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[[File:Oxyrhynchus papyrus with Euclid's Elements.jpg|right|thumb|250px|One of the oldest surviving fragments of [[Euclid's Elements|Euclid's ''Elements'']], found at [[Oxyrhynchus]] and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.<ref>{{cite web
|url=http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html
|title=One of the Oldest Extant Diagrams from Euclid
|author=Bill Casselman
|publisher=University of British Columbia
|accessdate=2008-09-26
}}</ref>]]
{{Use dmy dates|date=October 2010}}
{{List of publications intro|[[mathematics]]}} <!-- do not remove or change! -->
<!-- to edit introduction, see "Template:List of publications intro" -->
 
Among published compilations of important publications in mathematics are ''Landmark writings in Western mathematics 1640–1940'' by [[Ivor Grattan-Guinness]]<ref>[[Ivor Grattan-Guinness]], ''Landmark writings in Western mathematics 1640–1940'', Elsevier Science, 2005</ref> and ''A Source Book in Mathematics'' by [[David Eugene Smith]].<ref>[[David Eugene Smith]], ''A Source Book in Mathematics'', Dover Publications, 1984</ref>
{{TOC limit|limit=3}}
 
==Algebra==
 
===[[Theory of equations]]===
 
====''[[Baudhayana]] [[Sulba Sutras|Sulba Sutra]]''====
 
* [[Baudhayana]] (8th century BC)
'''Description:''' Believed to have been written around the 8th century BC, this is one of the oldest mathematical texts. It laid the foundations of [[Indian mathematics]] and was influential in [[South Asia]] and its surrounding regions, and [[Indian mathematics#Charges of Eurocentrism|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 ax<sup>2</sup> = c and ax<sup>2</sup> + bx = c, and integral solutions of simultaneous [[Diophantine equation]]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.
'''Description''':Contains the earliest description of [[Gaussian elimination]] for solving
system of linear equations, it also contains method for finding square root and cubic root.
 
====''[[The Sea Island Mathematical Manual]]''====
 
*[[Liu Hui]] (220-280)
 
'''Description''', contains the application of right angle triangles for survey of depth or height of distant objects.
 
====''[[The Mathematical Classic of Sun Zi]]''====
 
*Sunzi (5th century)
 
'''Description:''' Contains the earlist description of [[Chinese remainder theorem]].
 
====''[[Jigu Suanjing]]''====
 
[[Jigu Suanjing]] (626AD)
 
'''Description:''' This book by Tang dynasty mathematician Wang Xiaotong Contains the world's earliest third order equation.
 
====''[[Aryabhatiya]]''====
 
*[[Aryabhata]] (499 AD)
 
'''Description:'''  Aryabhatia 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.
 
====''[[Brāhmasphuṭasiddhānta]]''====
 
*[[Brahmagupta]] (628 AD)
 
'''Description:'''  Contained rules for manipulating both negative and positive numbers, a method for computing square roots, and general methods of solving linear and some quadratic equations.
<ref name=Sharma>{{cite book |title=Mathematics & Astronomers of Ancient India |author=Shashi S. Sharma |publisher=Pitambar |isbn=978-81-209-1421-6 |page=29 |quote=Brahmagupta is believed to have composed many important works of mathematics and astronomy. However, two of his most important works are: Brahmasphutasiddhanta (BSS) written in 628 AD, and the Khandakhadyaka...}}</ref>
<ref name="Petković">{{cite book |title=Famous puzzles of great mathematicians |author=Miodrag Petković |publisher=[[American Mathematical Society]] |year=2009 |isbn=978-0-8218-4814-2 |pages=77, 299 |quote=many important results from astronomy, arithmetic and algebra", "major work}}</ref>
<ref name=Selaine>{{cite book |title=Encyclopaedia of the history of science, technology, and medicine in non-western cultures |editor=Helaine Selin |editor-link=Helaine Selin |publisher=Springer |year=1997 |isbn=978-0-7923-4066-9 |page=162 |quote=holds a remarkable place in the history of Eastern civilzation", "most important work", "remarkably modern in outlook", "marvelous piece of pure mathematics", "more remarkable algebraic contributions", "important step towards the integral solutions of [second-order indeterminate] equations", "In geometry, Brahmagupta's achievements were equally praiseworthy.}}</ref>
<ref name=Tabak>{{cite book |title=Algebra: sets, symbols, and the language of thought |author=John Tabak |publisher=Infobase Publishing |year=2004 |isbn=978-0-8160-4954-7 |pages=38''ff'' |quote=Brahmagupta's masterpiece", "a great deal of important algebra", "The ''Brahma-sphuta-siddhānta'' was quickly recognized by Brahmagupta's contemporaries as an important and imaginative work. It inspired numerous commentaries by many generations of mathematicians.}}</ref>
 
====''[[The Compendious Book on Calculation by Completion and Balancing|Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa'l-muqābala]]''====
 
* [[Muhammad ibn Mūsā al-Khwārizmī]] (820)
 
'''Description:''' The first book on the systematic [[algebra]]ic solutions of [[linear equation|linear]] and [[quadratic equation]]s by the [[Persian people|Persian]] scholar [[Muhammad ibn Mūsā al-Khwārizmī]].  The book is considered to be the foundation of modern [[algebra]] and [[Islamic mathematics]].{{Citation needed|date=June 2011}}  The word "algebra" itself is derived from the ''al-Jabr'' in the title of the book.<ref name=clark>{{Cite book|title=Elements of abstract algebra|first=Allan|last=Clark|publisher=Courier Dover Publications|location=United States|year=1984|isbn=978-0-486-64725-8|ref=harv|page=ix}}</ref>
 
====[[Yigu yanduan]]====
 
*Liu Yi (12th century)
 
Contains the earliest invention of 4th order polynomial equation.
 
