This is a list of publications in

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 and ''A Source Book in Mathematics'' by David Eugene Smith.

Algebra

Theory of equations

'' Baudhayana Sulba Sutra''

* Baudhayana (8th century BCE) With linguistic evidence suggesting this text to have been written between the 8th and 5th centuries century BCE, this is one of the oldest mathematical texts. It laid the foundations of

Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, ...

and was influential in

South Asia South Asia is the southern Subregion#Asia, subregion of Asia that is defined in both geographical and Ethnicity, ethnic-Culture, cultural terms. South Asia, with a population of 2.04 billion, contains a quarter (25%) of the world's populatio ...

. It was primarily a geometrical text and also contained some important developments, including the list of Pythagorean triples, geometric solutions of linear and quadratic equations and square root of 2.

'' The Nine Chapters on the Mathematical Art''

* '' The Nine Chapters on the Mathematical Art'' (10th–2nd century BCE) Contains the earliest description of

Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can a ...

for solving system of linear equations, it also contains method for finding square root and cubic root.

''
Arithmetica Diophantus of Alexandria () (; ) was a Greek mathematics, Greek mathematician who was the author of the ''Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations ...
''

Diophantus Diophantus of Alexandria () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations. Although Jose ...

(3rd century CE) Contains the collection of 130 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) and indeterminate equations.

'' Haidao Suanjing''

Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu ( The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state ...

(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 In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...

'' Aryabhatiya''

Aryabhata Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the '' Āryabhaṭīya'' (which mentions that in 3600 ' ...

(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 ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real n ...

'' Jigu Suanjing''

Jigu Suanjing (626 CE) This book by Tang dynasty mathematician Wang Xiaotong contains the world's earliest third order equation.

'' Brāhmasphuṭasiddhānta''

Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...

(628 CE) Contained rules for manipulating both negative and positive numbers, rules for dealing with the number zero, a method for computing square roots, and general methods of solving linear and some quadratic equations, solution to Pell's equation.

'' Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa'l-muqābala''

* Muhammad ibn Mūsā al-Khwārizmī (820 CE) The first book on the systematic

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

ic solutions of

linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...

and

quadratic equation In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...

s by the Persian scholar Muhammad ibn Mūsā al-Khwārizmī. The book is considered to be the foundation of modern

and Islamic mathematics. The word "algebra" itself is derived from the ''al-Jabr'' in the title of the book.

'' 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''

* Li Ye (13th century) Contains the earliest invention of 4th order polynomial equation.

'' Mathematical Treatise in Nine Sections''

* Qin Jiushao (1247) This 13th-century book contains the earliest complete solution of 19th-century Horner's method of solving high order polynomial equations (up to 10th order). It also contains a complete solution of

, which predates Euler and

Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...

by several centuries.

'' 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 Gerolamo Cardano (; also Girolamo or Geronimo; ; ; 24 September 1501– 21 September 1576) was an Italian polymath whose interests and proficiencies ranged through those of mathematician, physician, biologist, physicist, chemist, astrologer, as ...

(1545) Otherwise known as ''The Great Art'', provided the first published methods for solving

cubic Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system w ...

and quartic equations (due to Scipione del Ferro, Niccolò Fontana Tartaglia, and Lodovico Ferrari), and exhibited the first published calculations involving non-real

complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

''Vollständige Anleitung zur Algebra''

Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

(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 equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real n ...

s. The last section contains a proof of

Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...

for the case ''n'' = 3, making some valid assumptions regarding

\mathbb(\sqrt)

that Euler did not prove.

''Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse''

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...

(1799) Gauss's doctoral dissertation, which contained a widely accepted (at the time) but incomplete proof of the fundamental theorem of algebra.

Abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

Group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

=''Réflexions sur la résolution algébrique des équations''

= *

Joseph Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLagrange resolvent In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group ''G'' is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial ''p'' and has, roughly speaking, a rat ...

of a polynomial equation are tied to permutations of the roots of the original equation, laying a more general foundation for what had previously been an ad hoc analysis and helping motivate the later development of the theory of

permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, and

Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...

. The Lagrange resolvent also introduced the

discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discre ...

of order 3.

''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 Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by Nth root, ...

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 Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated at ...

(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

simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...

and epimorphism (which he called ), proved part of the Jordan–Hölder theorem, and discussed matrix groups over finite fields as well as the Jordan normal form.

''Theorie der Transformationsgruppen''

Sophus Lie Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. He also made substantial cont ...

, Friedrich Engel (1888–1893). Publication data: 3 volumes, B.G. Teubner, Verlagsgesellschaft, mbH, Leipzig, 1888–1893
Volume 1Volume 2Volume 3
The first comprehensive work on transformation groups, serving as the foundation for the modern theory of

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

''Solvability of groups of odd order''

* Walter Feit and 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 In mathematics, the classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic group, cyclic, or alternating gro ...

Homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

''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 algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

s, and groups into a single theory.

" Sur Quelques Points d'Algèbre Homologique"

Alexander Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...

(1957) Often referred to as the "Tôhoku paper", it revolutionized

homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

by introducing abelian categories and providing a general framework for Cartan and Eilenberg's notion of

derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...

Algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

''Theorie der Abelschen Functionen''

Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...

(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 Abel ( ''Hébel'', in pausa ''Hā́ḇel''; ''Hábel''; , ''Hābēl'') is a biblical figure in the Book of Genesis within the Abrahamic religions. Born as the second son of Adam and Eve, the first two humans created by God in Judaism, God, he ...

and Jacobi.

André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was one of the most influential mathematicians of the twentieth century. His influence is du ...

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.''"

