Oxyrhynchus papyrus with Euclid's Elements

This is a list of important publications in

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 Ivor Owen Grattan-Guinness (23 June 1941 – 12 December 2014) was a historian of mathematics and logic. Life Grattan-Guinness was born in Bakewell, England; his father was a mathematics teacher and educational administrator. He gained his bac ...

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

Algebra

Theory of equations

'' Baudhayana Sulba Sutra''

* Baudhayana (8th century BCE) Believed to have been written around the 8th century BCE, this is one of the oldest mathematical texts. It laid the foundations of

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

and was influential in South Asia 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 equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...

s with up to four unknowns.

'' The Nine Chapters on the Mathematical Art''

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

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

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 (220-280 CE) Contains the application of right angle triangles for survey of depth or height of distant objects.

'' Sunzi Suanjing''

*Sunzi (5th century CE) Contains the earliest description of

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

''
Aryabhatiya ''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Sanskrit astronomical treatise, is the ''magnum opus'' and only known surviving work of the 5th century Indian mathematician Aryabhata. Philosopher of astronomy Roger Billard estimates that th ...
''

* Aryabhata (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'' ( zh, 緝古算經, ''Continuation of Ancient Mathematics'') was the work of early Tang dynasty calendarist and mathematician Wang Xiaotong, written some time before the year 626, when he presented his work to the Emperor. ''Jig ...
''

Jigu Suanjing ''Jigu suanjing'' ( zh, 緝古算經, ''Continuation of Ancient Mathematics'') was the work of early Tang dynasty calendarist and mathematician Wang Xiaotong, written some time before the year 626, when he presented his work to the Emperor. ''Jig ...

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

'' Brāhmasphuṭasiddhānta''

Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...

(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 Persians, Persian polymath from Khwarazm, who produced vastly influential works in Mathematics ...

(820 CE) The first book on the systematic algebraic solutions of linear and quadratic equations by the Persian scholar

. The book is considered to be the foundation of modern algebra and

Islamic mathematics Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important progress was made, such as full ...

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

''
Līlāvatī ''Līlāvatī'' is Indian mathematician Bhāskara II's treatise on mathematics, written in 1150 AD. It is the first volume of his main work, the ''Siddhānta Shiromani'', alongside the ''Bijaganita'', the ''Grahaganita'' and the ''Golādhyāya' ...
'', ''
Siddhānta Shiromani ''Siddhānta Śiromaṇi'' (Sanskrit: सिद्धान्त शिरोमणि for "Crown of treatises") is the major treatise of Indian mathematician Bhāskara II. He wrote the ''Siddhānta Śiromaṇi'' in 1150 when he was 36 years old ...
'' and ''
Bijaganita ''Bijaganita'' ( IAST: ') was treatise on algebra by the Indian mathematician Bhāskara II. It is the second volume of his main work '' Siddhānta Shiromani (''"Crown of treatises") alongside '' Lilāvati'', ''Grahaganita'' and ''Golādhyāya''.< ...
''

One of the major treatises on mathematics by

Bhāskara II Bhāskara II (c. 1114–1185), also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian mathematician and astronomer. From verses, in his main work, Siddhānta Shiroman ...

provides the solution for indeterminate equations of 1st and 2nd order.

''
Yigu yanduan ''Yigu yanduan'' (益古演段 Old Mathematics in Expanded Sections) is a 13th-century mathematical work by Yuan dynasty mathematician Li Zhi. Overview ''Yigu yanduan'' was based on Northern Song mathematician Jiang Zhou's (蒋周) ''Yigu Ji ...
''

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

'' Mathematical Treatise in Nine Sections''

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

, which predates

Euler 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 ...

and Gauss by several centuries.

''
Ceyuan haijing ''Ceyuan haijing'' () is a treatise on solving geometry problems with the algebra of Tian yuan shu written by the mathematician Li Zhi in 1248 in the time of the Mongol Empire. It is a collection of 692 formula and 170 problems, all derived from ...
''

* Li Zhi (1248) Contains the application of high order polynomial equation in solving complex geometry problems.

''
Jade Mirror of the Four Unknowns ''Jade Mirror of the Four Unknowns'', ''Siyuan yujian'' (), also referred to as ''Jade Mirror of the Four Origins'', is a 1303 mathematical monograph by Yuan dynasty mathematician Zhu Shijie. Zhu advanced Chinese algebra with this Magnum opus. ...
''

* Zhu Shijie (1303) Contains the method of establishing system of high order polynomial equations of up to four unknowns.

'' Ars Magna''

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

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

and quartic equations (due to

Scipione del Ferro 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, and Lodovico Ferrari), and exhibited the first published calculations involving non-real complex numbers.

''Vollständige Anleitung zur Algebra''

* Leonhard Euler (1770) Also known as Elements of Algebra, Euler's textbook on elementary algebra is one of the first to set out algebra in the modern form we would recognize today. The first volume deals with determinate equations, while the second part deals with

s. The last section contains a proof of Fermat's Last Theorem 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 (1799) Gauss' doctoral dissertation, which contained a widely accepted (at the time) but incomplete proof of the fundamental theorem of algebra.

Abstract algebra

Group theory

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

= * Joseph Louis Lagrange (1770) The title means "Reflections on the algebraic solutions of equations". Made the prescient observation that the roots of the Lagrange resolvent of a polynomial equation are tied to permutations of the roots of the original equation, laying a more general foundation for what had previously been an ad hoc analysis and helping motivate the later development of the theory of

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

s, group theory, and Galois theory. The Lagrange resolvent also introduced the discrete Fourier transform 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 by Joseph Liouville. Included are Galois' papers ''Mémoire sur les conditions de résolubilité des équations par radicaux'' and ''Des équations primitives qui sont solubles par radicaux''.

''Traité des substitutions et des équations algébriques''

* Camille Jordan (1870) Online version:''
Online version
Traité des substitutions et des équations algébriques (Treatise on Substitutions and Algebraic Equations). The first book on group theory, giving a then-comprehensive study of permutation groups and Galois theory. In this book, Jordan introduced the notion of a simple group and epimorphism (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 natura ...

, and discussed matrix groups over finite fields as well as the Jordan normal form.

''Theorie der Transformationsgruppen''

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

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

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

''Solvability of groups of odd order''

Walter Feit Walter Feit (October 26, 1930 – July 29, 2004) was an Austrian-born American mathematician who worked in finite group theory and representation theory. His contributions provided elementary infrastructure used in algebra, geometry, topology, n ...

and

John Thompson John Thompson may refer to: Academics * J. A. Thompson (1913–2002), Australian biblical scholar * John D. Thompson (1917–1992), nurse and professor at the Yale School of Public Health * John G. Thompson (born 1932), American mathematician * ...

(1960) Description: Gave a complete proof of the solvability of finite groups of odd order, establishing the long-standing Burnside conjecture that all finite non-abelian simple groups are of even order. Many of the original techniques used in this paper were used in the eventual classification of finite simple groups.

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

''Homological Algebra''

Henri Cartan Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of co ...

and Samuel Eilenberg (1956) Provided the first fully worked out treatment of abstract homological algebra, unifying previously disparate presentations of homology and cohomology for

associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

s, and groups into a single theory.

" Sur Quelques Points d'Algèbre Homologique"

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

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

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

Algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

''Theorie der Abelschen Functionen''

Bernhard 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 ...

