Mémoire sur la propagation de la chaleur dans les corps solides
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This is a list of important publications in mathematics, organized by field. Some reasons why 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 David Eugene Smith (January 21, 1860 – July 29, 1944) was an American mathematician, educator, and editor. Education and career David Eugene Smith is considered one of the founders of the field of mathematics education. Smith was born in Cortl ...
.


Algebra


Theory of equations In algebra, the theory of equations is the study of algebraic equations (also called "polynomial equations"), which are equations defined by a polynomial. The main problem of the theory of equations was to know when an algebraic equation has an ...


''

Baudhayana The (Sanskrit: बौधायन) are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics and is one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. Th ...
Sulba Sutra''

*
Baudhayana The (Sanskrit: बौधायन) are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics and is one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. Th ...
(8th century BCE) Believed to have been written around the 8th century BCE, this is one of the oldest mathematical texts. It laid the foundations of Indian mathematics and was influential in
South Asia South Asia is the southern subregion of Asia, which is defined in both geographical Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth descr ...
and its surrounding regions, and perhaps even Greece. Though this was primarily a geometrical text, it also contained some important algebraic developments, including the earliest list of Pythagorean triples discovered algebraically, geometric solutions of linear equations, the earliest use of quadratic equations of the forms ax2 = c and ax2 + bx = c, and integral solutions of simultaneous Diophantine equations with up to four unknowns.


''

The Nine Chapters on the Mathematical Art ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...
''

* ''
The Nine Chapters on the Mathematical Art ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...
'' from the 10th–2nd century BCE. Contains the earliest description of Gaussian elimination for solving system of linear equations, it also contains method for finding square root and cubic root.


''

Haidao Suanjing ''Haidao Suanjing'' (; ''The Sea Island Mathematical Manual'') was written by the Chinese mathematician Liu Hui of the Three Kingdoms era (220–280) as an extension of chapter 9 of ''The Nine Chapters on the Mathematical Art''. L. van. He ...
''

*
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 Suanjing'' () was a mathematical treatise written during 3rd to 5th centuries AD which was listed as one of the Ten Computational Canons during the Tang dynasty. The specific identity of its author Sunzi (lit. "Master Sun") is still ...
''

*Sunzi (5th century CE) Contains the earliest description of Chinese remainder theorem.


'' Aryabhatiya''

*
Aryabhata Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which ...
(499 CE) Aryabhata introduced the method known as "Modus Indorum" or the method of the Indians that has become our algebra today. This algebra came along with the Hindu Number system to Arabia and then migrated to Europe. The text contains 33 verses covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations. It also gave the modern standard algorithm for solving first-order diophantine equations.


'' Jigu Suanjing''

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


''

Brāhmasphuṭasiddhānta The ''Brāhmasphuṭasiddhānta'' ("Correctly Established Doctrine of Brahma", abbreviated BSS) is the main work of Brahmagupta, written c. 628. This text of mathematical astronomy contains significant mathematical content, including a good underst ...
''

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


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

*
Muhammad ibn Mūsā al-Khwārizmī Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...
(820 CE) The first book on the systematic
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
ic solutions of
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not q ...
s by the
Persian Persian may refer to: * People and things from Iran, historically called ''Persia'' in the English language ** Persians, the majority ethnic group in Iran, not to be conflated with the Iranic peoples ** Persian language, an Iranian language of the ...
scholar
Muhammad ibn Mūsā al-Khwārizmī Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...
. The book is considered to be the foundation of modern
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
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''

*Liu Yi (12th century) Contains the earliest invention of 4th order polynomial equation.


''

Mathematical Treatise in Nine Sections The ''Mathematical Treatise in Nine Sections'' () is a mathematical text written by Chinese Southern Song dynasty mathematician Qin Jiushao in the year 1247. The mathematical text has a wide range of topics and is taken from all aspects of th ...
''

*
Qin Jiushao Qin Jiushao (, ca. 1202–1261), courtesy name Daogu (道古), was a Chinese mathematician, meteorologist, inventor, politician, and writer. He is credited for discovering Horner's method as well as inventing Tianchi basins, a type of rain gau ...
(1247) This 13th century book contains the earliest complete solution of 19th century
Horner's method In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horn ...
of solving high order polynomial equations (up to 10th order). It also contains a complete solution of Chinese remainder theorem, which predates Euler and
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
by several centuries.


'' Ceyuan haijing''

*
Li Zhi Li Zhi may refer to: *Emperor Gaozong of Tang (628–683), named Li Zhi, Emperor of China *Li Ye (mathematician) (1192–1279), Chinese mathematician and scholar, birth name Li Zhi *Li Zhi (philosopher) (1527–1602), Chinese philosopher from the M ...
(1248) Contains the application of high order polynomial equation in solving complex geometry problems.


'' Jade Mirror of the Four Unknowns''

*
Zhu Shijie Zhu Shijie (, 1249–1314), courtesy name Hanqing (), pseudonym Songting (), was a Chinese mathematician and writer. He was a Chinese mathematician during the Yuan Dynasty. Zhu was born close to today's Beijing. Two of his mathematical works ha ...
(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; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
(1545) Otherwise known as ''The Great Art'', provided the first published methods for solving cubic and
quartic equation In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is :ax^4+bx^3+cx^2+dx+e=0 \, where ''a'' ≠ 0. The quartic is the highest order polynomi ...
s (due to
Scipione del Ferro Scipione del Ferro (6 February 1465 – 5 November 1526) was an Italian mathematician who first discovered a method to solve the depressed cubic equation. Life Scipione del Ferro was born in Bologna, in northern Italy, to Floriano and Filip ...
,
Niccolò Fontana Tartaglia Niccolò Fontana Tartaglia (; 1499/1500 – 13 December 1557) was an Italian mathematician, engineer (designing fortifications), a surveyor (of topography, seeking the best means of defense or offense) and a bookkeeper from the then Republi ...
, and
Lodovico Ferrari Lodovico de Ferrari (2 February 1522 – 5 October 1565) was an Italian mathematician. Biography Born in Bologna, Lodovico's grandfather, Bartolomeo Ferrari, was forced out of Milan to Bologna. Lodovico settled in Bologna, and he began his ...
), 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 ...
.