''Faisceaux Algébriques Cohérents''

* Jean-Pierre Serre Publication data: ''Annals of Mathematics'', 1955 ''FAC'', as it is usually called, was foundational for the use of sheaves in algebraic geometry, extending beyond the case of

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

s. Serre introduced

Čech cohomology In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open set, open cover (topology), covers of a topological space. It is named for the mathematician Eduard Čech. Moti ...

of sheaves in this paper, and, despite some technical deficiencies, revolutionized formulations of algebraic geometry. For example, the long exact sequence in sheaf cohomology allows one to show that some surjective maps of sheaves induce surjective maps on sections; specifically, these are the maps whose kernel (as a sheaf) has a vanishing first cohomology group. The dimension of a vector space of sections of a coherent sheaf is finite, in

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...

, and such dimensions include many discrete invariants of varieties, for example Hodge numbers. While Grothendieck's

cohomology has replaced Čech cohomology for technical reasons, actual calculations, such as of the cohomology of projective space, are usually carried out by Čech techniques, and for this reason Serre's paper remains important.

'' Géométrie Algébrique et Géométrie Analytique''

* Jean-Pierre Serre (1956) In

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

and

analytic geometry In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and als ...

are closely related subjects, where ''analytic geometry'' is the theory of

s and the more general analytic spaces defined locally by the vanishing of

analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...

s of several complex variables. A (mathematical) theory of the relationship between the two was put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory. (''NB'': While

as use of Cartesian coordinates is also in a sense included in the scope of algebraic geometry, that is not the topic being discussed in this article.) The major paper consolidating the theory was ''Géometrie Algébrique et Géométrie Analytique'' by Serre, now usually referred to as ''GAGA''. A ''GAGA-style result'' would now mean any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, and a well-defined subcategory of analytic geometry objects and holomorphic mappings.

''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 Riemann–Roch theorem, published after Grothendieck made it clear that he was not interested in writing up his own result. Grothendieck reinterpreted both sides of the formula that Hirzebruch proved in 1953 in the framework of

morphisms In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

between varieties, resulting in a sweeping generalization. In his proof, Grothendieck broke new ground with his concept of

Grothendieck group In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a group homomorp ...

s, which led to the development of K-theory.

'' Éléments de géométrie algébrique''

(1960–1967) Written with the assistance of Jean Dieudonné, this is Grothendieck's exposition of his reworking of the foundations of algebraic geometry. It has become the most important foundational work in modern algebraic geometry. The approach expounded in EGA, as these books are known, transformed the field and led to monumental advances.

'' Séminaire de géométrie algébrique''

et al. These seminar notes on Grothendieck's reworking of the foundations of algebraic geometry report on work done at IHÉS starting in the 1960s. SGA 1 dates from the seminars of 1960–1961, and the last in the series, SGA 7, dates from 1967 to 1969. In contrast to EGA, which is intended to set foundations, SGA describes ongoing research as it unfolded in Grothendieck's seminar; as a result, it is quite difficult to read, since many of the more elementary and foundational results were relegated to EGA. One of the major results building on the results in SGA is

Pierre Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoor ...

's proof of the last of the open Weil conjectures in the early 1970s. Other authors who worked on one or several volumes of SGA include Michel Raynaud, Michael Artin, Jean-Pierre Serre, Jean-Louis Verdier,

, and Nicholas Katz.

Number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

'' Brāhmasphuṭasiddhānta''

(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''

(1744) First presented in 1737, this paper provided the first then-comprehensive account of the properties of continued fractions. It also contains the first proof that the number e is irrational.

''Recherches d'Arithmétique''

Joseph Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiabinary quadratic forms to handle the general problem of when an integer is representable by the form

ax^2 + by^2 + cxy

. This included a reduction theory for binary quadratic forms, where he proved that every form is equivalent to a certain canonically chosen reduced form.

'' Disquisitiones Arithmeticae''

(1801) The '' Disquisitiones Arithmeticae'' is a profound and masterful book on

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

written by German

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

and first published in 1801 when Gauss was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as

Fermat Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...

, Euler,

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLegendre and adds many important new results of his own. Among his contributions was the first complete proof known of the

Fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 is prime or can be represented uniquely as a product of prime numbers, ...

, the first two published proofs of the law of

quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...

, a deep investigation of binary quadratic forms going beyond Lagrange's work in ''Recherches d'Arithmétique'', a first appearance of Gauss sums, cyclotomy, and the theory of constructible polygons with a particular application to the constructibility of the regular 17-gon. Of note, in section V, article 303 of Disquisitiones, Gauss summarized his calculations of class numbers of imaginary quadratic number fields, and in fact found all imaginary quadratic number fields of class numbers 1, 2, and 3 (confirmed in 1986) as he had conjectured. In section VII, article 358, Gauss proved what can be interpreted as the first non-trivial case of the Riemann Hypothesis for curves over finite fields (the Hasse–Weil theorem).

"Beweis des Satzes, daß jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält"

Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In analysis, he advanced the theory o ...

(1837) Pioneering paper in

analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dir ...

, which introduced Dirichlet characters and their L-functions to establish Dirichlet's theorem on arithmetic progressions. In subsequent publications, Dirichlet used these tools to determine, among other things, the class number for quadratic forms.

" Über die Anzahl der Primzahlen unter einer gegebenen Grösse"

(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

. It also contains the famous

Riemann Hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...

, one of the most important open problems in mathematics.

'' Vorlesungen über Zahlentheorie''

and Richard Dedekind '' Vorlesungen über Zahlentheorie'' (''Lectures on Number Theory'') is a textbook of

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

, Jacobi and

, and the modern number theory of Dedekind, Riemann and

Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosophy of mathematics, philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad ...