(1857) Publication data: ''Journal für die Reine und Angewandte Mathematik'' Developed the concept of Riemann surfaces and their topological properties beyond Riemann's 1851 thesis work, proved an index theorem for the genus (the original formulation of the Riemann–Hurwitz formula), proved the Riemann inequality for the dimension of the space of meromorphic functions with prescribed poles (the original formulation of the Riemann–Roch theorem), discussed birational transformations of a given curve and the dimension of the corresponding moduli space of inequivalent curves of a given genus, and solved more general inversion problems than those investigated by Abel and

Jacobi Jacobi may refer to: * People with the surname Jacobi (surname), Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenva ...

André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...

once wrote that this paper "''is one of the greatest pieces of mathematics that has ever been written; there is not a single word in it that is not of consequence.''"

''Faisceaux Algébriques Cohérents''

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

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

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

coherent sheaf 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 refe ...

is finite, in projective geometry, and such dimensions include many discrete invariants of varieties, for example

Hodge number In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...

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

'' Géométrie Algébrique et Géométrie Analytique''

* Jean-Pierre Serre (1956) In

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

and

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

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

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

several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...

. 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 In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...

. (''NB'' While

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

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

Armand Borel 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 (1958) Borel and Serre's exposition of Grothendieck's version of the Riemann–Roch theorem, published after Grothendieck made it clear that he was not interested in writing up his own result. Grothendieck reinterpreted both sides of the formula that Hirzebruch proved in 1953 in the framework of morphisms between varieties, resulting in a sweeping generalization. In his proof, Grothendieck broke new ground with his concept of

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

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

'' Éléments de géométrie algébrique''

* Alexander Grothendieck (1960–1967) Written with the assistance of

Jean Dieudonné Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymo ...

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

Jean-Louis Verdier Jean-Louis Verdier (; 2 February 1935 – 25 August 1989) was a French mathematician who worked, under the guidance of his doctoral advisor Alexander Grothendieck, on derived categories and Verdier duality. He was a close collaborator of Grothe ...

, Pierre Deligne, and

Nicholas Katz Nicholas Michael Katz (born December 7, 1943) is an American mathematician, working in arithmetic geometry, particularly on ''p''-adic methods, monodromy and moduli problems, and number theory. He is currently a professor of Mathematics at P ...

Number theory

'' Brāhmasphuṭasiddhānta''

(628) Brahmagupta's Brāhmasphuṭasiddhānta is the first book that mentions zero as a number, hence Brahmagupta is considered the first to formulate the concept of zero. The current system of the four fundamental operations (addition, subtraction, multiplication and division) based on the Hindu-Arabic number system also first appeared in Brahmasphutasiddhanta. It was also one of the first texts to provide concrete ideas on positive and negative numbers.

''De fractionibus continuis dissertatio''

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

''Recherches d'Arithmétique''

* Joseph Louis Lagrange (1775) Developed a general theory of binary quadratic forms to handle the general problem of when an integer is representable by the form

ax^2 + by^2 + cxy

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

'' Disquisitiones Arithmeticae''

* Carl Friedrich Gauss (1801) The '' Disquisitiones Arithmeticae'' is a profound and masterful book on number theory written by

German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ger ...

mathematician Carl Friedrich Gauss 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,

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLegendre and adds many important new results of his own. Among his contributions was the first complete proof known of the Fundamental theorem of arithmetic, the first two published proofs of the law of quadratic reciprocity, a deep investigation of binary quadratic forms going beyond Lagrange's work in ''Recherches d'Arithmétique'', a first appearance of

Gauss sums In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically :G(\chi) := G(\chi, \psi)= \sum \chi(r)\cdot \psi(r) where the sum is over elements of some finite commutative ring , is a ...

cyclotomy In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...

, and the theory of constructible polygons 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 heptadecagon'' is represented by the Schläfli symbol . Construction As 17 is a Fermat prime, the regular heptadecagon is ...

. Of note, in section V, article 303 of Disquisitiones, Gauss summarized his calculations of class numbers of imaginary quadratic number fields, and in fact found all imaginary quadratic number fields of class numbers 1, 2, and 3 (confirmed in 1986) as he had conjectured. In section VII, article 358, Gauss proved what can be interpreted as the first non-trivial case of the Riemann Hypothesis for curves over finite fields (the Hasse–Weil theorem).

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

* Peter Gustav Lejeune Dirichlet (1837) Pioneering paper in

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

, which introduced

Dirichlet characters In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b: :1) \ch ...

and their

L-functions In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ri ...

to establish

Dirichlet's theorem on arithmetic progressions In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is als ...

. In subsequent publications, Dirichlet used these tools to determine, among other things, the class number for quadratic forms.

" Über die Anzahl der Primzahlen unter einer gegebenen Grösse"

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

. It also contains the famous

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

, one of the most important open problems in mathematics.

'' Vorlesungen über Zahlentheorie''

* Peter Gustav Lejeune Dirichlet and

Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...

'' Vorlesungen über Zahlentheorie'' (''Lectures on Number Theory'') is a textbook of number theory written by

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,

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

''
Zahlbericht 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 w ...
''

David 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 a ...

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

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

made during the nineteenth century. Although criticized by

(who stated "''more than half of his famous Zahlbericht is little more than an account of

Kummer Kummer is a German surname. Notable people with the surname include: *Bernhard Kummer (1897–1962), German Germanist *Clare Kummer (1873—1958), American composer, lyricist and playwright *Clarence Kummer (1899–1930), American jockey * Christo ...

's number-theoretical work, with inessential improvements''") and Emmy Noether, it was highly influential for many years following its publication.

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

John Tate John Tate may refer to: * John Tate (mathematician) (1925–2019), American mathematician * John Torrence Tate Sr. (1889–1950), American physicist * John Tate (Australian politician) (1895–1977) * John Tate (actor) (1915–1979), Australian act ...

(1950) Generally referred to simply as '' Tate's Thesis'', Tate's Princeton PhD thesis, under

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

, is a reworking of Erich Hecke's theory of zeta- and ''L''-functions in terms of

Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...

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 form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...

"
Automorphic Forms on GL(2) ''Automorphic Forms on GL(2)'' is a mathematics book by where they rewrite Erich Hecke's theory of modular forms in terms of the representation theory of GL(2) over local fields and adele rings of global fields and prove the Jacquet–Langlands co ...
"

* Hervé Jacquet and Robert Langlands (1970) This publication offers evidence towards Langlands' conjectures by reworking and expanding the classical theory of

modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...

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

"La conjecture de Weil. I."

* Pierre Deligne (1974) Proved the Riemann hypothesis for varieties over finite fields, settling the last of the open Weil conjectures.

"Endlichkeitssätze für abelsche Varietäten über Zahlkörpern"

Gerd Faltings 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 Louis Joel Mordell (28 January 1888 – 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory. He was born in Philadelphia, United States, in a Jewish family of Lithuanian extraction. Educati ...

(a conjecture dating back to 1922). Other theorems proved in this paper include an instance of the Tate conjecture (relating the homomorphisms between two abelian varieties over a number field to the homomorphisms between their Tate modules) and some finiteness results concerning abelian varieties over number fields with certain properties.

"Modular Elliptic Curves and Fermat's Last Theorem"

* Andrew Wiles (1995) This article proceeds to prove a special case of the

Shimura–Taniyama conjecture The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. Andr ...

through the study of the

deformation theory In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesim ...

Galois representations In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring i ...