====''[[Mathematical Treatise in Nine Sections]]''====
 
*[[Qin Jiushao]] (1247)
 
'''Description''': 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 [[Chinese remainder theorem]], predates [[Euler]] and [[Gauss]] by several centuries.
 
====''[[Ceyuan haijing]]''====
 
*[[Li Zhi (mathematician)|Li Zhi]] (1248)
 
'''Description''':Contains the application of high order polynomial equation  in solving complex geometry problems.
 
====''[[Zhu Shijie#Jade Mirror of the Four Unknowns|Jade Mirror of the Four Unknowns]]''====
 
*[[Zhu Shijie]] (1303)
 
'''Description''' Contains the method of establishing  system of high order polynomial equations of up to four unknowns.
 
====''[[Ars Magna (Gerolamo Cardano)|Ars Magna]]''====
 
* [[Gerolamo Cardano]] (1545)
'''Description:'''  Otherwise known as ''The Great Art'', provided the first published methods for solving [[cubic equation|cubic]] and [[quartic equation]]s (due to [[Scipione del Ferro]], [[Niccolò Fontana Tartaglia]], and [[Lodovico Ferrari]]), and exhibited the first published calculations involving non-real [[complex numbers]].<ref name=arsMagna>{{Cite web| last1 = O'Connor | first1 = J. J. | last2 = Robertson  | first2 = E. F. | title=Girolamo Cardano  | url=http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Cardan.html| year=1998}}</ref><ref name=Fierz>{{cite book |title=Girolamo Cardano:
1501-1576. Physician, Natural Philosopher, Mathematician|author=Markus Fierz|year=1983|publisher=Birkhäuser Boston|isbn=978-0-8176-3057-7}}</ref>
 
====''Vollständige Anleitung zur Algebra''====
 
* [[Leonhard Euler]] (1770)
'''Description:''' Also known as [[Elements of Algebra]], Euler's textbook on elementary algebra is one of the first to set out algebra in the modern form we would recognize today.  The first volume deals with determinate equations, while the second part deals with [[Diophantine equation]]s.  The last section contains a proof of [[Fermat's Last Theorem]] for the case ''n''&nbsp;=&nbsp;3, making some valid assumptions regarding '''Q'''(√−3) that Euler did not prove.<ref name=Weil239>{{Cite book| last = Weil | first = André  | title = Number Theory: An approach through history From Hammurapi to Legendre | publisher = Birkhäuser | year = 1984 | pages = 239–242 | isbn = 0-8176-3141-0}}</ref>
 
====''Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse''====
 
* [[Carl Friedrich Gauss]] (1799)
'''Description:''' Gauss' doctoral dissertation,<ref name=fta>{{Cite web| last = Gauss | first = J.C.F. | title=Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse|url=http://www.thelatinlibrary.com/gauss.html| year=1799}}</ref> which contained a widely accepted (at the time) but incomplete proof<ref name=mactutor>{{Cite web| last1 = O'Connor | first1 = J. J. | last2 = Robertson  | first2 = E. F. | title=The fundamental theorem of algebra  | url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fund_theorem_of_algebra.html| year=1996}}</ref> of the [[fundamental theorem of algebra]].
 
===[[Abstract algebra]]===
 
====[[group (mathematics)|Group theory]]====
 
=====''Réflexions sur la résolution algébrique des équations''=====
*[[Joseph Louis Lagrange]] (1770)
'''Description:''' 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 groups]], [[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)
'''Description:''' 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:''' [http://books.google.com/books?id=TzQAAAAAQAAJ Online version]
 
'''Description:''' 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''),<ref name=19thcentury>{{Cite book| title = Mathematics of the 19th Century: Mathematical Logic, Algebra, Number Theory, and Probability Theory | publisher =  Birkhäuser Verlag | year = 2001 | pages = 39, 63, 66–68 | isbn = 3-7643-6441-6 | author = ed. by A. N. Kolmogorov...}}</ref> proved part of the [[Jordan–Hölder theorem]], and discussed matrix groups over finite fields as well as the [[Jordan normal form]].<ref name=traitedessubstitutions>{{Cite web| last1 = O'Connor | first1 = J. J. | last2 = Robertson  | first2 = E. F. | title= Marie Ennemond Camille Jordan  | url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Jordan.html| year=2001}}</ref>
 
====''Theorie der Transformationsgruppen''====
 
*[[Sophus Lie]], [[Friedrich Engel (mathematician)|Friedrich Engel]] (1888–1893).
 
