Retrieved 11 July 2020.
The first mathematical monograph on the subject of
Produits Tensoriels Topologiques et Espaces Nucléaires
- OCLC 1315788.
Grothendieck's thesis introduced the notion of a nuclear space, tensor products of locally convex topological vector spaces, and the start of Grothendieck's work on tensor products of Banach spaces.[45]
Alexander Grothendieck also wrote a textbook on topological vector spaces:
- OCLC 886098.
Sur certains espaces vectoriels topologiques
- OCLC 17499190.
Fourier analysis
Mémoire sur la propagation de la chaleur dans les corps solides
- Joseph Fourier (1807)[46]
Introduced Fourier analysis, specifically Fourier series. Key contribution was to not simply use trigonometric series, but to model all functions by trigonometric series:
Multiplying both sides by , and then integrating from to yields:
When Fourier submitted his paper in 1807, the committee (which included
Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données
- Peter Gustav Lejeune Dirichlet (1829, expanded German edition in 1837)
In his habilitation thesis on Fourier series, Riemann characterized this work of Dirichlet as "the first profound paper about the subject".[47] This paper gave the first rigorous proof of the convergence of Fourier series under fairly general conditions (piecewise continuity and monotonicity) by considering partial sums, which Dirichlet transformed into a particular Dirichlet integral involving what is now called the Dirichlet kernel. This paper introduced the nowhere continuous Dirichlet function and an early version of the Riemann–Lebesgue lemma.[48]
On convergence and growth of partial sums of Fourier series
- Lennart Carleson (1966)
Settled
Geometry
Sulba Sutra
- Baudhayana
Believed to have been written around the 8th century BCE, this is one of the oldest mathematical texts. It laid the foundations of Indian mathematics and was influential in South Asia . Though this was primarily a geometrical text, it also contained some important algebraic developments, including the list of Pythagorean triples discovered algebraically, geometric solutions of linear equations, the use of quadratic equations and square root of 2.
Euclid's Elements
Publication data: c. 300 BC
Online version: Interactive Java version
This is often regarded as not only the most important work in geometry but one of the most important works in mathematics. It contains many important results in plane and solid geometry, algebra (books II and V), and number theory (book VII, VIII, and IX).[49] More than any specific result in the publication, it seems that the major achievement of this publication is the promotion of an axiomatic approach as a means for proving results. Euclid's Elements has been referred to as the most successful and influential textbook ever written.[50]
The Nine Chapters on the Mathematical Art
- Unknown author
This was a Chinese
The Conics
The Conics was written by Apollonius of Perga, a
Surya Siddhanta
- Unknown (400 CE)
Contains the roots of modern trigonometry. It describes the archeo-astronomy theories, principles and methods of the ancient Hindus. This siddhanta is supposed to be the knowledge that the Sun god gave to an Asura called Maya. It uses sine (jya), cosine (kojya or "perpendicular sine") and inverse sine (otkram jya) for the first time, and also contains the earliest use of the tangent and secant. Later Indian mathematicians such as Aryabhata made references to this text, while later Arabic and Latin translations were very influential in Europe and the Middle East.
Aryabhatiya
- Aryabhata (499 CE)
This was a highly influential text during the Golden Age of mathematics in India. The text was highly concise and therefore elaborated upon in commentaries by later mathematicians. It made significant contributions to geometry and astronomy, including introduction of sine/ cosine, determination of the approximate value of pi and accurate calculation of the earth's circumference.
La Géométrie
La Géométrie was published in 1637 and written by René Descartes. The book was influential in developing the Cartesian coordinate system and specifically discussed the representation of points of a plane, via real numbers; and the representation of curves, via equations.
Grundlagen der Geometrie
Online version: English
Publication data: Hilbert, David (1899). Grundlagen der Geometrie. Teubner-Verlag Leipzig.
Hilbert's axiomatization of geometry, whose primary influence was in its pioneering approach to metamathematical questions including the use of models to prove axiom independence and the importance of establishing the consistency and completeness of an axiomatic system.