''Vollständige Anleitung zur Algebra''

*
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
(1770) Also known as
Elements of Algebra ''Elements of Algebra'' is an elementary mathematics textbook written by mathematician Leonhard Euler around 1765 in German. It was first published in Russian as "''Universal Arithmetic''" (''Универсальная арифметика''), tw ...
, Euler's textbook on elementary algebra is one of the first to set out algebra in the modern form we would recognize today. The first volume deals with determinate equations, while the second part deals with Diophantine equations. The last section contains a proof of
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...
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 (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
(1799) Gauss' doctoral dissertation, which contained a widely accepted (at the time) but incomplete proof of the
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
.


Abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...


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 rati ...
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 groups,
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 theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
. The Lagrange resolvent also introduced the
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
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 radicals, ...
by
Joseph Liouville Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...
. 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 In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f ...
(which he called ''l'isomorphisme mériédrique''), proved part of the
Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natur ...
, and discussed matrix groups over finite fields as well as the
Jordan normal form In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...
.


''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. Life and career Marius Soph ...
, 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 group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the gr ...
s, serving as the foundation for the modern theory of Lie groups.


''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 the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...
.


Homological algebra


''Homological Algebra''

* Henri Cartan and
Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to ...
(1956) Provided the first fully worked out treatment of abstract homological algebra, unifying previously disparate presentations of homology and cohomology for associative algebras, Lie algebras, and
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
s into a single theory.


" Sur Quelques Points d'Algèbre Homologique"

* Alexander Grothendieck (1957) Often referred to as the "Tôhoku paper", it revolutionized homological algebra by introducing
abelian categories In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
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 ...
s.


Algebraic geometry


''Theorie der Abelschen Functionen''

* Bernhard Riemann (1857) Publication data: ''Journal für die Reine und Angewandte Mathematik'' Developed the concept of Riemann surfaces and their topological properties beyond Riemann's 1851 thesis work, proved an index theorem for the genus (the original formulation of the
Riemann–Hurwitz formula In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ''ramified covering'' of the other. It therefore connects ramif ...
), proved the Riemann inequality for the dimension of the space of meromorphic functions with prescribed poles (the original formulation of the
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
), 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 ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepherd ...
and Jacobi. André Weil once wrote that this paper "''is one of the greatest pieces of mathematics that has ever been written; there is not a single word in it that is not of consequence.''"


''Faisceaux Algébriques Cohérents''

*
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
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 manifolds. Serre introduced
Čech cohomology In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech. Motivation Let ''X'' be a topol ...
of sheaves in this paper, and, despite some technical deficiencies, revolutionized formulations of algebraic geometry. For example, the
long exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
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 In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with ref ...
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, ...
, and such dimensions include many discrete invariants of varieties, for example Hodge numbers. While Grothendieck's
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 ...
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 Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
(1956) In mathematics, algebraic geometry and analytic geometry are closely related subjects, where ''analytic geometry'' is the theory of complex manifolds and the more general
analytic space An analytic space is a generalization of an analytic manifold that allows singularities. An analytic space is a space that is locally the same as an analytic variety. They are prominent in the study of several complex variables, but they also ...
s 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 analytic geometry 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 Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in alg ...
,
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
(1958) Borel and Serre's exposition of Grothendieck's version of the
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
, 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 Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as ...
proved in 1953 in the framework of
morphisms In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
between varieties, resulting in a sweeping generalization. In his proof, Grothendieck broke new ground with his concept of Grothendieck groups, which led to the development of
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
.


''

Éléments de géométrie algébrique The ''Éléments de géométrie algébrique'' ("Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or ''EGA'' for short, is a rigorous treatise, in French, on algebraic geometry that was published (in eight ...
''

* Alexander Grothendieck (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''

* Alexander Grothendieck 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 Crafoord P ...
's proof of the last of the open
Weil conjectures In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. Th ...
in the early 1970s. Other authors who worked on one or several volumes of SGA include
Michel Raynaud Michel Raynaud (; 16 June 1938 – 10 March 2018 Décès de Michel Raynaud
So ...
,
Michael Artin Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
, Jean-Louis Verdier,
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 Crafoord P ...
, and Nicholas Katz.


Number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...


''

Brāhmasphuṭasiddhānta The ''Brāhmasphuṭasiddhānta'' ("Correctly Established Doctrine of Brahma", abbreviated BSS) is the main work of Brahmagupta, written c. 628. This text of mathematical astronomy contains significant mathematical content, including a good underst ...
''

* Brahmagupta (628) Brahmagupta's
Brāhmasphuṭasiddhānta The ''Brāhmasphuṭasiddhānta'' ("Correctly Established Doctrine of Brahma", abbreviated BSS) is the main work of Brahmagupta, written c. 628. This text of mathematical astronomy contains significant mathematical content, including a good underst ...
is the first book that mentions zero as a number, hence Brahmagupta is considered the first to formulate the concept of zero. The current system of the four fundamental operations (addition, subtraction, multiplication and division) based on the Hindu-Arabic number system also first appeared in Brahmasphutasiddhanta. It was also one of the first texts to provide concrete ideas on positive and negative numbers.