'''Publication data:''' 3 volumes, B.G. Teubner, Verlagsgesellschaft, mbH, Leipzig, 1888–1893.  [http://www.archive.org/details/theotransformation01liesrich Volume 1], [http://www.archive.org/details/theotransformation02liesrich Volume 2], [http://www.archive.org/details/theotransformation03liesrich Volume 3].
 
'''Description:''' The first comprehensive work on [[transformation group]]s, serving as the foundation for the modern theory of [[Lie group]]s.
 
====''Solvability of groups of odd order''====
 
*[[Walter Feit]] and [[John Griggs Thompson|John Thompson]] (1960)
'''Description:''' Gave a complete proof of the [[Feit–Thompson theorem|solvability of finite groups of odd order]], establishing the long-standing Burnside conjecture that all finite non-abelian simple groups are of even order. Many of the original techniques used in this paper were used in the eventual [[classification of finite simple groups]].
 
====[[Homological algebra]]====
 
====Homological Algebra====
 
* [[Henri Cartan]] and [[Samuel Eilenberg]] (1956)
'''Description:''' Provided the first fully worked out treatment of abstract homological algebra, unifying previously disparate presentations of homology and cohomology for [[associative algebra]]s, [[Lie algebra]]s, and [[group (mathematics)|group]]s into a single theory.
 
====Sur Quelques Points d'Algèbre Homologique====
 
* [[Alexander Grothendieck]] (1957)
'''Description:''' Revolutionized [[homological algebra]] by introducing [[abelian category|abelian categories]] and providing a general framework for Cartan and Eilenberg's notion of [[derived functor]]s.
 
==[[Algebraic geometry]]==
 
===Theorie der Abelschen Functionen===
 
* [[Bernhard Riemann]] (1857)
'''Publication data:''' ''Journal für die Reine und Angewandte Mathematik''
 
'''Description:''' 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 [[Niels Henrik Abel|Abel]] and [[Carl Gustav Jacob Jacobi|Jacobi]].  [[André Weil]] once wrote that this paper "''is one of the greatest pieces of mathematics that has ever been written; there is not a single word in it that is not of consequence.''" <ref name=Krieger2007>
{{Cite journal
  | last = Krieger
  | first = Martin H.
  |date=March 2007
  | title = A 1940 Letter of André Weil on Analogy in Mathematics
  | journal = [[Notices of the American Mathematical Society]]
  | volume = 52
  | issue = 3
  | page = 338
  | url = http://www.ams.org/notices/200503/fea-weil.pdf
  | format = PDF
  | accessdate = 13 January 2008}}</ref>
 
===''Faisceaux Algébriques Cohérents''===
 
* [[Jean-Pierre Serre]]
'''Publication data:''' ''Annals of Mathematics'', 1955
 
'''Description:''' ''FAC'', as it is usually called, was foundational for the use of [[sheaf (mathematics)|sheaves]] in algebraic geometry, extending beyond the case of [[complex manifold]]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]] is finite, in [[projective geometry]], and such dimensions include many discrete invariants of varieties, for example [[Hodge number]]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.
 
===''[[Algebraic geometry and analytic geometry|Géométrie Algébrique et Géométrie Analytique]]''===
 
* [[Jean-Pierre Serre]] (1956)
'''Description:''' In [[mathematics]], [[algebraic geometry]] and [[analytic geometry]] are closely related subjects, where ''analytic geometry'' is the theory of [[complex manifold]]s and the more general [[analytic geometry|analytic space]]s defined locally by the vanishing of [[analytic function]]s of [[several complex variables]]. A (mathematical) theory of the relationship between the two was put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from [[Hodge theory]]. (''NB'' While [[analytic geometry]] as use of Cartesian coordinates is also in a sense included in the scope of algebraic geometry, that is not the topic being discussed in this article.) The major paper consolidating the theory was ''Géometrie Algébrique et Géométrie Analytique'' by [[Jean-Pierre Serre|Serre]], now usually referred to as ''GAGA''. A ''GAGA-style result'' would now mean any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, and a well-defined subcategory of analytic geometry objects and holomorphic mappings.
 
===Le théorème de Riemann–Roch, d'après A. Grothendieck===
 
* [[Armand Borel]], [[Jean-Pierre Serre]] (1958)
'''Description:''' Borel and Serre's exposition of Grothendieck's version of the [[Grothendieck–Hirzebruch–Riemann–Roch theorem|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.<ref name=Jackson2004>{{Cite journal| last = Jackson  | first = Allyn  |date=October 2004 | title = Comme Appelé du Néant — As If Summoned from the Void: The Life of Alexandre Grothendieck  | journal = [[Notices of the American Mathematical Society]]  | volume = 51  | issue = 9  | pages = 1045–1046
  | url = http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf  | format = PDF  | accessdate = 13 January 2008}}</ref>  In his proof, Grothendieck broke new ground with his concept of [[Grothendieck group]]s, which led to the development of [[K-theory]].<ref name=Dieudonné598>{{Cite book| last = Dieudonné | first = Jean | title = A history of algebraic and differential topology 1900–1960 | publisher = Birkhäuser | year = 1989 | pages = 598–600 | isbn = 0-8176-3388-X}}</ref>
 
===''[[Éléments de géométrie algébrique]]''===
 
* [[Alexander Grothendieck]] (1960–1967)
'''Description:''' Written with the assistance of [[Jean Dieudonné]], this is [[Grothendieck]]'s exposition of his reworking of the foundations of algebraic geometry.  It has become the most important foundational work in modern algebraic geometry.  The approach expounded in EGA, as these books are known, transformed the field and led to monumental advances.
 