Regular Polytopes
- H.S.M. Coxeter
Regular Polytopes is a comprehensive survey of the geometry of regular polytopes, the generalisation of regular polygons and regular polyhedra to higher dimensions. Originating with an essay entitled Dimensional Analogy written in 1923, the first edition of the book took Coxeter 24 years to complete. Originally written in 1947, the book was updated and republished in 1963 and 1973.
Differential geometry
Recherches sur la courbure des surfaces
- Leonhard Euler (1760)
Publication data: Mémoires de l'académie des sciences de Berlin 16 (1760) pp. 119–143; published 1767. (Full text and an English translation available from the Dartmouth Euler archive.)
Established the theory of
Disquisitiones generales circa superficies curvas
- Carl Friedrich Gauss (1827)
Publication data: "Disquisitiones generales circa superficies curvas", Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores Vol. VI (1827), pp. 99–146; "General Investigations of Curved Surfaces" (published 1965) Raven Press, New York, translated by A.M.Hiltebeitel and J.C.Morehead.
Groundbreaking work in differential geometry, introducing the notion of Gaussian curvature and Gauss' celebrated Theorema Egregium.
Über die Hypothesen, welche der Geometrie zu Grunde Liegen
- Bernhard Riemann (1854)
Publication data: "Über die Hypothesen, welche der Geometrie zu Grunde Liegen", Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Vol. 13, 1867. English translation
Riemann's famous Habiltationsvortrag, in which he introduced the notions of a
Leçons sur la théorie génerale des surfaces et les applications géométriques du calcul infinitésimal
- Gaston Darboux
Publication data: Darboux, Gaston (1887,1889,1896) (1890). Leçons sur la théorie génerale des surfaces. Gauthier-Villars.{{cite book}}
: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link) Volume I, Volume II, Volume III, Volume IV
Leçons sur la théorie génerale des surfaces et les applications géométriques du calcul infinitésimal (on the General Theory of Surfaces and the Geometric Applications of Infinitesimal Calculus). A treatise covering virtually every aspect of the 19th century differential geometry of surfaces.
Topology
Analysis situs
- Henri Poincaré (1895, 1899–1905)
Description: Poincaré's
L'anneau d'homologie d'une représentation, Structure de l'anneau d'homologie d'une représentation
- Jean Leray (1946)
These two
Quelques propriétés globales des variétés differentiables
- René Thom (1954)
In this paper, Thom proved the
Category theory
"General Theory of Natural Equivalences"
- Samuel Eilenberg and Saunders Mac Lane (1945)
The first paper on category theory. Mac Lane later wrote in Categories for the Working Mathematician that he and Eilenberg introduced categories so that they could introduce functors, and they introduced functors so that they could introduce natural equivalences. Prior to this paper, "natural" was used in an informal and imprecise way to designate constructions that could be made without making any choices. Afterwards, "natural" had a precise meaning which occurred in a wide variety of contexts and had powerful and important consequences.
Categories for the Working Mathematician
- Saunders Mac Lane (1971, second edition 1998)
Saunders Mac Lane, one of the founders of category theory, wrote this exposition to bring categories to the masses. Mac Lane brings to the fore the important concepts that make category theory useful, such as
Higher Topos Theory
- Jacob Lurie (2010)
This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck topoi. A few applications to classical topology are included. (see arXiv.)
Set theory
"Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"
- Georg Cantor (1874)
Online version: Online version
Contains the first proof that the set of all real numbers is uncountable; also contains a proof that the set of algebraic numbers is countable. (See
Grundzüge der Mengenlehre
First published in 1914, this was the first comprehensive introduction to set theory. Besides the systematic treatment of known results in set theory, the book also contains chapters on
"The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory"
- Kurt Gödel (1938)
Gödel proves the results of the title. Also, in the process, introduces the class L of constructible sets, a major influence in the development of axiomatic set theory.
"The Independence of the Continuum Hypothesis"
- Paul J. Cohen(1963, 1964)
Cohen's breakthrough work proved the independence of the continuum hypothesis and axiom of choice with respect to Zermelo–Fraenkel set theory. In proving this Cohen introduced the concept of forcing which led to many other major results in axiomatic set theory.