''De fractionibus continuis dissertatio''

*
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
(1744) First presented in 1737, this paper provided the first then-comprehensive account of the properties of
continued fractions In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
. 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 form In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables : q(x,y)=ax^2+bxy+cy^2, \, where ''a'', ''b'', ''c'' are the coefficients. When the coefficients can be arbitrary complex numbers, most results are ...
s 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 The (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on th ...
''

*
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
(1801) The ''
Disquisitiones Arithmeticae The (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on th ...
'' is a profound and masterful book on
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
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, structure, space, models, and change. History On ...
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
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 (; between 31 October and 6 December 1607 – 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 ...
, Euler, Lagrange and Legendre 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 can be represented uniquely as a product of prime numbers, up to the ord ...
, 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 form In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables : q(x,y)=ax^2+bxy+cy^2, \, where ''a'', ''b'', ''c'' are the coefficients. When the coefficients can be arbitrary complex numbers, most results are ...
s going beyond Lagrange's work in ''Recherches d'Arithmétique'', a first appearance of Gauss sums, cyclotomy, and the theory of
constructible polygon In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinite ...
s with a particular application to the constructibility of the regular
17-gon In geometry, a heptadecagon, septadecagon or 17-gon is a seventeen-sided polygon. Regular heptadecagon A ''regular polygon, regular heptadecagon'' is represented by the Schläfli symbol . Construction As 17 is a Fermat prime, the regular he ...
. 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 mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 199 ...
. 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 who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
(1837) Pioneering paper in analytic number theory, 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"

* Bernhard Riemann (1859) "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" (or "On the Number of Primes Less Than a Given Magnitude") is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the ''Monthly Reports of the Berlin Academy''. Although it is the only paper he ever published on number theory, it contains ideas which influenced dozens of researchers during the late 19th century and up to the present day. The paper consists primarily of definitions, heuristic arguments, sketches of proofs, and the application of powerful analytic methods; all of these have become essential concepts and tools of modern analytic number theory. It also contains the famous Riemann Hypothesis, one of the most important open problems in mathematics.


''

Vorlesungen über Zahlentheorie (German for ''Lectures on Number Theory'') is the name of several different textbooks of number theory. The best known was written by Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863. Others were written by Leopold Krone ...
''

*
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
and Richard Dedekind ''
Vorlesungen über Zahlentheorie (German for ''Lectures on Number Theory'') is the name of several different textbooks of number theory. The best known was written by Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863. Others were written by Leopold Krone ...
'' (''Lectures on Number Theory'') is a textbook of
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
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
Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 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 ...
, Jacobi and
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, and the modern number theory of Dedekind,
Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
and
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
. Dirichlet does not explicitly recognise the concept of the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
that is central to
modern algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ...
, but many of his proofs show an implicit understanding of group theory.


''

Zahlbericht In mathematics, the ''Zahlbericht'' (number report) was a report on algebraic number theory by . History In 1893 the German mathematical society invited Hilbert and Minkowski to write reports on the theory of numbers. They agreed that Minkowski ...
''

* David Hilbert (1897) Unified and made accessible many of the developments in algebraic number theory made during the nineteenth century. Although criticized by André Weil (who stated "''more than half of his famous Zahlbericht is little more than an account of Kummer's number-theoretical work, with inessential improvements''") and
Emmy Noether Amalie Emmy NoetherEmmy is the '' Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...
, 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 In number theory, Tate's thesis is the 1950 PhD thesis of completed under the supervision of Emil Artin at Princeton University. In it, Tate used a translation invariant integration on the locally compact group of ideles to lift the zeta function ...
'', Tate's
Princeton Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ni ...
PhD thesis, under Emil Artin, is a reworking of
Erich Hecke Erich Hecke (20 September 1887 – 13 February 1947) was a German mathematician known for his work in number theory and the theory of modular forms. Biography Hecke was born in Buk, Province of Posen, German Empire (now Poznań, Poland). He o ...
'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 Hervé Jacquet is a French American mathematician, working in automorphic forms. He is considered one of the founders of the theory of automorphic representations and their associated L-functions, and his results play a central role in modern num ...
and
Robert Langlands Robert Phelan Langlands, (; born October 6, 1936) is a Canadian mathematician. He is best known as the founder of the Langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study o ...
(1970) This publication offers evidence towards Langlands' conjectures by reworking and expanding the classical theory of modular forms and their ''L''-functions through the introduction of representation theory.


"La conjecture de Weil. I."

*
Pierre Deligne 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 Crafoord P ...
(1974) Proved the Riemann hypothesis for varieties over finite fields, settling the last of the open
Weil conjectures In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. Th ...
.


"Endlichkeitssätze für abelsche Varietäten über Zahlkörpern"

*
Gerd Faltings Gerd Faltings (; born 28 July 1954) is a German mathematician known for his work in arithmetic geometry. Education From 1972 to 1978, Faltings studied mathematics and physics at the University of Münster. In 1978 he received his PhD in mathema ...
(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 c ...
(relating the
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
s between two
abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law th ...
over a
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
to the homomorphisms between their
Tate module In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group ''A''. Often, this construction is made in the following situation: ''G'' is a commutative group scheme over a field ''K'', ...
s) 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, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awa ...
(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
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 integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...
. The proof's method of identification of a
deformation ring In mathematics, a deformation ring is a ring that controls liftings of a representation of a Galois group from a finite field to a local field. In particular for any such lifting problem there is often a universal deformation ring that classifies a ...
with a
Hecke algebra In mathematics, the Hecke algebra is the algebra generated by Hecke operators. Properties The algebra is a commutative ring. In the classical elliptic modular form theory, the Hecke operators ''T'n'' with ''n'' coprime to the level acting o ...
(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 Ngô Bảo Châu (, born June 28, 1972) is a Vietnamese-French mathematician at the University of Chicago, best known for proving the fundamental lemma for automorphic forms (proposed by Robert Langlands and Diana Shelstad). He is the first ...
(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 Peter Scholze (; born 11 December 1987) is a German mathematician known for his work in arithmetic geometry. He has been a professor at the University of Bonn since 2012 and director at the Max Planck Institute for Mathematics since 2018. He ha ...
(2012) Peter Scholze introduced Perfectoid space.