. Dirichlet does not explicitly recognise the concept of the group that is central to modern algebra, but many of his proofs show an implicit understanding of group theory.

'' Zahlbericht''

David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...

(1897) Unified and made accessible many of the developments in

algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

made during the nineteenth century. Although criticized by

(who stated "''more than half of his famous Zahlbericht is little more than an account of Kummer's number-theoretical work, with inessential improvements''") and

Emmy Noether Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...

, 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 '' Tate's Thesis'', Tate's Princeton PhD thesis, under

Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number t ...

, is a reworking of Erich Hecke's theory of zeta- and ''L''-functions in terms of Fourier analysis on the adeles. The introduction of these methods into number theory made it possible to formulate extensions of Hecke's results to more general ''L''-functions such as those arising from automorphic forms.

" 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 form In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...

s and their ''L''-functions through the introduction of representation theory.

"La conjecture de Weil. I."

(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 Mordell conjecture (a conjecture dating back to 1922). Other theorems proved in this paper include an instance of the

Tate conjecture In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The ...

(relating the

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

s between two abelian varieties over a number field to the homomorphisms between their Tate modules) and some finiteness results concerning abelian varieties over number fields with certain properties.

"Modular Elliptic Curves and Fermat's Last Theorem"

Andrew Wiles Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for Wiles's proof of Fermat's Last Theorem, proving Ferma ...

(1995) This article proceeds to prove a special case of the Shimura–Taniyama conjecture through the study of the deformation theory of Galois representations. This in turn implies the famed

. The proof's method of identification of a deformation ring with a Hecke algebra (now referred to as an ''R=T'' theorem) to prove modularity lifting theorems has been an influential development in algebraic number theory.

''The geometry and cohomology of some simple Shimura varieties''

* Michael Harris and Richard Taylor (2001) Harris and Taylor provide the first proof of the local Langlands conjecture for GL(''n''). As part of the proof, this monograph also makes an in depth study of the geometry and cohomology of certain Shimura varieties at primes of bad reduction.

"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''

(1748) The eminent historian of mathematics Carl Boyer once called Euler's ''

Introductio in analysin infinitorum ''Introductio in analysin infinitorum'' (Latin: ''Introduction to the Analysis of the Infinite'') is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the ''Introducti ...

'' the greatest modern textbook in mathematics. Published in two volumes, this book more than any other work succeeded in establishing

analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...

as a major branch of mathematics, with a focus and approach distinct from that used in geometry and algebra. Notably, Euler identified functions rather than curves to be the central focus in his book. Logarithmic, exponential, trigonometric, and transcendental functions were covered, as were expansions into partial fractions, evaluations of for a positive integer between 1 and 13, infinite series and infinite product formulas, continued fractions, and partitions of integers. In this work, Euler proved that every rational number can be written as a finite continued fraction, that the continued fraction of an irrational number is infinite, and derived continued fraction expansions for and

\textstyle\sqrt

. This work also contains a statement of Euler's formula and a statement of the pentagonal number theorem, which he had discovered earlier and would publish a proof for in 1751.

'' Yuktibhāṣā''

* Jyeshtadeva (1501) Written in

India India, officially the Republic of India, is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area; the List of countries by population (United Nations), most populous country since ...

in 1530, and served as a summary of the Kerala School's achievements in infinite series,

trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...

and

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

, most of which were earlier discovered by the 14th century mathematician Madhava. Some of its important developments in calculus include infinite series and

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

expansion of some trigonometry functions.

Calculus

''Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illi calculi genus''

Gottfried Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, Sir Isaac Newton, with the creation of calculus in ad ...

(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 Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...

(1687) The ''Philosophiae Naturalis Principia Mathematica'' (

Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...

: "mathematical principles of natural philosophy", often ''Principia'' or ''Principia Mathematica'' for short) is a three-volume work by

published on 5 July 1687. Perhaps the most influential scientific book ever published, it contains the statement of

Newton's laws of motion Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...

forming the foundation of

classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...

as well as his law of universal gravitation, and derives Kepler's laws for the motion of the

planet A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...

s (which were first obtained empirically). Here was born the practice, now so standard we identify it with science, of explaining nature by postulating mathematical axioms and demonstrating that their conclusion are observable phenomena. In formulating his physical theories, Newton freely used his unpublished work on calculus. When he submitted Principia for publication, however, Newton chose to recast the majority of his proofs as geometric arguments.

''Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum''

(1755) Published in two books, Euler's textbook on differential calculus presented the subject in terms of the function concept, which he had introduced in his 1748 ''Introductio in analysin infinitorum''. This work opens with a study of the calculus of finite differences and makes a thorough investigation of how differentiation behaves under substitutions. Also included is a systematic study of Bernoulli polynomials and the Bernoulli numbers (naming them as such), a demonstration of how the Bernoulli numbers are related to the coefficients in the Euler–Maclaurin formula and the values of ζ(2n), a further study of Euler's constant (including its connection to the

gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

), and an application of partial fractions to differentiation.

''Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe''

(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 In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Gö ...

, allowing some functions with dense subsets of discontinuities on an interval to be integrated (which he demonstrated by an example). He also stated the Riemann series theorem, proved the Riemann–Lebesgue lemma for the case of bounded Riemann integrable functions, and developed the Riemann localization principle.

''Intégrale, longueur, aire''

* Henri Lebesgue (1901) Lebesgue's doctoral dissertation, summarizing and extending his research to date regarding his development of

measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...

and the

Lebesgue integral In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...

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 map In mathematics, a conformal map is a function (mathematics), function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-prese ...

ping, simple connectivity, the

Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...