===''[[Séminaire de géométrie algébrique]]''===
 
* [[Alexander Grothendieck]] et al.
'''Description:''' 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]], [[Pierre Deligne]], and [[Nicholas Katz]].
 
==[[Number theory]]==
 
===''[[Brāhmasphuṭasiddhānta]]''===
 
*[[Brahmagupta]] (628)
'''Description:''' 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)
'''Description:''' First presented in 1737, this paper <ref name=fractionibuscontinuis>{{Cite web| last = Euler | first = L.| title=De fractionibus continuis dissertatio|url=http://www.math.dartmouth.edu/~euler/docs/originals/E071.pdf| year=1744|format=PDF|accessdate=23 June 2009}}</ref> provided the first then-comprehensive account of the properties of [[continued fractions]].  It also contains the first proof that the number [[e (mathematical constant)|e]] is irrational.<ref name=sandifer>{{Cite journal| last = Sandifer | first = Ed | title = How Euler Did It: Who proved e is irrational? | journal = MAA Online |date=February 2006 | url = http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2028%20e%20is%20irrational.pdf|format=PDF|accessdate=23 June 2009}}</ref>
 
===''Recherches d'Arithmétique''===
 
*[[Joseph Louis Lagrange]] (1775)
'''Description:''' Developed a general theory of [[binary quadratic form]]s to handle the general problem of when an integer is representable by the form <math>ax^2 + by^2 + cxy</math>. This included a reduction theory for binary quadratic forms, where he proved that every form is equivalent to a certain canonically chosen reduced form.<ref name=Goldfeld>{{Cite journal| last = Goldfeld  | first = Dorian  |date=July 1985 | title = Gauss' Class Number Problem For Imaginary Quadratic Fields  | journal = [[Bulletin of the American Mathematical Society]]  | volume = 13 | issue = 1  | page = 24  | url = http://www.ams.org/bull/1985-13-01/S0273-0979-1985-15352-2/S0273-0979-1985-15352-2.pdf  | format = PDF  | doi = 10.1090/S0273-0979-1985-15352-2}}</ref><ref name=Weil316>{{Cite book| last = Weil | first = André  | title = Number Theory: An approach through history From Hammurapi to Legendre | publisher = Birkhäuser | year = 1984 | pages = 316–322 | isbn = 0-8176-3141-0}}</ref>
 
===''[[Disquisitiones Arithmeticae]]''===
 
*[[Carl Friedrich Gauss]] (1801)
'''Description:''' The ''[[Disquisitiones Arithmeticae]]'' is a profound and masterful book on [[number theory]] written by [[Germany|German]] [[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]], [[Euler]], [[Joseph Louis Lagrange|Lagrange]] and [[Adrien-Marie Legendre|Legendre]] and adds many important new results of his own.  Among his contributions was the first complete proof known of the [[Fundamental theorem of arithmetic]], 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]], [[cyclotomy#Cyclotomic fields|cyclotomy]], and the theory of [[constructible polygon]]s with a particular application to the constructibility of the regular [[Heptadecagon|17-gon]].  Of note, in section V, article 303 of Disquisitiones, Gauss summarized his calculations of [[Class number (number theory)|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 [[Class number problem|conjectured]].<ref name=irelandrosen>{{Cite book| last1 = Ireland | first1 = K. |last2 = Rosen | first2 = M. | title = A Classical Introduction to Modern Number Theory  | publisher = Springer-Verlag | year = 1993  | location = New York, New York  | pages = 358–361  | isbn = 0-387-97329-X}}</ref>  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]]).<ref name=silvermantate>{{Cite book| last1 = Silverman | first1 = J. |last2 = Tate | first2 = J. | title = Rational Points on Elliptic Curves  | publisher = Springer-Verlag | year = 1992 | location = New York, New York  | isbn = 0-387-97825-9  | page = 110}}</ref>
 
===''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)
'''Description:''' Pioneering paper in [[analytic number theory]], which introduced [[Dirichlet characters]] and their [[Dirichlet L-function|L-functions]] to establish [[Dirichlet's theorem on arithmetic progressions]].<ref name=Elstrodt>{{Cite journal| last = Elstrodt | first = Jürgen | journal = Clay Mathematics Proceedings  | title = The Life and Work of Gustav Lejeune Dirichlet (1805–1859) | publisher = | year = 2007 | url = http://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf | format = PDF |pages = 21–22}}</ref>  In subsequent publications, Dirichlet used these tools to determine, among other things, the class number for quadratic forms.
 