Analysis


''Introductio in analysin infinitorum''

*
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
(1748) The eminent historian of mathematics
Carl Boyer Carl Benjamin Boyer (November 3, 1906 – April 26, 1976) was an American historian of sciences, and especially mathematics. Novelist David Foster Wallace called him the " Gibbon of math history". It has been written that he was one of few hist ...
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 (3 ...
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 In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
, 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 Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that fo ...
and a statement of the
pentagonal number theorem In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that :\prod_^\left(1-x^\right)=\sum_^\left(-1\right)^x^=1+\sum_^\infty(-1)^k\left(x^+x^\right ...
, which he had discovered earlier and would publish a proof for in 1751.


Calculus


'' Yuktibhāṣā''

* Jyeshtadeva (1501) Written in
India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...
in 1530, this was the world's first calculus text. "This work laid the foundation for a complete system of fluxions" and served as a summary of the Kerala School's achievements in calculus,
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies ...
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 (m ...
, most of which were earlier discovered by the 14th century mathematician
Madhava Mādhava means Lord Krishna an incarnation of Vishnu. It may also refer to: *a Sanskrit patronymic, "descendant of Madhu (a man of the Yadu tribe)". ** especially of Krishna, see Madhava (Vishnu) *** an icon of Krishna ** Madhava of Sangamagrama, ...
. It is possible that this text influenced the later development of calculus in Europe. Some of its important developments in calculus include: the fundamental ideas of differentiation and integration, the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
,
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s, term by term integration, numerical integration by means of infinite series, the relationship between the area of a curve and its integral, and the mean value theorem.


''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 (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
(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 (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
(1687) The ''Philosophiae Naturalis Principia Mathematica'' (
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
: "mathematical principles of natural philosophy", often ''Principia'' or ''Principia Mathematica'' for short) is a three-volume work by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
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 basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...
forming the foundation of
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
as well as his law of universal gravitation, and derives
Kepler's laws In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbi ...
for the motion of the
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
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''

*
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
(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 A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
and makes a thorough investigation of how differentiation behaves under substitutions. Also included is a systematic study of Bernoulli polynomials and the
Bernoulli numbers In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
(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 , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
), and an application of partial fractions to differentiation.


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

* Bernhard Riemann (1867) Written in 1853, Riemann's work on trigonometric series was published posthumously. In it, he extended Cauchy's definition of the integral to that of the
Riemann integral 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 G ...
, 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 In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms ...
, proved the
Riemann–Lebesgue lemma In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an ''L''1 function vanishes at infinity. It is of importance in harmonic analysis and asymptot ...
for the case of bounded Riemann integrable functions, and developed the Riemann localization principle.


''Intégrale, longueur, aire''

*
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
(1901) Lebesgue's
doctoral dissertation A thesis ( : theses), or dissertation (abbreviated diss.), is a document submitted in support of candidature for an academic degree or professional qualification presenting the author's research and findings.International Standard ISO 7144: ...
, summarizing and extending his research to date regarding his development of measure theory and the
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
.


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 In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
, conformal mapping, simple connectivity, the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
, 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 C which is not all of C, then there exists a biholomorphic mapping ''f'' (i.e. a bijective holomorphi ...
.


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 origina ...
(1932; originally published 1931 in Polish under the title ''Teorja operacyj''.) * The first mathematical monograph on the subject of
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
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 (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
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 spaces 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 In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous. Graphs and m ...
, and
Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
.


''Produits Tensoriels Topologiques et Espaces Nucléaires''

* Grothendieck's thesis introduced the notion of a
nuclear space In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces ...
, tensor products of locally convex topological vector spaces, and the start of Grothendieck's work on tensor products of Banach spaces. Alexander Grothendieck 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 (1807) Introduced Fourier analysis, specifically Fourier series. Key contribution was to not simply use
trigonometric series In mathematics, a trigonometric series is a infinite series of the form : \frac+\displaystyle\sum_^(A_ \cos + B_ \sin), an infinite version of a trigonometric polynomial. It is called the Fourier series of the integrable function f if the term ...
, but to model ''all'' functions by trigonometric series: When Fourier submitted his paper in 1807, the committee (which included Lagrange, Laplace,
Malus ''Malus'' ( or ) is a genus of about 30–55 species of small deciduous trees or shrubs in the family Rosaceae, including the domesticated orchard apple, crab apples, wild apples, and rainberries. The genus is native to the temperate zone ...
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
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
.


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

*
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
(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 Fourier series under fairly general conditions (piecewise continuity and monotonicity) by considering partial sums, which Dirichlet transformed into a particular Dirichlet integral involving what is now called the
Dirichlet kernel In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as D_n(x)= \sum_^n e^ = \left(1+2\sum_^n\cos(kx)\right)=\frac, where is any nonneg ...
. This paper introduced the nowhere continuous
Dirichlet function In mathematics, the Dirichlet function is the indicator function 1Q or \mathbf_\Q of the set of rational numbers Q, i.e. if ''x'' is a rational number and if ''x'' is not a rational number (i.e. an irrational number). \mathbf 1_\Q(x) = \begin 1 & ...
and an early version of the
Riemann–Lebesgue lemma In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an ''L''1 function vanishes at infinity. It is of importance in harmonic analysis and asymptot ...
.


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

*
Lennart Carleson Lennart Axel Edvard Carleson (born 18 March 1928) is a Swedish mathematician, known as a leader in the field of harmonic analysis. One of his most noted accomplishments is his proof of Lusin's conjecture. He was awarded the Abel Prize in 2006 fo ...
(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, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...