, the Laurent series expansion for functions having poles and branch points, and the

Riemann mapping theorem In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane \mathbb which is not all of \mathbb, then there exists a biholomorphic mapping f (i.e. a bijective hol ...

Functional analysis

''Théorie des opérations linéaires''

Stefan Banach Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...

(1932; originally published 1931 in Polish under the title ''Teorja operacyj''.) * The first mathematical monograph on the subject of

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

s, bringing the abstract study of

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...

to the wider mathematical community. The book introduced the ideas of a normed space and the notion of a so-called ''B''-space, a complete normed space. The ''B''-spaces are now called

Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...

s and are one of the basic objects of study in all areas of modern mathematical analysis. Banach also gave proofs of versions of the open mapping theorem, closed graph theorem, and

Hahn–Banach theorem In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient ...

''Produits Tensoriels Topologiques et Espaces Nucléaires''

* 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.

also wrote a textbook on

topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

s: *

''Sur certains espaces vectoriels topologiques''

Fourier analysis

''Mémoire sur la propagation de la chaleur dans les corps solides''

Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre, Burgundy and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analys ...

(1807) Introduced Fourier analysis, specifically

Fourier series A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...

. Key contribution was to not simply use trigonometric series, but to model ''all'' functions by trigonometric series: When Fourier submitted his paper in 1807, the committee (which included

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLaplace,

Malus ''Malus'' ( or ) is a genus of about 32–57 species of small deciduous trees or shrubs in the family Rosaceae, including the domesticated orchard apple, crab apples (sometimes known in North America as crabapples) and wild apples. The genus i ...

and Legendre, among others) concluded: ''...the manner in which the author arrives at these equations is not exempt of difficulties and ..his analysis to integrate them still leaves something to be desired on the score of generality and even rigour''. Making Fourier series rigorous, which in detail took over a century, led directly to a number of developments in analysis, notably the rigorous statement of the integral via the Dirichlet integral and later the

''Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données''

(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''". This paper gave the first rigorous proof of the convergence of

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 In mathematics, the Dirichlet function is the indicator function \mathbf_\Q of the set of rational numbers \Q, i.e. \mathbf_\Q(x) = 1 if is a rational number and \mathbf_\Q(x) = 0 if is not a rational number (i.e. is an irrational number). \mathb ...

and an early version of the Riemann–Lebesgue lemma.

''On convergence and growth of partial sums of Fourier series''

* Lennart Carleson (1966) Settled Lusin's conjecture that the Fourier expansion of any

L^2

function converges

almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...

Geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

'' Baudhayana 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

and was influential in

. 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''

Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...

Publication data: c. 300 BC Online version:''
Interactive Java version
This is often regarded as not only the most important work in

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

but one of the most important works in mathematics. It contains many important results in plane and solid

, algebra (books II and V), and

(book VII, VIII, and IX). 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.

'' The Nine Chapters on the Mathematical Art''

* Unknown author This was a Chinese

book, mostly geometric, composed during the

Han dynasty The Han dynasty was an Dynasties of China, imperial dynasty of China (202 BC9 AD, 25–220 AD) established by Liu Bang and ruled by the House of Liu. The dynasty was preceded by the short-lived Qin dynasty (221–206 BC ...

, perhaps as early as 200 BC. It remained the most important textbook in

China China, officially the People's Republic of China (PRC), is a country in East Asia. With population of China, a population exceeding 1.4 billion, it is the list of countries by population (United Nations), second-most populous country after ...

and

East Asia East Asia is a geocultural region of Asia. It includes China, Japan, Mongolia, North Korea, South Korea, and Taiwan, plus two special administrative regions of China, Hong Kong and Macau. The economies of Economy of China, China, Economy of Ja ...

for over a thousand years, similar to the position of Euclid's ''Elements'' in Europe. Among its contents: Linear problems solved using the principle known later in the West as the '' rule of false position''. Problems with several unknowns, solved by a principle similar to

. Problems involving the principle known in the West as the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

. The earliest solution of a

matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

using a method equivalent to the modern method.

'' The Conics''

* Apollonius of Perga The Conics was written by Apollonius of Perga, a Greek mathematician. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including

Ptolemy Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...

, Francesco Maurolico,

, and

René Descartes René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...

. It was Apollonius who gave the

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

, the

parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...

, and the hyperbola the names by which we know them.

'' Surya Siddhanta''

* Unknown (400 CE) 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 . 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''

(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 Publishing is the activities of making information, literature, music, software, and other content, physical or digital, available to the public for sale or free of charge. Traditionally, the term publishing refers to the creation and distribu ...

in 1637 and written by

. The book was influential in developing the

Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...

and specifically discussed the representation of points of a plane, via

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s; and the representation of

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

s, via

equation In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...

''Grundlagen der Geometrie''

Online version:''
English
Publication data: 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 polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitive group action, transitively on its flag (geometry), flags, thus giving it the highest degree of symmetry. In particular, all its elements or -faces (for all , w ...

s, the generalisation of regular

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

s and regular

polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

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 Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

''Recherches sur la courbure des surfaces''

(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 surfaces, and introduced the idea of principal curvatures, laying the foundation for subsequent developments in the

differential geometry of surfaces In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth manifold, smooth Surface (topology), surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensiv ...

''Disquisitiones generales circa superficies curvas''

(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 Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, introducing the notion of

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...

and Gauss's 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

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

Riemannian metric In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

, and curvature tensor. Richard Dedekind reported on the reaction of the then 77 year old

to Riemann's presentation, stating that it had "surpassed all his expectations" and that he spoke "with the greatest appreciation, and with an excitement rare for him, about the depth of the ideas presented by Riemann."