Logic
The Laws of Thought
- George Boole (1854)
Published in 1854, The Laws of Thought was the first book to provide a mathematical foundation for logic. Its aim was a complete re-expression and extension of Aristotle's logic in the language of mathematics. Boole's work founded the discipline of algebraic logic and would later be central for Claude Shannon in the development of digital logic.
Begriffsschrift
- Gottlob Frege (1879)
Published in 1879, the title Begriffsschrift is usually translated as concept writing or concept notation; the full title of the book identifies it as "a
Formulario mathematico
- Giuseppe Peano (1895)
First published in 1895, the Formulario mathematico was the first mathematical book written entirely in a formalized language. It contained a description of mathematical logic and many important theorems in other branches of mathematics. Many of the notations introduced in the book are now in common use.
Principia Mathematica
- Bertrand Russell and Alfred North Whitehead (1910–1913)
The Principia Mathematica is a three-volume work on the foundations of
"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I"
(On Formally Undecidable Propositions of Principia Mathematica and Related Systems)
- Kurt Gödel (1931)
Online version: Online version
In mathematical logic, Gödel's incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1931. The first incompleteness theorem states:
For any formal system such that (1) it is -consistent (
of the system.
Systems of Logic Based on Ordinals
- Alan Turing's PhD thesis (1938)
Combinatorics
"On sets of integers containing no k elements in arithmetic progression"
- Endre Szemerédi (1975)
Settled a conjecture of Paul Erdős and Pál Turán (now known as Szemerédi's theorem) that if a sequence of natural numbers has positive upper density then it contains arbitrarily long arithmetic progressions. Szemerédi's solution has been described as a "masterpiece of combinatorics"[55] and it introduced new ideas and tools to the field including a weak form of the Szemerédi regularity lemma.[56]
Graph theory
Solutio problematis ad geometriam situs pertinentis
- Leonhard Euler (1741)
- Euler's original publication (in Latin)
Euler's solution of the
"On the evolution of random graphs"
- Paul Erdős and Alfréd Rényi (1960)
Provides a detailed discussion of sparse
"Network Flows and General Matchings"
- L. R. Ford, Jr. & D. R. Fulkerson
- Flows in Networks. Prentice-Hall, 1962.
Presents the Ford–Fulkerson algorithm for solving the maximum flow problem, along with many ideas on flow-based models.
Computational complexity theory
See List of important publications in theoretical computer science.
Probability theory and statistics
See list of important publications in statistics.
Game theory
"Zur Theorie der Gesellschaftsspiele"
- John von Neumann (1928)
Went well beyond Émile Borel's initial investigations into strategic two-person game theory by proving the minimax theorem for two-person, zero-sum games.
Theory of Games and Economic Behavior
- Oskar Morgenstern, John von Neumann (1944)
This book led to the investigation of modern game theory as a prominent branch of mathematics. This work contained the method for finding optimal solutions for two-person zero-sum games.
"Equilibrium Points in N-person Games"
- PMID 16588946.
On Numbers and Games
- John Horton Conway (1976)
The book is in two, {0,1|}, parts. The zeroth part is about numbers, the first part about games – both the values of games and also some real games that can be played such as Nim, Hackenbush, Col and Snort amongst the many described.
Winning Ways for your Mathematical Plays
- Elwyn Berlekamp, John Conway and Richard K. Guy (1982)
A compendium of information on
Information theory
A Mathematical Theory of Communication
- Claude Shannon (1948)
An article, later expanded into a book, which developed the concepts of
Fractals
How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
- Benoît Mandelbrot(1967)
A discussion of self-similar curves that have fractional dimensions between 1 and 2. These curves are examples of fractals, although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. Shows Mandelbrot's early thinking on fractals, and is an example of the linking of mathematical objects with natural forms that was a theme of much of his later work.
Numerical analysis
Optimization
Method of Fluxions
- Isaac Newton (1736)
Method of Fluxions was a book written by Isaac Newton. The book was completed in 1671, and published in 1736. Within this book, Newton describes a method (the Newton–Raphson method) for finding the real zeroes of a function.
Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies
- Joseph Louis Lagrange(1761)
Major early work on the
"Математические методы организации и планирования производства"
- Leonid Kantorovich (1939) "[The Mathematical Method of Production Planning and Organization]" (in Russian).
Kantorovich wrote the first paper on production planning, which used Linear Programs as the model. He received the Nobel prize for this work in 1975.
"Decomposition Principle for Linear Programs"
- George Dantzig and P. Wolfe
- Operations Research 8:101–111, 1960.
Dantzig's is considered the father of linear programming in the western world. He independently invented the simplex algorithm. Dantzig and Wolfe worked on decomposition algorithms for large-scale linear programs in factory and production planning.
"How Good is the Simplex Algorithm?"
- Victor Klee and George J. Minty
- MR 0332165.
Klee and Minty gave an example showing that the
"Полиномиальный алгоритм в линейном программировании"
- Doklady Akademii Nauk SSSR(in Russian). 244: 1093–1096..
Khachiyan's work on the ellipsoid method. This was the first polynomial time algorithm for linear programming.
Early manuscripts
The examples and perspective in this article may not represent a worldwide view of the subject. (November 2009) |
These are publications that are not necessarily relevant to a mathematician nowadays, but are nonetheless important publications in the history of mathematics.
Moscow Mathematical Papyrus
This is one of the earliest mathematical treatises that still survives today. The Papyrus contains 25 problems involving arithmetic, geometry, and algebra, each with a solution given. Written in Ancient Egypt at approximately 1850 BC.[58]
Rhind Mathematical Papyrus
One of the oldest mathematical texts, dating to the
Archimedes Palimpsest
Although the only mathematical tools at its author's disposal were what we might now consider secondary-school
The Sand Reckoner
Online version: Online version
The first known (European) system of number-naming that can be expanded beyond the needs of everyday life.
Textbooks
Abstract Algebra
Dummit and Foote has become the modern dominant abstract algebra textbook following Jacobson's Basic Algebra.
Arithmetika Horvatzka
Arithmetika Horvatzka (1758) was the first Croatian language arithmetic textbook, written in the vernacular Kajkavian dialect of Croatian language. It established a complete system of arithmetic terminology in Croatian, and vividly used examples from everyday life in Croatia to present mathematical operations.[59] Although it was clear that Šilobod had made use of words that were in dictionaries, this was clearly insufficient for his purposes; and he made up some names by adapting Latin terminology to Kaikavian use.[60] Full text of Arithmetika Horvatszka is available via archive.org.
Synopsis of Pure Mathematics
Contains over 6000 theorems of mathematics, assembled by George Shoobridge Carr for the purpose of training his students for the Cambridge Mathematical Tripos exams. Studied extensively by
Éléments de mathématique
One of the most influential books in French mathematical literature. It introduces some of the notations and definitions that are now usual (the symbol ∅ or the term bijective for example). Characterized by an extreme level of rigour, formalism and generality (up to the point of being highly criticized for that), its publication started in 1939 and is still unfinished today.
Arithmetick: or, The Grounde of Arts
Written in 1542, it was the first really popular arithmetic book written in the English Language.
Cocker's Arithmetick
- Edward Cocker (authorship disputed)
Textbook of arithmetic published in 1678 by John Hawkins, who claimed to have edited manuscripts left by Edward Cocker, who had died in 1676. This influential mathematics textbook used to teach arithmetic in schools in the United Kingdom for over 150 years.
The Schoolmaster's Assistant, Being a Compendium of Arithmetic both Practical and Theoretical
An early and popular English arithmetic textbook published in
Geometry
- Andrei Kiselyov
Publication data: 1892
The most widely used and influential textbook in Russian mathematics. (See Kiselyov page.)
A Course of Pure Mathematics
A classic textbook in introductory
Moderne Algebra
The first introductory textbook (graduate level) expounding the abstract approach to algebra developed by Emil Artin and Emmy Noether. First published in German in 1931 by Springer Verlag. A later English translation was published in 1949 by Frederick Ungar Publishing Company.
Algebra
A definitive introductory text for abstract algebra using a category theoretic approach. Both a rigorous introduction from first principles, and a reasonably comprehensive survey of the field.