''

Baudhayana The (Sanskrit: बौधायन) are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics and is one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. Th ...
Sulba Sutra''

*
Baudhayana The (Sanskrit: बौधायन) are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics and is one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. Th ...
Written around the 8th century BC, this is one of the oldest geometrical texts. It laid the foundations of Indian mathematics and was influential in
South Asia South Asia is the southern subregion of Asia, which is defined in both geographical Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth descr ...
and its surrounding regions, and perhaps even Greece. Among the important geometrical discoveries included in this text are: the earliest list of Pythagorean triples discovered algebraically, the earliest statement of the Pythagorean theorem, geometric solutions of linear equations, several approximations of π, the first use of irrational numbers, and an accurate computation of the square root of 2, correct to a remarkable five decimal places. Though this was primarily a geometrical text, it also contained some important algebraic developments, including the earliest use of quadratic equations of the forms ax2 = c and ax2 + bx = c, and integral solutions of simultaneous Diophantine equations with up to four unknowns.


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

*
Euclid Euclid (; grc-gre, Εὐκλείδης; 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 ...
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, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
but one of the most important works in mathematics. It contains many important results in plane and solid
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, algebra (books II and V), and
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
(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 ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...
''

* Unknown author This was a Chinese mathematics book, mostly geometric, composed during the
Han Dynasty The Han dynasty (, ; ) was an imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by the short-lived Qin dynasty (221–207 BC) and a warr ...
, perhaps as early as 200 BC. It remained the most important textbook in China and
East Asia East Asia is the eastern region of Asia, which is defined in both Geography, geographical and culture, ethno-cultural terms. The modern State (polity), states of East Asia include China, Japan, Mongolia, North Korea, South Korea, and Taiwan. ...
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 Gaussian elimination. Problems involving the principle known in the West as the Pythagorean theorem. The earliest solution of a matrix using a method equivalent to the modern method.


'' The Conics''

* Apollonius of Perga The Conics was written by Apollonius of Perga, a
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
mathematician. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importance ...
, Francesco Maurolico,
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
, and
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 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 science. Ma ...
. It was Apollonius who gave the ellipse, 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 exact ...
, and the
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
the names by which we know them.


'' Surya Siddhanta''

* Unknown (400 CE) Contains the roots of modern trigonometry. It describes the archeo-astronomy theories, principles and methods of the ancient Hindus. This siddhanta is supposed to be the knowledge that the Sun god gave to an Asura called Maya. It uses sine (jya), cosine (kojya or "perpendicular sine") and inverse sine (otkram jya) for the first time, and also contains the earliest use of the tangent and secant. Later Indian mathematicians such as Aryabhata made references to this text, while later Arabic and Latin translations were very influential in Europe and the Middle East.


'' Aryabhatiya''

*
Aryabhata Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which ...
(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''

*
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 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 science. Ma ...
La Géométrie was
published Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for free. Traditionally, the term refers to the creation and distribution of printed works, such as books, news ...
in 1637 and
written Writing is a medium of human communication which involves the representation of a language through a system of physically inscribed, mechanically transferred, or digitally represented symbols. Writing systems do not themselves constitute h ...
by
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 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 science. Ma ...
. The book was influential in developing the Cartesian coordinate system and specifically discussed the representation of points of a
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
, 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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 equations.


''Grundlagen der Geometrie''

* David Hilbert 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 In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
''

* 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 transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
s, the generalisation of regular
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
s and regular
polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...
to higher dimensions. Originating with an essay entitled ''Dimensional Analogy'' written in 1923, the first edition of the book took Coxeter 24 years to complete. Originally written in 1947, the book was updated and republished in 1963 and 1973.


Differential geometry


''Recherches sur la courbure des surfaces''

*
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
(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
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s, 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 surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspective ...
.


''Disquisitiones generales circa superficies curvas''

*
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
(1827) Publication data:''
"Disquisitiones generales circa superficies curvas"
''Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores'' Vol. VI (1827), pp. 99–146;
General Investigations of Curved Surfaces
(published 1965) Raven Press, New York, translated by A.M.Hiltebeitel and J.C.Morehead. Groundbreaking work in differential geometry, introducing the notion of Gaussian curvature and Gauss' celebrated
Theorema Egregium Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determi ...
.


''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,
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
, and curvature tensor.


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

*
Gaston Darboux Jean-Gaston Darboux FAS MIF FRS FRSE (14 August 1842 – 23 February 1917) was a French mathematician. Life According this birth certificate he was born in Nîmes in France on 14 August 1842, at 1 am. However, probably due to the midnigh ...
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 differential geometry of
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s.


Topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...


''Analysis situs''

* Henri Poincaré (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 invariants that classify topological spaces up to homeomorphism, though usually most classify ...
. In these papers, Poincaré introduced the notions of homology and the fundamental group, provided an early formulation of
Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
, gave the Euler–Poincaré characteristic for
chain complexes In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
, and mentioned several important conjectures including the
Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured ...
, demonstrated by
Grigori Perelman Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
in 2003.