===''[[On the Number of Primes Less Than a Given Magnitude|Über die Anzahl der Primzahlen unter einer gegebenen Grösse]]''===
 
*[[Bernhard Riemann]] (1859)
'''Description:''' ''Über die Anzahl der Primzahlen unter einer gegebenen Grösse'' (or ''On the Number of Primes Less Than a Given Magnitude'') is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the ''Monthly Reports of the Berlin Academy''. Although it is the only paper he ever published on number theory, it contains ideas which influenced dozens of researchers during the late 19th century and up to the present day. The paper consists primarily of definitions, heuristic arguments, sketches of proofs, and the application of powerful analytic methods; all of these have become essential concepts and tools of modern [[analytic number theory]].  It also contains the famous [[Riemann Hypothesis]], one of the most important open problems in mathematics.<ref>[[Harold Edwards (mathematician)|H. M. Edwards]], ''Riemann's Zeta Function'', Academic Press, 1974</ref>
 
===''[[Vorlesungen über Zahlentheorie]]''===
 
* [[Peter Gustav Lejeune Dirichlet]] and [[Richard Dedekind]]
'''Description:''' ''[[Vorlesungen über Zahlentheorie]]'' (''Lectures on Number Theory'') is a textbook of [[number theory]] written by [[Germany|German]] mathematicians P. G. Lejeune Dirichlet and R. Dedekind, and published in 1863.
The ''Vorlesungen'' can be seen as a watershed between the classical number theory of [[Fermat]], [[Carl Gustav Jakob Jacobi|Jacobi]] and [[Carl Friedrich Gauss|Gauss]], and the modern number theory of Dedekind, [[Bernhard Riemann|Riemann]] and [[David Hilbert|Hilbert]]. Dirichlet does not explicitly recognise the concept of the [[group theory|group]] that is central to [[Abstract algebra|modern algebra]], but many of his proofs show an implicit understanding of group theory
 
===''Zahlbericht''===
 
{{main|Zahlbericht}}
* [[David Hilbert]] (1897)
'''Description:''' Unified and made accessible many of the developments in [[algebraic number theory]] made during the nineteenth century. Although criticized by [[André Weil]] (who stated "''more than half of his famous Zahlbericht is little more than an account of [[Ernst Kummer|Kummer]]'s number-theoretical work, with inessential improvements''")<ref name=Zahlbericht1>
{{Cite web
  | last = Lemmermeyer
  | first = Franz
  | coauthors = Schappacher, Norbert
  | title=Introduction to the English Edition of Hilbert’s Zahlbericht
  | page = 3
  | url=http://www.fen.bilkent.edu.tr/~franz/publ/hil.pdf
  | format = PDF
  | accessdate=13 January 2008}}</ref> and [[Emmy Noether]],<ref name=Zahlbericht2>{{Cite web
  | last = Lemmermeyer
  | first = Franz
  | coauthors = Schappacher, Norbert
  | title=Introduction to the English Edition of Hilbert’s Zahlbericht
  | page = 5
  | url=http://www.fen.bilkent.edu.tr/~franz/publ/hil.pdf
  | format = PDF
  | accessdate=13 January 2008}}</ref> it was highly influential for many years following its publication.
 
===Fourier Analysis in Number Fields and Hecke's Zeta-Functions===
 
* [[John Tate]] (1950)
'''Description:''' Generally referred to simply as ''[[Tate's Thesis]]'', Tate's [[Princeton University|Princeton]] Ph.D. thesis, under [[Emil Artin]], is a reworking of [[Erich Hecke]]'s theory of zeta- and ''L''-functions in terms of [[Fourier analysis]] on the [[Adele ring|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]]s.
 
===Automorphic Forms on GL(2)===
 
* [[Hervé Jacquet]] and [[Robert Langlands]] (1970)
'''Description:''' This publication offers evidence towards Langlands' conjectures by reworking and expanding the classical theory of [[modular form]]s and their ''L''-functions through the introduction of representation theory.
 
===La conjecture de Weil. I.===
 
* [[Pierre Deligne]] (1974)
'''Description:''' Proved the Riemann hypothesis for varieties over finite fields, settling the last of the open [[Weil conjectures]].
 
===Endlichkeitssätze für abelsche Varietäten über Zahlkörpern===
 
* [[Gerd Faltings]] (1983)
'''Description:''' Faltings proves a collection of important results in this paper, the most famous of which is the first proof of the [[Mordell conjecture]] (a conjecture dating back to 1922). Other theorems proved in this paper include an instance of the [[Tate conjecture]] (relating the [[homomorphism]]s between two [[abelian varieties]] over a [[number field]] to the homomorphisms between their [[Tate module]]s) and some finiteness results concerning abelian varieties over number fields with certain properties.
 