Calculus, Vol. 1
Algebraic Geometry
The first comprehensive introductory (graduate level) text in algebraic geometry that used the language of schemes and cohomology. Published in 1977, it lacks aspects of the scheme language which are nowadays considered central, like the functor of points.
Naive Set Theory
An undergraduate introduction to not-very-naive set theory which has lasted for decades. It is still considered by many to be the best introduction to set theory for beginners. While the title states that it is naive, which is usually taken to mean without axioms, the book does introduce all the axioms of Zermelo–Fraenkel set theory and gives correct and rigorous definitions for basic objects. Where it differs from a "true" axiomatic set theory book is its character: There are no long-winded discussions of axiomatic minutiae, and there is next to nothing about topics like large cardinals. Instead it aims, and succeeds, in being intelligible to someone who has never thought about set theory before.
Cardinal and Ordinal Numbers
The nec plus ultra reference for basic facts about cardinal and ordinal numbers. If you have a question about the cardinality of sets occurring in everyday mathematics, the first place to look is this book, first published in the early 1950s but based on the author's lectures on the subject over the preceding 40 years.
Set Theory: An Introduction to Independence Proofs
This book is not really for beginners, but graduate students with some minimal experience in set theory and formal logic will find it a valuable self-teaching tool, particularly in regard to forcing. It is far easier to read than a true reference work such as Jech, Set Theory. It may be the best textbook from which to learn forcing, though it has the disadvantage that the exposition of forcing relies somewhat on the earlier presentation of Martin's axiom.
Topologie
- Pavel Sergeevich Alexandrov
- Heinz Hopf
First published round 1935, this text was a pioneering "reference" text book in topology, already incorporating many modern concepts from set-theoretic topology, homological algebra and homotopy theory.
General Topology
First published in 1955, for many years the only introductory graduate level textbook in the US, teaching the basics of point set, as opposed to algebraic, topology. Prior to this the material, essential for advanced study in many fields, was only available in bits and pieces from texts on other topics or journal articles.
Topology from the Differentiable Viewpoint
This short book introduces the main concepts of differential topology in Milnor's lucid and concise style. While the book does not cover very much, its topics are explained beautifully in a way that illuminates all their details.
Number Theory, An approach through history from Hammurapi to Legendre
An historical study of number theory, written by one of the 20th century's greatest researchers in the field. The book covers some thirty six centuries of arithmetical work but the bulk of it is devoted to a detailed study and exposition of the work of Fermat, Euler, Lagrange, and Legendre. The author wishes to take the reader into the workshop of his subjects to share their successes and failures. A rare opportunity to see the historical development of a subject through the mind of one of its greatest practitioners.
An Introduction to the Theory of Numbers
An Introduction to the Theory of Numbers was first published in 1938, and is still in print, with the latest edition being the 6th (2008). It is likely that almost every serious student and researcher into number theory has consulted this book, and probably has it on their bookshelf. It was not intended to be a textbook, and is rather an introduction to a wide range of differing areas of number theory which would now almost certainly be covered in separate volumes. The writing style has long been regarded as exemplary, and the approach gives insight into a variety of areas without requiring much more than a good grounding in algebra, calculus and complex numbers.
Foundations of Differential Geometry
- Shoshichi Kobayashi and Katsumi Nomizu (1963; 1969)
Hodge Theory and Complex Algebraic Geometry I
Hodge Theory and Complex Algebraic Geometry II
Popular writings
Gödel, Escher, Bach
Gödel, Escher, Bach: an Eternal Golden Braid is a Pulitzer Prize-winning book, first published in 1979 by Basic Books. It is a book about how the creative achievements of logician Kurt Gödel, artist M. C. Escher and composer Johann Sebastian Bach interweave. As the author states: "I realized that to me, Gödel and Escher and Bach were only shadows cast in different directions by some central solid essence. I tried to reconstruct the central object, and came up with this book."
The World of Mathematics
See also
References
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Brahmagupta is believed to have composed many important works of mathematics and astronomy. However, two of his most important works are: Brahmasphutasiddhanta (BSS) written in 628 AD, and the Khandakhadyaka...
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