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

*
Jean Leray Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. Life and career He was born in Chantenay-sur-Loire (today part of Nantes). He studied at Éc ...
(1946) These two Comptes Rendus notes of Leray from 1946 introduced the novel concepts of sheafs, sheaf cohomology, and
spectral sequences In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they hav ...
, 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 Jean-Louis Koszul (; January 3, 1921 – January 12, 2018) was a French mathematician, best known for studying geometry and discovering the Koszul complex. He was a second generation member of Bourbaki. Biography Koszul was educated at the in ...
,
Armand Borel Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in alg ...
,
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
, 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 In differential topology, the transversality theorem, also known as the Thom transversality theorem after French mathematician René Thom, is a major result that describes the transverse intersection properties of a smooth family of smooth maps. It ...
, introduced the notions of oriented and unoriented cobordism, and demonstrated that cobordism groups could be computed as the homotopy groups of certain
Thom space In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact s ...
s. 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


"General Theory of Natural Equivalences"

*
Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to ...
and
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville ...
(1945) The first paper on category theory. Mac Lane later wrote in ''Categories for the Working Mathematician'' that he and Eilenberg introduced categories so that they could introduce functors, and they introduced functors so that they could introduce natural equivalences. Prior to this paper, "natural" was used in an informal and imprecise way to designate constructions that could be made without making any choices. Afterwards, "natural" had a precise meaning which occurred in a wide variety of contexts and had powerful and important consequences.


'' Categories for the Working Mathematician''

*
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville ...
(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 functor In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
s and universal properties.


'' Higher Topos Theory''

*
Jacob Lurie Jacob Alexander Lurie (born December 7, 1977) is an American mathematician who is a professor at the Institute for Advanced Study. Lurie is a 2014 MacArthur Fellow. Life When he was a student in the Science, Mathematics, and Computer Science ...
(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 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 mathematics, is mostly conce ...


"Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"

*
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
(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 who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, an ...
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 measure theory 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 (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 In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
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 such ...
. 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 science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...


''

The Laws of Thought ''An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities'' by George Boole, published in 1854, is the second of Boole's two monographs on algebraic logic. Boole was a professor of mathem ...
''

*
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 ...
(1854) Published in 1854,
The Laws of Thought ''An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities'' by George Boole, published in 1854, is the second of Boole's two monographs on algebraic logic. Boole was a professor of mathem ...
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, and cryptographer known as a "father of information theory". As a 21-year-old master's degree student at the Massachusetts Inst ...
in the development of digital logic.


''

Begriffsschrift ''Begriffsschrift'' (German for, roughly, "concept-script") 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 notatio ...
''

*
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 ph ...
(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
language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of ...
, modelled on that of arithmetic, of pure thought''". Frege's motivation for developing his
formal logical system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A form ...
was similar to
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
's desire for a ''
calculus ratiocinator The ''calculus ratiocinator'' is a theoretical universal logical calculation framework, a concept described in the writings of Gottfried Leibniz, usually paired with his more frequently mentioned ''characteristica universalis'', a universal conce ...
''. Frege defines a logical calculus to support his research in the
foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
. ''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 science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
since
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...
.


''

Formulario mathematico ''Formulario Mathematico'' (Latino sine flexione: ''Formulary for Mathematics'') is a book There are many editions. Here are two: * (French) Published 1901 by Gauthier-Villars, Paris. 230p.OpenLibrary OL15255022WGiuseppe 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 notation. The sta ...
(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 formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
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 mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
''

*
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
and Alfred North Whitehead (1910–1913) The ''Principia Mathematica'' is a three-volume work on the foundations of mathematics, written by
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
and Alfred North Whitehead 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. 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.


'' 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. Turing was highly influential in the development of theoretical co ...
's PhD thesis


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

(
On Formally Undecidable Propositions of Principia Mathematica and Related Systems "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems I") is a paper in mathematical logic by Kurt Gödel. Submitted November ...
) * Kurt Gödel (1931) Online version:''
Online version
In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
, Gödel's incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1931. The first incompleteness theorem states:
For any formal system such that (1) it is \omega-consistent ( omega-consistent), (2) it has a recursively definable set of
axioms An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
and rules of derivation, and (3) every
recursive Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
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, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
of the system.


Combinatorics


"On sets of integers containing no k elements in arithmetic progression"

*
Endre Szemerédi Endre Szemerédi (; born August 21, 1940) is a Hungarian-American mathematician and computer scientist, working in the field of combinatorics and theoretical computer science. He has been the State of New Jersey Professor of computer science ...
(1975) Settled a conjecture of Paul Erdős and
Pál Turán Pál Turán (; 18 August 1910 – 26 September 1976) also known as Paul Turán, was a Hungarian mathematician who worked primarily in extremal combinatorics. He had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting ...
(now known as
Szemerédi's theorem In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured that every set of integers ''A'' with positive natural density contains a ''k''-ter ...
) 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, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...


''Solutio problematis ad geometriam situs pertinentis''

*
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
(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, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
.


"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 In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate. The maximum flow problem can be seen as a special case of more complex network flow problems, such ...
, along with many ideas on flow-based models.


Computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved ...

''See
List of important publications in theoretical computer science This is a list of important publications in theoretical computer science, organized by field. Some reasons why a particular publication might be regarded as important: *Topic creator – A publication that created a new topic *Breakthrough ...
.''


Probability theory and statistics

''See
list of important publications in statistics This is a list of important publications in statistics, organized by field. Some reasons why a particular publication might be regarded as important: *Topic creator – A publication that created a new topic *Breakthrough – A publicatio ...
.''


Game theory


"Zur Theorie der Gesellschaftsspiele"

*
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
(1928) Went well beyond
Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Biography Borel was ...
's initial investigations into strategic two-person game theory by proving the
minimax theorem In the mathematical area of game theory, a minimax theorem is a theorem providing conditions that guarantee that the max–min inequality is also an equality. The first theorem in this sense is von Neumann's minimax theorem from 1928, which was c ...
for two-person, zero-sum games.