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

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

* Michael Harris and Richard Taylor (2001) Harris and Taylor provide the first proof of the

local Langlands conjecture In mathematics, the local Langlands conjectures, introduced by , are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group ''G'' over a local field ''F'', and representation ...

for GL(''n''). As part of the proof, this monograph also makes an in depth study of the geometry and cohomology of certain Shimura varieties at primes of bad reduction.

"Le lemme fondamental pour les algèbres de Lie"

* Ngô Bảo Châu (2008) Ngô Bảo Châu proved a long-standing unsolved problem in the classical Langlands program, using methods from the Geometric Langlands program.

"Perfectoid space"

* Peter Scholze (2012) Peter Scholze introduced

Perfectoid space In mathematics, perfectoid spaces are adic spaces of special kind, which occur in the study of problems of "mixed characteristic", such as local fields of characteristic zero which have residue fields of characteristic prime ''p''. A perfecto ...

Analysis

''Introductio in analysin infinitorum''

* Leonhard Euler (1748) The eminent historian of mathematics Carl Boyer once called Euler's '' Introductio in analysin infinitorum'' the greatest modern textbook in mathematics. Published in two volumes, this book more than any other work succeeded in establishing analysis as a major branch of mathematics, with a focus and approach distinct from that used in geometry and algebra. Notably, Euler identified functions rather than curves to be the central focus in his book. Logarithmic, exponential, trigonometric, and transcendental functions were covered, as were expansions into partial fractions, evaluations of for a positive integer between 1 and 13, infinite series and infinite product formulas, continued fractions, and

partitions Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...

of integers. In this work, Euler proved that every rational number can be written as a finite continued fraction, that the continued fraction of an irrational number is infinite, and derived continued fraction expansions for and

\textstyle\sqrt

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

Calculus

''
Yuktibhāṣā ''Yuktibhāṣā'' ( ml, യുക്തിഭാഷ, lit=Rationale), also known as Gaṇita-yukti-bhāṣā and (''Compendium of Astronomical Rationale''), is a major treatise on Indian mathematics, mathematics and Hindu astronomy, astronomy, ...
''

* Jyeshtadeva (1501) Written in India 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 and mathematical analysis, most of which were earlier discovered by the 14th century mathematician Madhava. 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 Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...

, the derivative, differential equations, 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 In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...

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

* Gottfried Leibniz (1684) Leibniz's first publication on differential calculus, containing the now familiar notation for differentials as well as rules for computing the derivatives of powers, products and quotients.

''
Philosophiae Naturalis Principia Mathematica Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
''

* Isaac Newton (1687) The ''Philosophiae Naturalis Principia Mathematica'' ( Latin: "mathematical principles of natural philosophy", often ''Principia'' or ''Principia Mathematica'' for short) is a three-volume work by Isaac Newton published on 5 July 1687. Perhaps the most influential scientific book ever published, it contains the statement of Newton's laws of motion forming the foundation of classical mechanics as well as his

law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...

, and derives Kepler's laws for the motion of the planets (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 (1755) Published in two books, Euler's textbook on differential calculus presented the subject in terms of the function concept, which he had introduced in his 1748 ''Introductio in analysin infinitorum''. This work opens with a study of the calculus of finite differences and makes a thorough investigation of how differentiation behaves under substitutions. Also included is a systematic study of

Bernoulli polynomials In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur in ...

and the Bernoulli numbers (naming them as such), a demonstration of how the Bernoulli numbers are related to the coefficients in the

Euler–Maclaurin formula In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using ...

and the values of ζ(2n), a further study of

Euler's constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural ...

(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''

(1867) Written in 1853, Riemann's work on trigonometric series was published posthumously. In it, he extended Cauchy's definition of the integral to that of the Riemann integral, allowing some functions with dense subsets of discontinuities on an interval to be integrated (which he demonstrated by an example). He also stated the Riemann series theorem, proved the Riemann–Lebesgue lemma for the case of bounded Riemann integrable functions, and developed the Riemann localization principle.

''Intégrale, longueur, aire''

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

measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...

and the Lebesgue integral.

Complex analysis

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

* Bernhard Riemann (1851) Riemann's doctoral dissertation introduced the notion of a Riemann surface,

conformal map In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...

ping, simple connectivity, the Riemann sphere, the Laurent series expansion for functions having poles and branch points, and the Riemann mapping theorem.

Functional analysis

''Théorie des opérations linéaires''

* Stefan Banach (1932; originally published 1931 in

Polish Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Poles, people from Poland or of Polish descent * Polish chicken *Polish brothers (Mark Polish and Michael Polish, born 1970), American twin screenwr ...

under the title ''Teorja operacyj''.) * The first mathematical monograph on the subject of linear metric spaces, bringing the abstract study of functional analysis to the wider mathematical community. The book introduced the ideas of a

normed space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...

and the notion of a so-called ''B''-space, a

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

normed space. The ''B''-spaces are now called

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

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

''Produits Tensoriels Topologiques et Espaces Nucléaires''

* Grothendieck's thesis introduced the notion of a nuclear space, tensor products of locally convex topological vector spaces, and the start of Grothendieck's work on tensor products of Banach spaces. Alexander Grothendieck also wrote a textbook on topological vector spaces: *

''Sur certains espaces vectoriels topologiques''

Fourier analysis

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

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

(1807) Introduced

, specifically

Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...

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

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLaplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...

, Malus 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 In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line: : \int_0^\in ...

and later the Lebesgue integral.

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

* Peter Gustav Lejeune Dirichlet (1829, expanded German edition in 1837) In his habilitation thesis on Fourier series, Riemann characterized this work of Dirichlet as "''the first profound paper about the subject''". This paper gave the first rigorous proof of the convergence of

under fairly general conditions (piecewise continuity and monotonicity) by considering partial sums, which Dirichlet transformed into a particular

involving what is now called the Dirichlet kernel. This paper introduced the nowhere continuous Dirichlet function and an early version of the Riemann–Lebesgue lemma.

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

* Lennart Carleson (1966) Settled

Lusin's conjecture Carleson's theorem is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of functions, proved by . The name is also often used to refer to the extension of the re ...

that the Fourier expansion of any

L^2

function converges almost everywhere.

Geometry

'' Baudhayana Sulba Sutra''

* Baudhayana Written around the 8th century BC, this is one of the oldest geometrical texts. It laid the foundations of

and was influential in South Asia 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 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...

, 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

s with up to four unknowns.

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

* Euclid Publication data: c. 300 BC Online version:''
Interactive Java version
This is often regarded as not only the most important work in geometry but one of the most important works in mathematics. It contains many important results in plane and solid geometry, algebra (books II and V), and number theory (book VII, VIII, and IX). More than any specific result in the publication, it seems that the major achievement of this publication is the promotion of an axiomatic approach as a means for proving results. Euclid's ''Elements'' has been referred to as the most successful and influential textbook ever written.

'' The Nine Chapters on the Mathematical Art''

* Unknown author This was a Chinese

book, mostly geometric, composed during the Han Dynasty, perhaps as early as 200 BC. It remained the most important textbook in

China China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's most populous country, with a population exceeding 1.4 billion, slightly ahead of India. China spans the equivalent of five time zones and ...

and East Asia 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 In mathematics, the ''regula falsi'', method of false position, or false position method is a very old method for solving an equation with one unknown; this method, in modified form, is still in use. In simple terms, the method is the trial and e ...

''. Problems with several unknowns, solved by a principle similar to

. Problems involving the principle known in the West as the

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

. The earliest solution of a

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

using a method equivalent to the modern method.