''Leçons sur la théorie génerale des surfaces et les applications géométriques du calcul infinitésimal''

* Gaston Darboux Publication data:
Volume IVolume IIVolume IIIVolume 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

of surfaces.

Topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

''Analysis situs''

Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...

(1895, 1899–1905) Description: Poincaré's Analysis Situs and his Compléments à l'Analysis Situs laid the general foundations for

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

. In these papers, Poincaré introduced the notions of homology and the fundamental group, provided an early formulation of Poincaré duality, gave the Euler–Poincaré characteristic for chain complexes, and mentioned several important conjectures including the Poincaré conjecture, demonstrated by Grigori Perelman in 2003.

''L'anneau d'homologie d'une représentation'', ''Structure de l'anneau d'homologie d'une représentation''

* Jean Leray (1946) These two Comptes Rendus notes of Leray from 1946 introduced the novel concepts of sheafs, sheaf cohomology, and spectral sequences, which he had developed during his years of captivity as a prisoner of war. Leray's announcements and applications (published in other Comptes Rendus notes from 1946) drew immediate attention from other mathematicians. Subsequent clarification, development, and generalization by Henri Cartan, Jean-Louis Koszul, Armand Borel, Jean-Pierre Serre, and Leray himself allowed these concepts to be understood and applied to many other areas of mathematics. Dieudonné would later write that these notions created by Leray "''undoubtedly rank at the same level in the history of mathematics as the methods invented by Poincaré and Brouwer''".

Quelques propriétés globales des variétés differentiables

* René Thom (1954) In this paper, Thom proved the Thom transversality theorem, introduced the notions of oriented and unoriented cobordism, and demonstrated that cobordism groups could be computed as the homotopy groups of certain Thom spaces. Thom completely characterized the unoriented cobordism ring and achieved strong results for several problems, including Steenrod's problem on the realization of cycles.

Category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. 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 ''Categories for the Working Mathematician'' (''CWM'') is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. It was first published in 1971, and is based ...
''

* 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 adjoint functors and universal properties.

'' 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 Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

"Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"

Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...

(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 Georg Cantor's first set theory article.)

'' Grundzüge der Mengenlehre''

Felix Hausdorff Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician, pseudonym Paul Mongré (''à mogré' (Fr.) = "according to my taste"), who is considered to be one of the founders of modern topology and who contributed sig ...

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

and topology, which were then still considered parts of set theory. Here Hausdorff presents and develops highly original material which was later to become the basis for those areas.

"The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory"

Kurt Gödel Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...

(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 set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...

. In proving this Cohen introduced the concept of '' forcing'' which led to many other major results in axiomatic set theory.

Logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

'' The Laws of Thought''

George Boole George Boole ( ; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland. H ...

(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 Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electrical engineer, computer scientist, cryptographer and inventor known as the "father of information theory" and the man who laid the foundations of th ...

in the development of digital logic.

''
Begriffsschrift ''Begriffsschrift'' (German for, roughly, "concept-writing") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept notati ...
''

Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philos ...

(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

formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...

language Language is a structured system of communication that consists of grammar and vocabulary. It is the primary means by which humans convey meaning, both in spoken and signed language, signed forms, and may also be conveyed through writing syste ...

, modelled on that of

arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...

, of pure

thought In their most common sense, the terms thought and thinking refer to cognitive processes that can happen independently of sensory stimulation. Their most paradigmatic forms are judging, reasoning, concept formation, problem solving, and de ...

''". Frege's motivation for developing his formal logical system was similar to Leibniz's desire for a '' calculus ratiocinator''. Frege defines a logical calculus to support his research in the

foundations of mathematics Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theo ...

. ''Begriffsschrift'' is both the name of the book and the calculus defined therein. It was arguably the most significant publication in

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

since

Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...

''
Formulario mathematico ''Formulario Mathematico'' (Latino sine flexione: ''Formulary for Mathematics'') is a book by Giuseppe Peano which expresses fundamental theorems of mathematics in a Symbolic language (mathematics), symbolic language developed by Peano. The autho ...
''

Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much Mathematical notati ...

(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 Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

and many important theorems in other branches of mathematics. Many of the notations introduced in the book are now in common use.

''
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1 ...
''

Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...

and

Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He created the philosophical school known as process philosophy, which has been applied in a wide variety of disciplines, inclu ...

(1910–1913) The ''Principia Mathematica'' is a three-volume work on the foundations of

, written by

and

and published in 1910–1913. It is an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in

symbolic logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

. The questions remained whether a contradiction could be derived from the Principia's axioms, and whether there exists a mathematical statement which could neither be proven nor disproven in the system. These questions were settled, in a rather surprising way, by Gödel's incompleteness theorem in 1931.

"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I"

( On Formally Undecidable Propositions of Principia Mathematica and Related Systems) *

(1931) Online version:''
Online version
In

Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...

are two celebrated theorems proved by

in 1931. The first incompleteness theorem states:

For any formal system such that (1) it is $\omega$ -consistent ( omega-consistent), (2) it has a recursively definable set of axioms and rules of derivation, and (3) every recursive relation of natural numbers is definable in it, there exists a formula of the system such that, according to the intended interpretation of the system, it expresses a truth about natural numbers and yet it is not a
theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
of the system.

'' Systems of Logic Based on Ordinals''

Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical computer ...

's PhD thesis (1938)

Combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

"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" and it introduced new ideas and tools to the field including a weak form of the Szemerédi regularity lemma.

Graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

''Solutio problematis ad geometriam situs pertinentis''

(1741)
Euler's original publication
(in Latin) Euler's solution of the Königsberg bridge problem in ''Solutio problematis ad geometriam situs pertinentis'' (''The solution of a problem relating to the geometry of position'') is considered to be the first theorem of

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

"On the evolution of random graphs"

* Paul Erdős and Alfréd Rényi (1960) Provides a detailed discussion of sparse random graphs, including distribution of components, occurrence of small subgraphs, and phase transitions.

"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.

Probability theory and statistics

''See list of important publications in statistics.''

Game theory Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed ...

"Zur Theorie der Gesellschaftsspiele"

John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...

(1928) Went well beyond

Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French people, French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Biograp ...

'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 Oskar Morgenstern (; January 24, 1902 – July 26, 1977) was a German-born economist. In collaboration with mathematician John von Neumann, he is credited with founding the field of game theory and its application to social sciences and strategic ...

(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"

Nash equilibrium In game theory, the Nash equilibrium is the most commonly used solution concept for non-cooperative games. A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed) ...

''
On Numbers and Games ''On Numbers and Games'' is a mathematics book by John Horton Conway first published in 1976. The book is written by a pre-eminent mathematician, and is directed at other mathematicians. The material is, however, developed in a playful and unpr ...
''

* John Horton Conway (1976) The book is in two, , 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 mathematical games. It was first published in 1982 in two volumes, one focusing on Combinatorial game theory and surreal numbers, and the other concentrating on a number of specific games.

Information theory Information theory is the mathematical study of the quantification (science), quantification, Data storage, storage, and telecommunications, communication of information. The field was established and formalized by Claude Shannon in the 1940s, ...

''
A Mathematical Theory of Communication "A Mathematical Theory of Communication" is an article by mathematician Claude E. Shannon published in '' Bell System Technical Journal'' in 1948. It was renamed ''The Mathematical Theory of Communication'' in the 1949 book of the same name, a s ...
''

(1948) An article, later expanded into a book, which developed the concepts of

information entropy In information theory, the entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential states or possible outcomes. This measures the expected amount of information needed ...

and redundancy, and introduced the term bit (which Shannon credited to

John Tukey John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician and statistician, best known for the development of the fast Fourier Transform (FFT) algorithm and box plot. The Tukey range test, the Tukey lambda distributi ...

) as a unit of information.

Fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
s

'' How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension''

Benoît Mandelbrot Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of #Fractals and the ...

(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 Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

Optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...

''Method of Fluxions''

(1736) '' Method of Fluxions'' was a book written by

. 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 Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiacalculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

, building upon some of Lagrange's prior investigations as well as those of Euler. Contains investigations of minimal surface determination as well as the initial appearance of Lagrange multipliers.

"Математические методы организации и планирования производства"

* Leonid Kantorovich (1939) " he 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 Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear function#As a polynomia ...

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 * Klee and Minty gave an example showing that the simplex algorithm can take exponentially many steps to solve a linear program.

"Полиномиальный алгоритм в линейном программировании"

* . Khachiyan's work on the ellipsoid method. This was the first polynomial time algorithm for linear programming.

Early manuscripts

These are publications that are not necessarily relevant to a mathematician nowadays, but are nonetheless important publications in the

history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples ...

'' 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.

'' Rhind Mathematical Papyrus''

Ahmes Ahmes ( “, a common Egyptian name also transliterated Ahmose (disambiguation), Ahmose) was an ancient Egyptian scribe who lived towards the end of the 15th Dynasty, Fifteenth Dynasty (and of the Second Intermediate Period) and the beginning of t ...

(

scribe A scribe is a person who serves as a professional copyist, especially one who made copies of manuscripts before the invention of Printing press, automatic printing. The work of scribes can involve copying manuscripts and other texts as well as ...

) One of the oldest mathematical texts, dating to the Second Intermediate Period of

ancient Egypt Ancient Egypt () was a cradle of civilization concentrated along the lower reaches of the Nile River in Northeast Africa. It emerged from prehistoric Egypt around 3150BC (according to conventional Egyptian chronology), when Upper and Lower E ...

. It was copied by the scribe

(properly ''Ahmose'') from an older Middle Kingdom

papyrus Papyrus ( ) is a material similar to thick paper that was used in ancient times as a writing surface. It was made from the pith of the papyrus plant, ''Cyperus papyrus'', a wetland sedge. ''Papyrus'' (plural: ''papyri'' or ''papyruses'') can a ...

. It laid the foundations of Egyptian mathematics and in turn, later influenced Greek and Hellenistic mathematics. Besides describing how to obtain an approximation of π only missing the mark by less than one per cent, it is describes one of the earliest attempts at

squaring the circle Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square (geometry), square with the area of a circle, area of a given circle by using only a finite number of steps with a ...

and in the process provides persuasive evidence against the theory that the

Egyptians Egyptians (, ; , ; ) are an ethnic group native to the Nile, Nile Valley in Egypt. Egyptian identity is closely tied to Geography of Egypt, geography. The population is concentrated in the Nile Valley, a small strip of cultivable land stretchi ...

deliberately built their

pyramid A pyramid () is a structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a pyramid in the geometric sense. The base of a pyramid can be of any polygon shape, such as trian ...

s to enshrine the value of π in the proportions. Even though it would be a strong overstatement to suggest that the papyrus represents even rudimentary attempts at analytical geometry, Ahmes did make use of a kind of an analogue of the

cotangent In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...

'' Archimedes Palimpsest''

* Archimedes of Syracuse Although the only mathematical tools at its author's disposal were what we might now consider secondary-school

, he used those methods with rare brilliance, explicitly using

infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...

s to solve problems that would now be treated by integral calculus. Among those problems were that of the center of gravity of a solid hemisphere, that of the center of gravity of a frustum of a circular paraboloid, and that of the area of a region bounded by a

and one of its secant lines. For explicit details of the method used, see Archimedes' use of infinitesimals.