===Modular Elliptic Curves and Fermat's Last Theorem===
 
* [[Andrew Wiles]] (1995)
'''Description:''' This article proceeds to prove a special case of the [[Modularity theorem|Shimura–Taniyama conjecture]] through the study of the [[Galois deformation theory|deformation theory]] of [[Galois representations]]. This in turn implies the famed [[Fermat's Last Theorem]]. The proof's method of identification of a [[deformation ring]] with a [[Hecke operator|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 (mathematician)|Richard Taylor]] (2001)
'''Description:''' Harris and Taylor provide the first proof of the [[local Langlands conjecture]] for [[GL(n)|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]]
'''Description:''' Ngô Bảo Châu proved a long standing unsolved problem in the classical Langlands program, using methods from the Geometric Langlands program.
 
==Analysis==
 
===''Introductio in analysin infinitorum''===
 
* [[Leonhard Euler]] (1748)
'''Description:''' The eminent historian of mathematics [[Carl Boyer]] once called Euler's ''[[Introductio in analysin infinitorum]]'' the greatest modern textbook in mathematics.<ref name=Alexanderson>{{Cite journal
  | last = Alexanderson  | first = Gerald L.  |date=October 2007 | title = Euler’s Introductio In Analysin Infinitorum  | journal = [[Bulletin of the American Mathematical Society]]  | volume = 44
  | issue = 4  | pages = 635–639  | url = http://www.ams.org/bull/2007-44-04/S0273-0979-07-01183-4/S0273-0979-07-01183-4.pdf  | format = PDF
  | doi = 10.1090/S0273-0979-07-01183-4}}</ref>  Published in two volumes,<ref name=IntroductioI>{{Cite web| last = Euler | first = L.| title=E101 – Introductio in analysin infinitorum, volume 1|url=http://math.dartmouth.edu/~euler/pages/E101.html| accessdate=16 March 2008}}</ref><ref name=IntroductioII>{{Cite web| last = Euler | first = L.| title=E102 – Introductio in analysin infinitorum, volume 2|url=http://math.dartmouth.edu/~euler/pages/E102.html| accessdate=16 March 2008}}</ref> this book more than any other work succeeded in establishing [[Mathematical analysis|analysis]] as a major branch of mathematics, with a focus and approach distinct from that used in geometry and algebra.<ref name=calinger>{{Cite book
  | last = Calinger  | first = Ronald  | title = Classics of Mathematics  | publisher = Moore Publishing Company, Inc.  | year = 1982  | location = Oak Park, Illinois  | pages = 396–397  | isbn = 0-935610-13-8}}</ref>  Notably, Euler identified functions rather than curves to be the central focus in his book.<ref name=functions>{{Cite web| last1 = O'Connor | first1 = J. J. | last2 = Robertson  | first2 = E. F. | title=The function concept  | url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Functions.html| year=1995}}</ref>  Logarithmic, exponential, trigonometric, and transcendental functions were covered, as were  expansions into partial fractions, evaluations of {{math|ζ(2k)}} for {{math|k}} a positive integer between 1 and 13, infinite series-infinite product formulas,<ref name=Alexanderson/> [[continued fractions]], and [[Partition (number theory)|partitions]] of integers.<ref name=Andrews>{{Cite journal
  | last = Andrews  | first = George E.  |date=October 2007 | title = Euler’s "De Partitio Numerorum"  | journal = [[Bulletin of the American Mathematical Society]]  | volume = 44
  | issue = 4  | pages = 561–573  | url = http://www.ams.org/bull/2007-44-04/S0273-0979-07-01180-9/S0273-0979-07-01180-9.pdf  | format = PDF  | doi = 10.1090/S0273-0979-07-01180-9}}</ref>  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 {{math|e}} and <math alt="square root of e">\textstyle\sqrt{e}</math>.<ref name=IntroductioI/> 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]]''====
 
*[[Jyeshtadeva]] (1501)
'''Description:''' Written in [[Indian mathematics|India]] in 1501, this was the world's first calculus text. "This work laid the foundation for a complete system of fluxions"<ref>{{Cite journal
| author =Charles Whish
| year = 1834
| title = On the [[Hindu]] [[Quadrature of the circle]] and the [[infinite series]] of the proportion of the circumference to the diameter exhibited in the four [[Sastra]]s, the [[Tantrasangraha|Tantra Sahgraham]], [[Yuktibhāṣā|Yucti Bhasha]], [[Karanapaddhati|Carana Padhati]] and [[Sadratnamala]]
| journal = Transactions of the Royal Asiatic Society of Great Britain and Ireland
| publisher = [[Royal Asiatic Society of Great Britain and Ireland]]
| doi=10.1017/S0950473700001221
| volume=3
| issue=3
| pages=509–523
| jstor=25581775
| postscript =<!--None-->
| authorlink = C.M. Whish
}}</ref>
{{Citation needed|date=July 2010|reason=source cannot be verified}} and served as a summary of the [[Kerala school of astronomy and mathematics|Kerala School]]'s achievements in calculus, [[trigonometry]] and [[mathematical analysis]], most of which were earlier discovered by the 14th century  mathematician [[Madhava of Sangamagrama|Madhava]]. It's possible that this text influenced the later development of calculus in Europe. Some of its important developments in calculus include: the fundamental ideas of [[derivative|differentiation]] and [[Integral|integration]], the [[derivative]], [[differential equation]]s, term by term integration, numerical integration by means of infinite series, the relationship between the area of a curve and its integral, and the [[mean value theorem]].
 