'' The Conics''

Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...

The Conics was written by Apollonius of Perga, a Greek mathematician. His innovative methodology and terminology, especially in the field of

conics In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...

, influenced many later scholars including Ptolemy,

Francesco Maurolico Francesco Maurolico (Latin: ''Franciscus Maurolycus''; Italian: ''Francesco Maurolico''; gr, Φραγκίσκος Μαυρόλυκος, 16 September 1494 - 21/22 July 1575) was a mathematician and astronomer from Sicily. He made contributions t ...

, Isaac Newton, and René Descartes. It was Apollonius who gave the

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

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

''
Surya Siddhanta The ''Surya Siddhanta'' (; ) is a Sanskrit treatise in Indian astronomy dated to 505 CE,Menso Folkerts, Craig G. Fraser, Jeremy John Gray, John L. Berggren, Wilbur R. Knorr (2017)Mathematics Encyclopaedia Britannica, Quote: "(...) its Hindu inven ...
''

* 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 ''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Sanskrit astronomical treatise, is the ''magnum opus'' and only known surviving work of the 5th century Indian mathematician Aryabhata. Philosopher of astronomy Roger Billard estimates that th ...
''

* Aryabhata (499 CE) This was a highly influential text during the Golden Age of mathematics in India. The text was highly concise and therefore elaborated upon in commentaries by later mathematicians. It made significant contributions to geometry and astronomy, including introduction of sine/ cosine, determination of the approximate value of pi and accurate calculation of the earth's circumference.

''
La Géométrie ''La Géométrie'' was published in 1637 as an appendix to ''Discours de la méthode'' (''Discourse on the Method''), written by René Descartes. In the ''Discourse'', he presents his method for obtaining clarity on any subject. ''La Géométrie ...
''

* René Descartes La Géométrie was published in 1637 and written by René Descartes. The book was influential in developing the

Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

and specifically discussed the representation of

point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...

s 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 (gen ...

, via real numbers; and the representation of

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

s, via

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

''Grundlagen der Geometrie''

Online version:''
English
Publication data: Hilbert's axiomatization of geometry, whose primary influence was in its pioneering approach to metamathematical questions including the use of models to prove axiom independence and the importance of establishing the consistency and completeness of an axiomatic system.

'' Regular Polytopes''

H.S.M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

''Regular Polytopes'' is a comprehensive survey of the geometry of regular polytopes, the generalisation of regular polygons and regular polyhedra to higher dimensions. Originating with an essay entitled ''Dimensional Analogy'' written in 1923, the first edition of the book took Coxeter 24 years to complete. Originally written in 1947, the book was updated and republished in 1963 and 1973.

Differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

''Recherches sur la courbure des surfaces''

* Leonhard Euler (1760) Publication data: Mémoires de l'académie des sciences de Berlin 16 (1760) pp. 119–143; published 1767.
Full text
and an English translation available from the Dartmouth Euler archive.) Established the theory of surfaces, and introduced the idea of

principal curvatures In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by d ...

, laying the foundation for subsequent developments in the differential geometry of surfaces.

''Disquisitiones generales circa superficies curvas''

* Carl Friedrich Gauss (1827) Publication data:''
"Disquisitiones generales circa superficies curvas"
''Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores'' Vol. VI (1827), pp. 99–146;
General Investigations of Curved Surfaces
(published 1965) Raven Press, New York, translated by A.M.Hiltebeitel and J.C.Morehead. Groundbreaking work in

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, introducing the notion of

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...

and Gauss' celebrated Theorema Egregium.

''Über die Hypothesen, welche der Geometrie zu Grunde Liegen''

* Bernhard Riemann (1854) Publication data:''
"Über die Hypothesen, welche der Geometrie zu Grunde Liegen"
''Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen'', Vol. 13, 1867
English translation
Riemann's famous Habiltationsvortrag, in which he introduced the notions of a

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

, Riemannian metric, and curvature tensor.

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

* Gaston Darboux Publication data:
Volume IVolume IIVolume IIIVolume IV
Leçons sur la théorie génerale des surfaces et les applications géométriques du calcul infinitésimal (on the General Theory of Surfaces and the Geometric Applications of Infinitesimal Calculus). A treatise covering virtually every aspect of the 19th century

of surfaces.

Topology

''Analysis situs''

Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...

(1895, 1899–1905) Description: Poincaré's Analysis Situs and his Compléments à l'Analysis Situs laid the general foundations for algebraic topology. In these papers, Poincaré introduced the notions of

homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...

and the

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

, provided an early formulation of Poincaré duality, gave the

Euler–Poincaré characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...

for chain complexes, and mentioned several important conjectures including the Poincaré conjecture, demonstrated by Grigori Perelman in 2003.

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

* Jean Leray (1946) These two Comptes Rendus notes of Leray from 1946 introduced the novel concepts of sheafs,

sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when i ...

, and spectral sequences, which he had developed during his years of captivity as a prisoner of war. Leray's announcements and applications (published in other Comptes Rendus notes from 1946) drew immediate attention from other mathematicians. Subsequent clarification, development, and generalization by

, Jean-Louis Koszul,

, Jean-Pierre Serre, and Leray himself allowed these concepts to be understood and applied to many other areas of mathematics. Dieudonné would later write that these notions created by Leray "''undoubtedly rank at the same level in the history of mathematics as the methods invented by Poincaré and Brouwer''".

Quelques propriétés globales des variétés differentiables

René Thom René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became w ...

(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 In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...

and unoriented cobordism, and demonstrated that cobordism groups could be computed as the homotopy groups of certain Thom spaces. Thom completely characterized the unoriented cobordism ring and achieved strong results for several problems, including Steenrod's problem on the realization of cycles.

Category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

"General Theory of Natural Equivalences"

* Samuel Eilenberg and Saunders Mac Lane (1945) The first paper on category theory. Mac Lane later wrote in ''Categories for the Working Mathematician'' that he and Eilenberg introduced categories so that they could introduce functors, and they introduced functors so that they could introduce natural equivalences. Prior to this paper, "natural" was used in an informal and imprecise way to designate constructions that could be made without making any choices. Afterwards, "natural" had a precise meaning which occurred in a wide variety of contexts and had powerful and important consequences.

''
Categories for the Working Mathematician ''Categories for the Working Mathematician'' (''CWM'') is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. It was first published in 1971, and is based on ...
''

* Saunders Mac Lane (1971, second edition 1998) Saunders Mac Lane, one of the founders of category theory, wrote this exposition to bring categories to the masses. Mac Lane brings to the fore the important concepts that make category theory useful, such as adjoint functors and

universal properties In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...

''
Higher Topos Theory ''Higher Topos Theory'' is a treatise on the theory of ∞-categories written by American mathematician Jacob Lurie. In addition to introducing Lurie's new theory of ∞-topoi, the book is widely considered foundational to higher category theory ...
''

* Jacob Lurie (2010) ''This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck topoi. A few applications to classical topology are included.'' (see arXiv.)

Set theory

"Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"

* Georg Cantor (1874) Online version:''
Online version
Contains the first proof that the set of all real numbers is uncountable; also contains a proof that the set of algebraic numbers is countable. (See

Georg Cantor's first set theory article Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is uncou ...