'' The Sand Reckoner''

* Archimedes of Syracuse 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 In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
''

* David Dummit and Richard Foote '' Dummit and Foote'' has become the modern dominant abstract algebra textbook following Jacobson's Basic Algebra.

''Arithmetika Horvatzka''

* Mihalj Šilobod Bolšić ''Arithmetika Horvatzka'' (1758) was the first Croatian language arithmetic textbook, written in the vernacular

Kajkavian Kajkavian is a South Slavic languages, South Slavic supradialect or language spoken primarily by Croats in much of Central Croatia and Gorski Kotar. It is part of the South Slavic dialect continuum, being transitional to the supradialects of Č ...

dialect of

Croatian language Croatian (; ) is the standard language, standardised Variety (linguistics)#Standard varieties, variety of the Serbo-Croatian pluricentric language mainly used by Croats. It is the national official language and literary standard of Croatia, o ...

. It established a complete system of

terminology in Croatian, and vividly used examples from everyday life in

Croatia Croatia, officially the Republic of Croatia, is a country in Central Europe, Central and Southeast Europe, on the coast of the Adriatic Sea. It borders Slovenia to the northwest, Hungary to the northeast, Serbia to the east, Bosnia and Herze ...

to present mathematical operations. 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

terminology to Kaikavian use. Full text o
''Arithmetika Horvatszka''
is available via archive.org.

'' Synopsis of Pure Mathematics''

* G. S. Carr 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 Ramanujan
(first half here)

'' Éléments de mathématique''

Nicolas Bourbaki Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intende ...

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''

* Robert Recorde 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''

* Thomas Dilworth An early and popular English arithmetic textbook published in America in the 18th century. The book reached from the introductory topics to the advanced in five sections.

''Geometry''

* Andrei Kiselyov Publication data: 1892 The most widely used and influential textbook in Russian mathematics. (See Kiselyov page.)

'' A Course of Pure Mathematics''

G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...

A classic textbook in introductory

, written by

. It was first published in 1908, and went through many editions. It was intended to help reform mathematics teaching in the UK, and more specifically in the

University of Cambridge The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...

, and in schools preparing pupils to study mathematics at Cambridge. As such, it was aimed directly at "scholarship level" students – the top 10% to 20% by ability. The book contains a large number of difficult problems. The content covers introductory

calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

and the theory of infinite series.

'' Moderne Algebra''

* B. L. van der Waerden 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 Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
''

* Saunders Mac Lane and

Garrett Birkhoff Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician. He is best known for his work in lattice theory. The mathematician George Birkhoff (1884–1944) was his father. Life The son of the mathematician Ge ...

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''

* Tom M. Apostol

''
Algebraic Geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
''

* Robin Hartshorne 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''

Paul Halmos Paul Richard Halmos (; 3 March 1916 – 2 October 2006) was a Kingdom of Hungary, Hungarian-born United States, American mathematician and probabilist who made fundamental advances in the areas of mathematical logic, probability theory, operat ...

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

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 cardinal In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least ...

s. Instead it aims, and succeeds, in being intelligible to someone who has never thought about set theory before.

'' Cardinal and Ordinal Numbers''

Wacław Sierpiński Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions ...

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''

Kenneth Kunen Herbert Kenneth Kunen (August 2, 1943August 14, 2020) was a professor of mathematics at the University of Wisconsin–Madison who worked in set theory and its applications to various areas of mathematics, such as set-theoretic topology and ...

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''

* John L. Kelley 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''

* John Milnor 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''

and E. M. Wright ''

An Introduction to the Theory of Numbers ''An Introduction to the Theory of Numbers'' is a classic textbook in the field of number theory, by G. H. Hardy and E. M. Wright. It is on the list of 173 books essential for undergraduate math libraries. The book grew out of a series of le ...

'' 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''

* Claire Voisin

Handbooks

'' Bronshtein and Semendyayev''

* Ilya Nikolaevich Bronshtein and Konstantin Adolfovic Semendyayev '' Bronshtein and Semendyayev'' is the informal name of a comprehensive handbook of fundamental working knowledge of mathematics and table of formulas originally compiled by the Russian mathematician Ilya Nikolaevich Bronshtein and engineer Konstantin Adolfovic Semendyayev. The work was first published in 1945 in Russia and soon became a "standard" and frequently used guide for scientists, engineers, and technical university students. It has been translated into German, English, and many other languages. The latest edition was published in 2015 by

Springer Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinationa ...

'' CRC Standard Mathematical Tables''

* Editors-in-chief: Charles D. Hodgman (14th edition and earlier); Samuel M. Selby (15th–23rd editions); William H. Beyer (24th–29th editions); Daniel Zwillinger (30th–33rd editions) '' CRC Standard Mathematical Tables'' is a comprehensive one-volume handbook of fundamental working knowledge of mathematics and table of formulas. The handbook was originally published in 1928. The latest edition was published in 2018 by

CRC Press The CRC Press, LLC is an American publishing group that specializes in producing technical books. Many of their books relate to engineering, science and mathematics. Their scope also includes books on business, forensics and information technol ...

, with Daniel Zwillinger as the editor-in-chief.

Popular writings

''Gödel, Escher, Bach''

Douglas Hofstadter Douglas Richard Hofstadter (born 15 February 1945) is an American cognitive and computer scientist whose research includes concepts such as the sense of self in relation to the external world, consciousness, analogy-making, Strange loop, strange ...