====''Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illi calculi genus''====
 
*[[Gottfried Leibniz]] (1684)
'''Description:''' Leibniz's first publication on differential calculus, containing the now familiar notation for differentials as well as rules for computing the derivatives of powers, products and quotients.
 
====''[[Philosophiae Naturalis Principia Mathematica]]''====
 
*[[Isaac Newton]]
'''Description:''' 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 [[Gravity|law of universal gravitation]], and derives [[laws of Kepler|Kepler's laws]] for the motion of the [[planet]]s (which were first obtained empirically). Here was born the practice, now so standard we identify it with science, of explaining nature by postulating mathematical axioms and demonstrating that their conclusion are observable phenomena.  In formulating his physical theories, Newton freely used his unpublished work on calculus.  When he submitted Principia for publication, however, Newton chose to recast the majority of his proofs as geometric arguments.<ref name=Gray>{{Cite web| last = Gray | first = Jeremy| title=MAA Book Review: Reading the Principia: The Debate on Newton's Mathematical Methods for Natural Philosophy from 1687 to 1736 by Niccolò Guicciardini|url=http://www.maa.org/reviews/readnewton.html| year=2000}}</ref>
 
====''Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum''====
 
*[[Leonhard Euler]] (1755)
'''Description:''' Published in two books,<ref name=Institutiones>{{Cite web| last = Euler | first = L.| title=E212 – Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum |url=http://www.math.dartmouth.edu/~euler/pages/E212.html| accessdate=21 March 2008}}</ref> 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.<ref>{{Cite web| last1 = O'Connor | first1 = J. J. | last2 = Robertson  | first2 = E. F. | title=Leonhard Euler | url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Euler.html| year=1998}}</ref>  Also included is a systematic study of [[Bernoulli polynomials]] and the [[Bernoulli numbers]] (naming them as such), a demonstration of how the Bernoulli numbers are related to the coefficients in the [[Euler–Maclaurin formula]] and the values of ζ(2n),<ref name=sandiferSept2005>{{Cite journal| last = Sandifer | first = Ed | title = How Euler Did It: Bernoulli Numbers | journal = MAA Online |date=September 2005 | url = http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2023%20Bernoulli%20numbers.pdf|format=PDF|accessdate=23 June 2009}}</ref>  a further study of [[Euler–Mascheroni constant|Euler's constant]] (including its connection to the [[gamma function]]), and an application of partial fractions to differentiation.<ref name=sandiferJune2007>{{Cite journal| last = Sandifer | first = Ed | title = How Euler Did It: Partial Fractions | journal = MAA Online |date=June 2007 | url = http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2044%20partial%20fractions.pdf|format=PDF|accessdate=23 June 2009}}</ref>
 
====''Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe''====
 
*Bernhard Riemann (1867)
'''Description:''' 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).<ref name=Bressoud>{{Cite book| last = Bressoud | first = David |authorlink = David Bressoud | title = A Radical Approach to Real Analysis | publisher = Mathematical Association of America | year = 2007 | pages = 248–255 | isbn = 0-88385-747-2}}</ref>  He also stated the [[Riemann series theorem]],<ref name=Bressoud/> proved the [[Riemann-Lebesgue lemma]] for the case of bounded Riemann integrable functions,<ref name=Kline>{{Cite book| authorlink = Morris Kline | last = Kline | first = Morris | title = Mathematical Thought From Ancient to Modern Times | publisher = Oxford University Press | year = 1990 | pages = 1046–1047 | isbn = 0-19-506137-3}}</ref> and developed the Riemann localization principle.<ref name=Benedetto>{{Cite book| last = Benedetto | first = John | title = Harmonic Analysis and Applications | publisher = CRC Press | year = 1997 | pages = 170–171 | isbn = 0-8493-7879-6}}</ref>
 
====''Intégrale, longueur, aire''====
 
*[[Henri Lebesgue]] (1901)
 
'''Description:''' Lebesgue's [[Intégrale, longueur, aire|doctoral dissertation]], summarizing and extending his research to date regarding his development of [[measure theory]] and the [[Lebesgue integral]].
 
===Complex analysis===
 
====''Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse''====
 
* Bernhard Riemann (1851)
 
'''Description:''' Riemann's doctoral dissertation introduced the notion of a [[Riemann surface]], [[conformal map]]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 language|Polish]] under the title ''Teorja operacyj''.)
 
'''Description:'''  The first mathematical monograph on the subject of [[linear space|linear]] [[metric space]]s, bringing the abstract study of [[functional analysis]] to the wider mathematical community.  The book introduced the ideas of a [[normed space]] and the notion of a so-called ''B''-space, a [[complete metric space|complete]] normed space.  The ''B''-spaces are now called [[Banach space]]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 (functional analysis)|open mapping theorem]], [[closed graph theorem]], and [[Hahn–Banach theorem]].
 