''
Grundzüge der Mengenlehre ''Grundzüge der Mengenlehre'' (German for "Basics of Set Theory") is a book on set theory written by Felix Hausdorff. First published in April 1914, ''Grundzüge der Mengenlehre'' was the first comprehensive introduction to set theory. Besides th ...
''

* Felix Hausdorff First published in 1914, this was the first comprehensive introduction to set theory. Besides the systematic treatment of known results in set theory, the book also contains chapters on

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

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

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

(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 Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was award ...

(1963, 1964) Cohen's breakthrough work proved the independence of the continuum hypothesis and axiom of choice with respect to Zermelo–Fraenkel set theory. In proving this Cohen introduced the concept of ''

forcing Forcing may refer to: Mathematics and science * Forcing (mathematics), a technique for obtaining independence proofs for set theory *Forcing (recursion theory), a modification of Paul Cohen's original set theoretic technique of forcing to deal with ...

'' which led to many other major results in axiomatic set theory.

Logic

'' The Laws of Thought''

* George Boole (1854) Published in 1854, The Laws of Thought was the first book to provide a mathematical foundation for logic. Its aim was a complete re-expression and extension of Aristotle's logic in the language of mathematics. Boole's work founded the discipline of algebraic logic and would later be central for Claude Shannon in the development of digital logic.

'' Begriffsschrift''

* Gottlob Frege (1879) Published in 1879, the title ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept notation''; the full title of the book identifies it as "''a

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

language, modelled on that of

arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...

, of pure

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

''". Frege's motivation for developing his formal logical system was similar to Leibniz's desire for a '' calculus ratiocinator''. Frege defines a logical calculus to support his research in the foundations of mathematics. ''Begriffsschrift'' is both the name of the book and the calculus defined therein. It was arguably the most significant publication in logic since Aristotle.

'' Formulario mathematico''

* Giuseppe Peano (1895) First published in 1895, the Formulario mathematico was the first mathematical book written entirely in a formalized language. It contained a description of mathematical logic and many important theorems in other branches of mathematics. Many of the notations introduced in the book are now in common use.

'' Principia Mathematica''

* Bertrand Russell and

Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applicat ...

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

, written by Bertrand Russell and

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

symbolic logic 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 ...

. 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 ''Systems of Logic Based on Ordinals'' was the PhD dissertation of the mathematician Alan Turing. Turing's thesis is not about a new type of formal logic, nor was he interested in so-called ‘ranked logic’ systems derived from ordinal or relat ...
''

* Alan Turing's PhD thesis

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

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

(1931) Online version:''
Online version
In mathematical logic,

Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research i ...

are two celebrated theorems proved by

in 1931. The first incompleteness theorem states:

For any formal system such that (1) it is $\omega$ -consistent ( omega-consistent), (2) it has a
recursively definable In computability theory, a set of natural numbers is called computable, recursive, or decidable if there is an algorithm which takes a number as input, terminates after a finite amount of time (possibly depending on the given number) and correctly ...
set of axioms and
rules of derivation In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...
, and (3) every recursive relation of natural numbers is definable in it, there exists a formula of the system such that, according to the intended interpretation of the system, it expresses a truth about natural numbers and yet it is not a theorem of the system.

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

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

* Endre Szemerédi (1975) Settled a conjecture of

Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...

and Pál Turán (now known as Szemerédi's theorem) that if a sequence of natural numbers has positive upper density then it contains arbitrarily long arithmetic progressions. Szemerédi's solution has been described as a "masterpiece of combinatorics" and it introduced new ideas and tools to the field including a weak form of the

Szemerédi regularity lemma Szemerédi's regularity lemma is one of the most powerful tools in extremal graph theory, particularly in the study of large dense graphs. It states that the vertices of every large enough graph can be partitioned into a bounded number of parts so ...

Graph theory

''Solutio problematis ad geometriam situs pertinentis''

* Leonhard Euler (1741)
Euler's original publication
(in Latin) Euler's solution of the

Königsberg bridge problem Königsberg (, ) was the historic Prussian city that is now Kaliningrad, Russia. Königsberg was founded in 1255 on the site of the ancient Old Prussian settlement ''Twangste'' by the Teutonic Knights during the Northern Crusades, and was named ...

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.

"On the evolution of random graphs"

and

Alfréd Rényi Alfréd Rényi (20 March 1921 – 1 February 1970) was a Hungarian mathematician known for his work in probability theory, though he also made contributions in combinatorics, graph theory, and number theory. Life Rényi was born in Budapest to ...

(1960) Provides a detailed discussion of sparse

random graphs In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs li ...

, including distribution of components, occurrence of small subgraphs, and phase transitions.

"Network Flows and General Matchings"

L. R. Ford, Jr. Lester Randolph Ford Jr. (September 23, 1927 – February 26, 2017) was an American mathematician specializing in network flow problems. He was the son of mathematician Lester R. Ford Sr. Ford's paper with D. R. Fulkerson on the maximum flow p ...

D. R. Fulkerson Delbert Ray Fulkerson (; August 14, 1924 – January 10, 1976) was an American mathematician who co-developed the FordFulkerson algorithm, one of the most well-known algorithms to solve the maximum flow problem in Flow network, networks. Early l ...

* ''Flows in Networks''. Prentice-Hall, 1962. Presents the Ford–Fulkerson algorithm for solving the maximum flow problem, along with many ideas on flow-based models.

Computational complexity theory

''See List of important publications in theoretical computer science.''

Probability theory and statistics

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

Game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

"Zur Theorie der Gesellschaftsspiele"

* John von Neumann (1928) Went well beyond Émile Borel's initial investigations into strategic two-person game theory by proving the minimax theorem for two-person, zero-sum games.

''
Theory of Games and Economic Behavior ''Theory of Games and Economic Behavior'', published in 1944 by Princeton University Press, is a book by mathematician John von Neumann and economist Oskar Morgenstern which is considered the groundbreaking text that created the interdisciplinar ...
''

* Oskar Morgenstern, John von Neumann (1944) This book led to the investigation of modern game theory as a prominent branch of mathematics. This work contained the method for finding optimal solutions for two-person zero-sum games.

"Equilibrium Points in N-person Games"

Nash equilibrium In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...

'' On Numbers and Games''

* John Horton Conway (1976) The book is in two, , parts. The zeroth part is about numbers, the first part about games – both the values of games and also some real games that can be played such as

Nim Nim is a mathematical two player game. Nim or NIM may also refer to: * Nim (programming language) * Nim Chimpsky, a signing chimpanzee Acronyms * Network Installation Manager, an IBM framework * Nuclear Instrumentation Module * Negative index met ...

Hackenbush Hackenbush is a two-player game invented by mathematician John Horton Conway. It may be played on any configuration of colored line segments connected to one another by their endpoints and to a "ground" line. Gameplay The game starts with the p ...

, Col and Snort amongst the many described.

'' Winning Ways for your Mathematical Plays''

* Elwyn Berlekamp,

John 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 o ...

and

Richard K. Guy Richard Kenneth Guy (30 September 1916 – 9 March 2020) was a British mathematician. He was a professor in the Department of Mathematics at the University of Calgary. He is known for his work in number theory, geometry, recreational mathemati ...

(1982) A compendium of information on mathematical games. It was first published in 1982 in two volumes, one focusing on Combinatorial game theory and surreal numbers, and the other concentrating on a number of specific games.

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

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

* Benoît Mandelbrot (1967) A discussion of self-similar curves that have fractional dimensions between 1 and 2. These curves are examples of fractals, although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. Shows Mandelbrot's early thinking on fractals, and is an example of the linking of mathematical objects with natural forms that was a theme of much of his later work.