'' Theory of Games and Economic Behavior''

*
Oskar Morgenstern Oskar Morgenstern (January 24, 1902 – July 26, 1977) was an Austrian-American economist. In collaboration with mathematician John von Neumann, he founded the mathematical field of game theory as applied to the social sciences and strategic decis ...
,
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
(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


''

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 unpre ...
''

*
John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
(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 Elwyn Ralph Berlekamp (September 6, 1940 – April 9, 2019) was a professor of mathematics and computer science at the University of California, Berkeley.Contributors, ''IEEE Transactions on Information Theory'' 42, #3 (May 1996), p. 1048. DO10. ...
, John Conway and Richard K. Guy (1982) A compendium of information on
mathematical games A mathematical game is a game whose rules, strategies, and outcomes are defined by clear mathematical parameters. Often, such games have simple rules and match procedures, such as Tic-tac-toe and Dots and Boxes. Generally, mathematical games ne ...
. It was first published in 1982 in two volumes, one focusing on
Combinatorial game theory Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the player ...
and
surreal numbers In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surrea ...
, and the other concentrating on a number of specific games.


Fractals


''

How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by mathematician Benoit Mandelbrot, first published in ''Science'' on 5 May 1967. In this paper, Mandelbrot discusses self-similar curves that ...
''

*
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 roughness" of phy ...
(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 manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...


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


''Method of Fluxions''

*
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
(1736) ''
Method of Fluxions ''Method of Fluxions'' ( la, De Methodis Serierum et Fluxionum) is a mathematical treatise by Sir Isaac Newton which served as the earliest written formulation of modern calculus. The book was completed in 1671, and published in 1736. Fluxion ...
'' was a book written by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
. The book was completed in 1671, and published in 1736. Within this book, Newton describes a method (the
Newton–Raphson method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-va ...
) for finding the real zeroes of a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
.


''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, 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 In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied e ...
.


"Математические методы организации и планирования производства"

*
Leonid Kantorovich Leonid Vitalyevich Kantorovich ( rus, Леони́д Вита́льевич Канторо́вич, , p=lʲɪɐˈnʲit vʲɪˈtalʲjɪvʲɪtɕ kəntɐˈrovʲɪtɕ, a=Ru-Leonid_Vitaliyevich_Kantorovich.ogg; 19 January 19127 April 1986) was a Sovie ...
(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 George Bernard Dantzig (; November 8, 1914 – May 13, 2005) was an American mathematical scientist who made contributions to industrial engineering, operations research, computer science, economics, and statistics. Dantzig is known for his ...
and P. Wolfe * Operations Research 8:101–111, 1960. Dantzig's is considered the father of linear programming in the western world. He independently invented the
simplex algorithm In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are n ...
. 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 Victor LaRue Klee, Jr. (September 18, 1925 – August 17, 2007) was a mathematician specialising in convex sets, functional analysis, analysis of algorithms, optimization, and combinatorics. He spent almost his entire career at the University of ...
and George J. Minty * Klee and Minty gave an example showing that the
simplex algorithm In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are n ...
can take exponentially many steps to solve a
linear program 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 are represented by linear relationships. Linear programming i ...
.


"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 mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...
.


''

Moscow Mathematical Papyrus The Moscow Mathematical Papyrus, also named the Golenishchev Mathematical Papyrus after its first non-Egyptian owner, Egyptologist Vladimir Golenishchev, is an ancient Egyptian mathematical papyrus containing several problems in arithmetic, geom ...
''

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 ( egy, jꜥḥ-ms “, a common Egyptian name also transliterated Ahmose) was an ancient Egyptian scribe who lived towards the end of the Fifteenth Dynasty (and of the Second Intermediate Period) and the beginning of the Eighteenth Dyna ...
( scribe) One of the oldest mathematical texts, dating to the
Second Intermediate Period The Second Intermediate Period marks a period when ancient Egypt fell into disarray for a second time, between the end of the Middle Kingdom and the start of the New Kingdom. The concept of a "Second Intermediate Period" was coined in 1942 b ...
of ancient Egypt. It was copied by the scribe
Ahmes Ahmes ( egy, jꜥḥ-ms “, a common Egyptian name also transliterated Ahmose) was an ancient Egyptian scribe who lived towards the end of the Fifteenth Dynasty (and of the Second Intermediate Period) and the beginning of the Eighteenth Dyna ...
(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'') can also refer to 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 and in the process provides persuasive evidence against the theory that the Egyptians deliberately built their
pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...
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.


'' Archimedes Palimpsest''

*
Archimedes of Syracuse Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
Although the only mathematical tools at its author's disposal were what we might now consider secondary-school
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, he used those methods with rare brilliance, explicitly using infinitesimals to solve problems that would now be treated by integral calculus. Among those problems were that of the
center of gravity In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force ma ...
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
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 exact ...
and one of its secant lines. For explicit details of the method used, see
Archimedes' use of infinitesimals ''The Method of Mechanical Theorems'' ( el, Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as ''The Method'', is one of the major surviving works of the ancient Greece, ancient G ...
.


''

The Sand Reckoner ''The Sand Reckoner'' ( el, Ψαμμίτης, ''Psammites'') is a work by Archimedes, an Ancient Greek mathematician of the 3rd century BC, in which he set out to determine an upper bound for the number of grains of sand that fit into the unive ...
''

*
Archimedes of Syracuse Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
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. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
''

* 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 Mihalj Šilobod Bolšić (1 November 1724 – 4 April 1787) was a Croatian Roman Catholic priest, mathematician, writer, and musical theorist primarily known for writing the first Croatian arithmetics textbook '' Arithmetika Horvatzka'' (publish ...
''Arithmetika Horvatzka'' (1758) was the first Croatian language arithmetic textbook, written in the vernacular
Kajkavian Kajkavian (Kajkavian noun: ''kajkavščina''; Shtokavian adjective: ''kajkavski'' , noun: ''kajkavica'' or ''kajkavština'' ) is a South Slavic regiolect or language spoken primarily by Croats in much of Central Croatia, Gorski Kotar and no ...
dialect of Croatian language. It established a complete system of arithmetic terminology in Croatian, and vividly used examples from everyday life in
Croatia , image_flag = Flag of Croatia.svg , image_coat = Coat of arms of Croatia.svg , anthem = "Lijepa naša domovino"("Our Beautiful Homeland") , image_map = , map_caption = , capit ...
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
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
terminology to Kaikavian use. Full text o
''Arithmetika Horvatszka''
is available via archive.org.