Gödel, Escher, Bach ''Gödel, Escher, Bach: an Eternal Golden Braid'' (abbreviated as ''GEB'') is a 1979 nonfiction book by American cognitive scientist Douglas Hofstadter. By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. Esc ...

: 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''

* James R. Newman '' The World of Mathematics'' was specially designed to make mathematics more accessible to the inexperienced. It comprises nontechnical essays on every aspect of the vast subject, including articles by and about scores of eminent mathematicians, as well as literary figures, economists, biologists, and many other eminent thinkers. Includes the work of Archimedes, Galileo, Descartes, Newton, Gregor Mendel, Edmund Halley, Jonathan Swift, John Maynard Keynes, Henri Poincaré, Lewis Carroll, George Boole, Bertrand Russell, Alfred North Whitehead, John von Neumann, and many others. In addition, an informative commentary by distinguished scholar James R. Newman precedes each essay or group of essays, explaining their relevance and context in the history and development of mathematics. Originally published in 1956, it does not include many of the exciting discoveries of the later years of the 20th century but it has no equal as a general historical survey of important topics and applications.

References

{{DEFAULTSORT:Important Publications in Mathematics

Publications To publish is to make Content (media), content available to the general public.Berne Conv ...

Mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

Algebra

Theory of equations

'' Baudhayana Sulba Sutra''

'' The Nine Chapters on the Mathematical Art''

''Arithmetica Diophantus of Alexandria () (; ) was a Greek mathematics, Greek mathematician who was the author of the ''Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations ...''

'' Haidao Suanjing''

'' Sunzi Suanjing''

'' Aryabhatiya''

'' Jigu Suanjing''

'' Brāhmasphuṭasiddhānta''

'' Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa'l-muqābala''

'' Līlāvatī'', '' Siddhānta Shiromani'' and '' Bijaganita''

'' Yigu yanduan''

'' Mathematical Treatise in Nine Sections''

'' Ceyuan haijing''

'' Jade Mirror of the Four Unknowns''

'' Ars Magna''

''Vollständige Anleitung zur Algebra''

''Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse''

Abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

Group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

=''Réflexions sur la résolution algébrique des équations''

''Articles Publiés par Galois dans les Annales de Mathématiques''

''Traité des substitutions et des équations algébriques''

''Theorie der Transformationsgruppen''

''Solvability of groups of odd order''

Homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

''Homological Algebra''

" Sur Quelques Points d'Algèbre Homologique"

Algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

''Theorie der Abelschen Functionen''

''Faisceaux Algébriques Cohérents''

'' Géométrie Algébrique et Géométrie Analytique''

''Le théorème de Riemann–Roch, d'après A. Grothendieck''

'' Éléments de géométrie algébrique''

'' Séminaire de géométrie algébrique''

Number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

'' Brāhmasphuṭasiddhānta''

''De fractionibus continuis dissertatio''

''Recherches d'Arithmétique''

'' Disquisitiones Arithmeticae''

"Beweis des Satzes, daß jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält"

" Über die Anzahl der Primzahlen unter einer gegebenen Grösse"

'' Vorlesungen über Zahlentheorie''

'' Zahlbericht''

''Fourier Analysis in Number Fields and Hecke's Zeta-Functions''

" Automorphic Forms on GL(2)"

"La conjecture de Weil. I."

"Endlichkeitssätze für abelsche Varietäten über Zahlkörpern"

"Modular Elliptic Curves and Fermat's Last Theorem"

''The geometry and cohomology of some simple Shimura varieties''

"Le lemme fondamental pour les algèbres de Lie"

"Perfectoid space"

Analysis

''Introductio in analysin infinitorum''

'' Yuktibhāṣā''

Calculus

''Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illi calculi genus''

'' Philosophiae Naturalis Principia Mathematica''

''Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum''

''Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe''

''Intégrale, longueur, aire''

Complex analysis

''Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse''

Functional analysis

''Théorie des opérations linéaires''

''Produits Tensoriels Topologiques et Espaces Nucléaires''

''Sur certains espaces vectoriels topologiques''

Fourier analysis

''Mémoire sur la propagation de la chaleur dans les corps solides''

''Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données''

''On convergence and growth of partial sums of Fourier series''

Geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

'' Baudhayana Sulba Sutra''

''Euclid's'' ''Elements''

'' The Nine Chapters on the Mathematical Art''

'' The Conics''

'' Surya Siddhanta''

'' Aryabhatiya''

'' La Géométrie''

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Arithmetica Diophantus of Alexandria () (; ) was a Greek mathematics, Greek mathematician who was the author of the ''Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations ...
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Categories for the Working Mathematician ''Categories for the Working Mathematician'' (''CWM'') is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. It was first published in 1971, and is based ...
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Begriffsschrift ''Begriffsschrift'' (German for, roughly, "concept-writing") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept notati ...
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Formulario mathematico ''Formulario Mathematico'' (Latino sine flexione: ''Formulary for Mathematics'') is a book by Giuseppe Peano which expresses fundamental theorems of mathematics in a Symbolic language (mathematics), symbolic language developed by Peano. The autho ...
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Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1 ...
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On Numbers and Games ''On Numbers and Games'' is a mathematics book by John Horton Conway first published in 1976. The book is written by a pre-eminent mathematician, and is directed at other mathematicians. The material is, however, developed in a playful and unpr ...
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A Mathematical Theory of Communication "A Mathematical Theory of Communication" is an article by mathematician Claude E. Shannon published in '' Bell System Technical Journal'' in 1948. It was renamed ''The Mathematical Theory of Communication'' in the 1949 book of the same name, a s ...
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Fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
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Abstract Algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
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Algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
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