Numerical analysis

Optimization

''Method of Fluxions''

* Isaac Newton (1736) '' Method of Fluxions'' was a book written by Isaac Newton. The book was completed in 1671, and published in 1736. Within this book, Newton describes a method (the Newton–Raphson method) for finding the real zeroes of a function.

''Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies''

* Joseph Louis Lagrange (1761) Major early work on the

calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

, building upon some of Lagrange's prior investigations as well as those of

. Contains investigations of minimal surface determination as well as the initial appearance of Lagrange multipliers.

"Математические методы организации и планирования производства"

* Leonid Kantorovich (1939) "

he Mathematical Method of Production Planning and Organization He or HE may refer to: Language * He (pronoun), an English pronoun * He (kana), the romanization of the Japanese kana へ * He (letter), the fifth letter of many Semitic alphabets * He (Cyrillic), a letter of the Cyrillic script called ''He'' in ...

(in Russian). Kantorovich wrote the first paper on production planning, which used Linear Programs as the model. He received the Nobel prize for this work in 1975.

"Decomposition Principle for Linear Programs"

* George Dantzig and P. Wolfe * Operations Research 8:101–111, 1960. Dantzig's is considered the father of

linear programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, li ...

in the western world. He independently invented the simplex algorithm. Dantzig and Wolfe worked on decomposition algorithms for large-scale linear programs in factory and production planning.

"How Good is the Simplex Algorithm?"

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

"Полиномиальный алгоритм в линейном программировании"

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

Early manuscripts

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

'' Moscow Mathematical Papyrus''

This is one of the earliest mathematical treatises that still survives today. The Papyrus contains 25 problems involving arithmetic, geometry, and algebra, each with a solution given. Written in Ancient Egypt at approximately 1850 BC.

''
Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased ...
''

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 Dynas ...

(

scribe A scribe is a person who serves as a professional copyist, especially one who made copies of manuscripts before the invention of automatic printing. The profession of the scribe, previously widespread across cultures, lost most of its promi ...

) One of the oldest mathematical texts, dating to the Second Intermediate Period of ancient Egypt. It was copied by the scribe

(properly ''Ahmose'') from an older Middle Kingdom papyrus. It laid the foundations of

Egyptian mathematics Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt 3000 to c. , from the Old Kingdom of Egypt until roughly the beginning of Hellenistic Egypt. The ancient Egyptians utilized a numeral system for count ...

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

squaring the circle Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficulty ...

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

Egyptians Egyptians ( arz, المَصرِيُون, translit=al-Maṣriyyūn, ; arz, المَصرِيِين, translit=al-Maṣriyyīn, ; cop, ⲣⲉⲙⲛ̀ⲭⲏⲙⲓ, remenkhēmi) are an ethnic group native to the Nile, Nile Valley in Egypt. Egyptian ...

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

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

''
Archimedes Palimpsest The Archimedes Palimpsest is a parchment codex palimpsest, originally a Byzantine Greek copy of a compilation of Archimedes and other authors. It contains two works of Archimedes that were thought to have been lost (the ''Ostomachion'' and the ' ...
''

* Archimedes of Syracuse Although the only mathematical tools at its author's disposal were what we might now consider secondary-school geometry, he used those methods with rare brilliance, explicitly using

infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...

s to solve problems that would now be treated by integral calculus. Among those problems were that of the center of gravity of a solid hemisphere, that of the center of gravity of a frustum of a circular paraboloid, and that of the area of a region bounded by a parabola and one of its secant lines. For explicit details of the method used, see Archimedes' use of infinitesimals.

'' The Sand Reckoner''

* Archimedes of Syracuse Online version:''
Online version
The first known (European) system of number-naming that can be expanded beyond the needs of everyday life.

Textbooks

'' Abstract Algebra''

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

''Arithmetika Horvatzka''

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

Croatian language Croatian (; ' ) is the standardized variety of the Serbo-Croatian pluricentric language used by Croats, principally in Croatia, Bosnia and Herzegovina, the Serbian province of Vojvodina, and other neighboring countries. It is the official ...

. It established a complete system of

terminology in Croatian, and vividly used examples from everyday life in Croatia 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 terminology to Kaikavian use. Full text o
''Arithmetika Horvatszka''
is available via archive.org.

'' Synopsis of Pure Mathematics''

G. S. Carr George Shoobridge Carr (1837–1914) was a British mathematician. He wrote ''Synopsis of Pure Mathematics'' (1886). This book, first published in England in 1880, was read and studied closely by mathematician Srinivasa Ramanujan when he was a te ...

Contains over 6000 theorems of mathematics, assembled by George Shoobridge Carr for the purpose of training his students for the Cambridge Mathematical Tripos exams. Studied extensively by Ramanujan
(first half here)

'' Éléments de mathématique''

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

One of the most influential books in French mathematical literature. It introduces some of the notations and definitions that are now usual (the symbol ∅ or the term bijective for example). Characterized by an extreme level of rigour, formalism and generality (up to the point of being highly criticized for that), its publication started in 1939 and is still unfinished today.

'' Arithmetick: or, The Grounde of Arts''

* Robert Recorde Written in 1542, it was the first really popular arithmetic book written in the English Language.

'' Cocker's Arithmetick''

Edward Cocker Edward Cocker (163122 August 1676) was an English engraver, who also taught writing and arithmetic. Cocker was the reputed author of the famous ''Arithmetick'', the popularity of which has added a phrase ("according to Cocker") to the list of ...

(authorship disputed) Textbook of arithmetic published in 1678 by John Hawkins, who claimed to have edited manuscripts left by Edward Cocker, who had died in 1676. This influential mathematics textbook used to teach arithmetic in schools in the United Kingdom for over 150 years.

'' The Schoolmaster's Assistant, Being a Compendium of Arithmetic both Practical and Theoretical''

* Thomas Dilworth An early and popular English arithmetic textbook published in

America The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...

in the 18th century. The book reached from the introductory topics to the advanced in five sections.

''Geometry''

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

'' A Course of Pure Mathematics''

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

A classic textbook in introductory mathematical analysis, written by

. It was first published in 1908, and went through many editions. It was intended to help reform mathematics teaching in the UK, and more specifically in the University of Cambridge, 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 and the theory of infinite series.

''
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'', tho ...
''

B. L. van der Waerden Bartel Leendert van der Waerden (; 2 February 1903 – 12 January 1996) was a Dutch mathematician and historian of mathematics. Biography Education and early career Van der Waerden learned advanced mathematics at the University of Amsterd ...

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 Frederick Ungar Publishing Company was a New York publishing firm which was founded in 1940. History The Frederick Ungar Publishing Company published over 2,000 titles, including reference books such as the ''Encyclopedia of World Literature in ...

'' Algebra''

* Saunders Mac Lane and Garrett Birkhoff A definitive introductory text for abstract algebra using a

category theoretic Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...

approach. Both a rigorous introduction from first principles, and a reasonably comprehensive survey of the field.

''Calculus, Vol. 1''

Tom M. Apostol Tom Mike Apostol (August 20, 1923 – May 8, 2016) was an American analytic number theorist and professor at the California Institute of Technology, best known as the author of widely used mathematical textbooks. Life and career Apostol was bor ...

''
Algebraic Geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
''

* Robin Hartshorne The first comprehensive introductory (graduate level) text in algebraic geometry that used the language of schemes and cohomology. Published in 1977, it lacks aspects of the scheme language which are nowadays considered central, like the functor of points.