''

Synopsis of Pure Mathematics ''Synopsis of Pure Mathematics'' is a book by G. S. Carr, written in 1886. The book attempted to summarize the state of most of the basic mathematics known at the time. The book is noteworthy because it was a major source of information for the ...
''

* 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 ''Éléments de mathématique'' (English: ''Elements of Mathematics'') is a series of mathematics books written by the pseudonymous French collective Nicolas Bourbaki. Begun in 1939, the series has been published in several volumes, and remain ...
''

* Nicolas Bourbaki 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 Robert Recorde () was an Anglo-Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus sign (+) to English speakers in 1557. Biography Born around 1512, Robert Recorde was the second and las ...
Written in 1542, it was the first really popular arithmetic book written in the English Language.


''

Cocker's Arithmetick ''Cocker's Arithmetick'', also known by its full title "Cocker's Arithmetick: Being a Plain and Familiar Method Suitable to the Meanest Capacity for the Full Understanding of That Incomparable Art, As It Is Now Taught by the Ablest School-Masters ...
''

* 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 The Reverend Mr. Thomas Dilworth (died 1780) was an English cleric and author of a widely used schoolbook, both in Great Britain and America, ''A New Guide to the English Tongue.'' Noah Webster as a boy studied Dilworth's book, and was inspired par ...
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 ''A Course of Pure Mathematics'' is a classic textbook in introductory mathematical analysis, written by G. H. Hardy. It is recommended for people studying calculus. First published in 1908, it went through ten editions (up to 1952) and several r ...
''

*
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
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 (m ...
, written by
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 ...
. 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 collegiate research university in Cambridge, England. Founded in 1209 and granted a royal charter by Henry III in 1231, Cambridge is the world's third oldest surviving university and one of its most pr ...
, 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, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
and the theory of
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
.


''

Moderne Algebra ''Moderne Algebra'' is a two-volume German textbook on graduate abstract algebra by , originally based on lectures given by Emil Artin in 1926 and by from 1924 to 1928. The English translation of 1949–1950 had the title ''Modern algebra'', th ...
''

* 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 one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
''

*
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville ...
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''

*
Robin Hartshorne __NOTOC__ Robin Cope Hartshorne ( ; born March 15, 1938) is an American mathematician who is known for his work in algebraic geometry. Career Hartshorne was a Putnam Fellow in Fall 1958 while he was an undergraduate at Harvard University (under ...
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 In algebraic geometry, a functor represented by a scheme ''X'' is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme ''S'' is (up to natural bijections) the set of all morphisms S \to X. T ...
.


'' Naive Set Theory''

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Paul Halmos Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...
An undergraduate introduction to not-very-naive set theory which has lasted for decades. It is still considered by many to be the best introduction to set theory for beginners. While the title states that it is naive, which is usually taken to mean without axioms, the book does introduce all the axioms of
Zermelo–Fraenkel set theory 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 such ...
and gives correct and rigorous definitions for basic objects. Where it differs from a "true" axiomatic set theory book is its character: There are no long-winded discussions of axiomatic minutiae, and there is next to nothing about topics like large cardinals. Instead it aims, and succeeds, in being intelligible to someone who has never thought about set theory before.


''

Cardinal and Ordinal Numbers ''Cardinal and Ordinal Numbers'' is a book on transfinite numbers, by Polish mathematician Wacław Sierpiński. It was published in 1958 by Państwowe Wydawnictwo Naukowe, as volume 34 of the series Monografie Matematyczne of the Institute of Mathe ...
''

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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, and t ...
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''

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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 me ...
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 Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth ( ...
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''

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John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Univ ...
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''

* André Weil 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''

*
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 ...
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. The book grew out of a series of lectures by Hardy and Wright and was first published in 1938. The third edition ...
'' 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 ''Foundations of Differential Geometry'' is an influential 2-volume mathematics book on differential geometry written by Shoshichi Kobayashi and Katsumi Nomizu. The first volume was published in 1963 and the second in 1969, by Interscience Publis ...
''

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Shoshichi Kobayashi was a Japanese mathematician. He was the eldest brother of electrical engineer and computer scientist Hisashi Kobayashi. His research interests were in Riemannian and complex manifolds, transformation groups of geometric structures, and Lie alg ...
and
Katsumi Nomizu was a Japanese-American mathematician known for his work in differential geometry. Life and career Nomizu was born in Osaka, Japan on the first day of December, 1924. He studied mathematics at Osaka University, graduating in 1947 with a Maste ...
(1963; 1969)


''Hodge Theory and Complex Algebraic Geometry I''


''Hodge Theory and Complex Algebraic Geometry II''

* Claire Voisin


Popular writings


''Gödel, Escher, Bach''

* Douglas Hofstadter '' Gödel, Escher, Bach: an Eternal Golden Braid'' is a Pulitzer Prize-winning book, first published in 1979 by Basic Books. It is a book about how the creative achievements of logician Kurt Gödel, artist M. C. Escher and composer Johann Sebastian Bach interweave. As the author states: "I realized that to me, Gödel and Escher and Bach were only shadows cast in different directions by some central solid essence. I tried to reconstruct the central object, and came up with this book."


''The World of Mathematics''

* 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.


See also

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References

{{DEFAULTSORT:Important Publications in Mathematics
Publications To publish is to make content available to the general public.Berne Conv ...
Mathematics