''
Naive Set Theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike Set theory#Axiomatic set theory, axiomatic set theories, which are defined using Mathematical_logic#Formal_logical_systems, forma ...
''

* Paul Halmos An undergraduate introduction to not-very-naive set theory which has lasted for decades. It is still considered by many to be the best introduction to set theory for beginners. While the title states that it is naive, which is usually taken to mean without axioms, the book does introduce all the axioms of Zermelo–Fraenkel set theory and gives correct and rigorous definitions for basic objects. Where it differs from a "true" axiomatic set theory book is its character: There are no long-winded discussions of axiomatic minutiae, and there is next to nothing about topics like

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

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

'' Cardinal and Ordinal Numbers''

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

The ''nec plus ultra'' reference for basic facts about cardinal and ordinal numbers. If you have a question about the cardinality of sets occurring in everyday mathematics, the first place to look is this book, first published in the early 1950s but based on the author's lectures on the subject over the preceding 40 years.

'' Set Theory: An Introduction to Independence Proofs''

* Kenneth Kunen 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

. 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 Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, ma ...

* Heinz Hopf First published round 1935, this text was a pioneering "reference" text book in topology, already incorporating many modern concepts from set-theoretic topology, homological algebra and homotopy theory.

''General Topology''

John L. Kelley John L. Kelley (December 6, 1916, Kansas – November 26, 1999, Berkeley, California) was an American mathematician at the University of California, Berkeley, who worked in general topology and functional analysis. Kelley's 1955 text, ''General ...

First published in 1955, for many years the only introductory graduate level textbook in the US, teaching the basics of point set, as opposed to algebraic, topology. Prior to this the material, essential for advanced study in many fields, was only available in bits and pieces from texts on other topics or journal articles.

''Topology from the Differentiable Viewpoint''

* John Milnor This short book introduces the main concepts of differential topology in Milnor's lucid and concise style. While the book does not cover very much, its topics are explained beautifully in a way that illuminates all their details.

''
Number Theory, An approach through history from Hammurapi to Legendre ''Number Theory: An Approach Through History from Hammurapi to Legendre'' is a book on the history of number theory, written by André Weil and published in 1984. The book reviews over three millennia of research on numbers but the key focus is on ...
''

An historical study of number theory, written by one of the 20th century's greatest researchers in the field. The book covers some thirty six centuries of arithmetical work but the bulk of it is devoted to a detailed study and exposition of the work of Fermat, Euler, Lagrange, and Legendre. The author wishes to take the reader into the workshop of his subjects to share their successes and failures. A rare opportunity to see the historical development of a subject through the mind of one of its greatest practitioners.

''An Introduction to the Theory of Numbers''

and

E. M. Wright Sir Edward Maitland Wright (13 February 1906, Farnley – 2 February 2005, Reading) was an English mathematician, best known for co-authoring ''An Introduction to the Theory of Numbers'' with G. H. Hardy. Career He was born in Farnl ...

'' An Introduction to the Theory of Numbers'' was first published in 1938, and is still in print, with the latest edition being the 6th (2008). It is likely that almost every serious student and researcher into number theory has consulted this book, and probably has it on their bookshelf. It was not intended to be a textbook, and is rather an introduction to a wide range of differing areas of number theory which would now almost certainly be covered in separate volumes. The writing style has long been regarded as exemplary, and the approach gives insight into a variety of areas without requiring much more than a good grounding in algebra, calculus and complex numbers.

'' Foundations of Differential Geometry''

* Shoshichi Kobayashi and Katsumi Nomizu (1963; 1969)

''Hodge Theory and Complex Algebraic Geometry I''

''Hodge Theory and Complex Algebraic Geometry II''

Claire Voisin Claire Voisin (born 4 March 1962) is a French mathematician known for her work in algebraic geometry. She is a member of the French Academy of Sciences and holds the chair of Algebraic Geometry at the Collège de France. Work She is noted for ...

Popular writings

''Gödel, Escher, Bach''

Douglas Hofstadter Douglas Richard Hofstadter (born February 15, 1945) is an American scholar of cognitive science, physics, and comparative literature whose research includes concepts such as the sense of self in relation to the external world, consciousness, an ...

Gödel, Escher, Bach ''Gödel, Escher, Bach: an Eternal Golden Braid'', also known as ''GEB'', is a 1979 book by Douglas Hofstadter. By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. Escher, and composer Johann Sebastian Bach, t ...

: 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 James Roy Newman (1907–1966) was an American mathematician and mathematical historian. He was also a lawyer, practicing in the state of New York from 1929 to 1941. During and after World War II, he held several positions in the United States go ...

The World of Mathematics James Roy Newman (1907–1966) was an American mathematician and mathematical historian. He was also a lawyer, practicing in the state of New York from 1929 to 1941. During and after World War II, he held several positions in the United States g ...

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

References

{{DEFAULTSORT:Important Publications in Mathematics Publications

Mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

Algebra

'' Baudhayana Sulba Sutra''

'' The Nine Chapters on the Mathematical Art''

''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 ...''

'' Sunzi Suanjing''

''Aryabhatiya ''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Sanskrit astronomical treatise, is the ''magnum opus'' and only known surviving work of the 5th century Indian mathematician Aryabhata. Philosopher of astronomy Roger Billard estimates that th ...''

''Jigu Suanjing ''Jigu suanjing'' ( zh, 緝古算經, ''Continuation of Ancient Mathematics'') was the work of early Tang dynasty calendarist and mathematician Wang Xiaotong, written some time before the year 626, when he presented his work to the Emperor. ''Jig ...''

'' Brāhmasphuṭasiddhānta''

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

''Yigu yanduan ''Yigu yanduan'' (益古演段 Old Mathematics in Expanded Sections) is a 13th-century mathematical work by Yuan dynasty mathematician Li Zhi. Overview ''Yigu yanduan'' was based on Northern Song mathematician Jiang Zhou's (蒋周) ''Yigu Ji ...''

'' Mathematical Treatise in Nine Sections''

''Ceyuan haijing ''Ceyuan haijing'' () is a treatise on solving geometry problems with the algebra of Tian yuan shu written by the mathematician Li Zhi in 1248 in the time of the Mongol Empire. It is a collection of 692 formula and 170 problems, all derived from ...''

''Jade Mirror of the Four Unknowns ''Jade Mirror of the Four Unknowns'', ''Siyuan yujian'' (), also referred to as ''Jade Mirror of the Four Origins'', is a 1303 mathematical monograph by Yuan dynasty mathematician Zhu Shijie. Zhu advanced Chinese algebra with this Magnum opus. ...''

'' Ars Magna''

''Vollständige Anleitung zur Algebra''

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

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

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

''Traité des substitutions et des équations algébriques''

''Theorie der Transformationsgruppen''

''Solvability of groups of odd order''

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

''Homological Algebra''

" Sur Quelques Points d'Algèbre Homologique"

Algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

''Theorie der Abelschen Functionen''

''Faisceaux Algébriques Cohérents''

'' Géométrie Algébrique et Géométrie Analytique''

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

'' Éléments de géométrie algébrique''

'' Séminaire de géométrie algébrique''

'' Brāhmasphuṭasiddhānta''

''De fractionibus continuis dissertatio''

''Recherches d'Arithmétique''

'' Disquisitiones Arithmeticae''

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

" Über die Anzahl der Primzahlen unter einer gegebenen Grösse"

'' Vorlesungen über Zahlentheorie''

''Zahlbericht 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 w ...''

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

"Automorphic Forms on GL(2) ''Automorphic Forms on GL(2)'' is a mathematics book by where they rewrite Erich Hecke's theory of modular forms in terms of the representation theory of GL(2) over local fields and adele rings of global fields and prove the Jacquet–Langlands co ..."

"La conjecture de Weil. I."

"Endlichkeitssätze für abelsche Varietäten über Zahlkörpern"

"Modular Elliptic Curves and Fermat's Last Theorem"

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

"Le lemme fondamental pour les algèbres de Lie"

"Perfectoid space"

Analysis

''Introductio in analysin infinitorum''

Calculus

''Yuktibhāṣā ''Yuktibhāṣā'' ( ml, യുക്തിഭാഷ, lit=Rationale), also known as Gaṇita-yukti-bhāṣā and (''Compendium of Astronomical Rationale''), is a major treatise on Indian mathematics, mathematics and Hindu astronomy, astronomy, ...''

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

''Philosophiae Naturalis Principia Mathematica Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...''

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

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

''Intégrale, longueur, aire''

Complex analysis

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

Functional analysis

''Théorie des opérations linéaires''

''Produits Tensoriels Topologiques et Espaces Nucléaires''

''Sur certains espaces vectoriels topologiques''

Fourier analysis

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

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

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

'' Baudhayana Sulba Sutra''

'' The Nine Chapters on the Mathematical Art''

'' The Conics''

''Surya Siddhanta The ''Surya Siddhanta'' (; ) is a Sanskrit treatise in Indian astronomy dated to 505 CE,Menso Folkerts, Craig G. Fraser, Jeremy John Gray, John L. Berggren, Wilbur R. Knorr (2017)Mathematics Encyclopaedia Britannica, Quote: "(...) its Hindu inven ...''

''Aryabhatiya ''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Sanskrit astronomical treatise, is the ''magnum opus'' and only known surviving work of the 5th century Indian mathematician Aryabhata. Philosopher of astronomy Roger Billard estimates that th ...''

''La Géométrie ''La Géométrie'' was published in 1637 as an appendix to ''Discours de la méthode'' (''Discourse on the Method''), written by René Descartes. In the ''Discourse'', he presents his method for obtaining clarity on any subject. ''La Géométrie ...''

''Grundlagen der Geometrie''

'' Regular Polytopes''

Differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

''Recherches sur la courbure des surfaces''

''Disquisitiones generales circa superficies curvas''

''Über die Hypothesen, welche der Geometrie zu Grunde Liegen''

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

''Analysis situs''

''
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 ...
''

''
Aryabhatiya ''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Sanskrit astronomical treatise, is the ''magnum opus'' and only known surviving work of the 5th century Indian mathematician Aryabhata. Philosopher of astronomy Roger Billard estimates that th ...
''

''
Jigu Suanjing ''Jigu suanjing'' ( zh, 緝古算經, ''Continuation of Ancient Mathematics'') was the work of early Tang dynasty calendarist and mathematician Wang Xiaotong, written some time before the year 626, when he presented his work to the Emperor. ''Jig ...
''

''
Yigu yanduan ''Yigu yanduan'' (益古演段 Old Mathematics in Expanded Sections) is a 13th-century mathematical work by Yuan dynasty mathematician Li Zhi. Overview ''Yigu yanduan'' was based on Northern Song mathematician Jiang Zhou's (蒋周) ''Yigu Ji ...
''

''
Ceyuan haijing ''Ceyuan haijing'' () is a treatise on solving geometry problems with the algebra of Tian yuan shu written by the mathematician Li Zhi in 1248 in the time of the Mongol Empire. It is a collection of 692 formula and 170 problems, all derived from ...
''

''
Jade Mirror of the Four Unknowns ''Jade Mirror of the Four Unknowns'', ''Siyuan yujian'' (), also referred to as ''Jade Mirror of the Four Origins'', is a 1303 mathematical monograph by Yuan dynasty mathematician Zhu Shijie. Zhu advanced Chinese algebra with this Magnum opus. ...
''

''
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 w ...
''

"
Automorphic Forms on GL(2) ''Automorphic Forms on GL(2)'' is a mathematics book by where they rewrite Erich Hecke's theory of modular forms in terms of the representation theory of GL(2) over local fields and adele rings of global fields and prove the Jacquet–Langlands co ...
"

''
Yuktibhāṣā ''Yuktibhāṣā'' ( ml, യുക്തിഭാഷ, lit=Rationale), also known as Gaṇita-yukti-bhāṣā and (''Compendium of Astronomical Rationale''), is a major treatise on Indian mathematics, mathematics and Hindu astronomy, astronomy, ...
''

''
Philosophiae Naturalis Principia Mathematica Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
''

''
Surya Siddhanta The ''Surya Siddhanta'' (; ) is a Sanskrit treatise in Indian astronomy dated to 505 CE,Menso Folkerts, Craig G. Fraser, Jeremy John Gray, John L. Berggren, Wilbur R. Knorr (2017)Mathematics Encyclopaedia Britannica, Quote: "(...) its Hindu inven ...
''

''
Aryabhatiya ''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Sanskrit astronomical treatise, is the ''magnum opus'' and only known surviving work of the 5th century Indian mathematician Aryabhata. Philosopher of astronomy Roger Billard estimates that th ...
''

''
La Géométrie ''La Géométrie'' was published in 1637 as an appendix to ''Discours de la méthode'' (''Discourse on the Method''), written by René Descartes. In the ''Discourse'', he presents his method for obtaining clarity on any subject. ''La Géométrie ...
''

''
Categories for the Working Mathematician ''Categories for the Working Mathematician'' (''CWM'') is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. It was first published in 1971, and is based on ...
''

''
Higher Topos Theory ''Higher Topos Theory'' is a treatise on the theory of ∞-categories written by American mathematician Jacob Lurie. In addition to introducing Lurie's new theory of ∞-topoi, the book is widely considered foundational to higher category theory ...
''

''
Grundzüge der Mengenlehre ''Grundzüge der Mengenlehre'' (German for "Basics of Set Theory") is a book on set theory written by Felix Hausdorff. First published in April 1914, ''Grundzüge der Mengenlehre'' was the first comprehensive introduction to set theory. Besides th ...
''

''
Systems of Logic Based on Ordinals ''Systems of Logic Based on Ordinals'' was the PhD dissertation of the mathematician Alan Turing. Turing's thesis is not about a new type of formal logic, nor was he interested in so-called ‘ranked logic’ systems derived from ordinal or relat ...
''

''
Theory of Games and Economic Behavior ''Theory of Games and Economic Behavior'', published in 1944 by Princeton University Press, is a book by mathematician John von Neumann and economist Oskar Morgenstern which is considered the groundbreaking text that created the interdisciplinar ...
''

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

''
Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased ...
''

''
Archimedes Palimpsest The Archimedes Palimpsest is a parchment codex palimpsest, originally a Byzantine Greek copy of a compilation of Archimedes and other authors. It contains two works of Archimedes that were thought to have been lost (the ''Ostomachion'' and the ' ...
''

''
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'', tho ...
''

''
Algebraic Geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
''

''
Naive Set Theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike Set theory#Axiomatic set theory, axiomatic set theories, which are defined using Mathematical_logic#Formal_logical_systems, forma ...
''