===Fourier analysis===
 
====''Mémoire sur la propagation de la chaleur dans les corps solides''====
 
* [[Joseph Fourier]] (1807)<ref>{{Cite book
|title=Mémoire sur la propagation de la chaleur dans les corps solides, présenté le 21 décembre 1807 à l'Institut national – Nouveau Bulletin des sciences par la Société philomatique de Paris
|volume=I
|pages=112–116
|date=March 1808
|location=Paris
|publisher=Bernard
|issue=6
}}
Reprinted in
{{Cite book
|title=Joseph Fourier – Œuvres complètes, tome 2
|url=http://mathdoc.emath.fr/cgi-bin/oetoc?id=OE_FOURIER__2
|chapter=Mémoire sur la propagation de la chaleur dans les corps solides
|chapterurl=http://gallica.bnf.fr/ark:/12148/bpt6k33707/f220n7.capture
|pages=215–221
}}</ref>
 
'''Description:''' Introduced [[Fourier analysis]], specifically [[Fourier series]]. Key contribution was to not simply use [[trigonometric series]], but to model ''all'' functions by trigonometric series.
{{cquote|<math>\varphi(y)=a\cos\frac{\pi y}{2}+a'\cos 3\frac{\pi y}{2}+a''\cos5\frac{\pi y}{2}+\cdots.</math>
 
Multiplying both sides by <math>\cos(2i+1)\frac{\pi y}{2}</math>, and then integrating from <math>y=-1</math> to <math>y=+1</math> yields:
 
<math>a_i=\int_{-1}^1\varphi(y)\cos(2i+1)\frac{\pi y}{2}\,dy.</math>}}
When Fourier submitted his paper in 1807, the committee (which included [[Joseph Louis Lagrange|Lagrange]], [[Laplace]], [[Étienne-Louis Malus|Malus]] and [[Adrien-Marie Legendre|Legendre]], among others) concluded: ''...the manner in which the author arrives at these equations is not exempt of difficulties and [...]  his analysis to integrate them still leaves something to be desired on the score of generality and even rigour''. Making Fourier series rigorous, which in detail took over a century, led directly to a number of developments in analysis, notably the rigorous statement of the integral via the [[Dirichlet integral]] and later the [[Lebesgue integral]].
 
====''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)
 
'''Description:''' In his habilitation thesis on Fourier series, Riemann characterized this work of Dirichlet as "''the first profound paper about the subject''".<ref name=Koch>{{Cite book| last = Koch | first = Helmut  | title = Mathematics in Berlin: Gustav Peter Lejeune Dirichlet | publisher = Birkhäuser | year = 1998 | pages = 33–40 | isbn = 3-7643-5943-9}}</ref>  This paper gave the first rigorous proof of the convergence of [[Fourier series]] under fairly general conditions (piecewise continuity and monotonicity) by considering partial sums, which Dirichlet transformed into a particular [[Dirichlet integral]] involving what is now called the [[Dirichlet kernel]].  This paper introduced the nowhere continuous [[Dirichlet function]] and an early version of the [[Riemann–Lebesgue lemma]].<ref name=ElstrodtFourier>{{Cite journal| last = Elstrodt | first = Jürgen | journal = Clay Mathematics Proceedings  | title = The Life and Work of Gustav Lejeune Dirichlet (1805–1859) | publisher = | year = 2007 | url = http://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf | format = PDF |pages = 19–20}}</ref>
 
====''On convergence and growth of partial sums of Fourier series''====
 
* [[Lennart Carleson]] (1966)
 
'''Description:''' Settled [[Lusin's conjecture]] that the Fourier expansion of any <math>L^2</math> function converges [[almost everywhere]].
 
==[[Geometry]]==
{{See also|List of books in computational geometry|List of books about polyhedra}}
 
===''[[Baudhayana]] [[Sulba Sutras|Sulba Sutra]]''===
 
* [[Baudhayana]]
'''Description:''' Written around the 8th century BC{{Citation needed|date=July 2010|reason=no manuscript survives from anywhere near that early}}, this is one of the oldest geometrical texts. It laid the foundations of [[Indian mathematics]] and was influential in [[South Asia]] and its surrounding regions, and [[Indian mathematics#Charges of Eurocentrism|perhaps even Greece]]. Among the important geometrical discoveries included in this text are: the earliest list of Pythagorean triples discovered algebraically, the earliest statement of the Pythagorean theorem,  geometric solutions of linear equations, several approximations of [[π]], the first use of irrational numbers, and an accurate computation of the square root of 2, correct to a remarkable five decimal places. Though this was primarily a geometrical text, it also contained some important algebraic developments, including the earliest use of quadratic equations of the forms ax<sup>2</sup> = c and ax<sup>2</sup> + bx = c, and integral solutions of simultaneous [[Diophantine equation]]s with up to four unknowns.
 
===[[Euclid's Elements|''Euclid's'' ''Elements'']]===
 
* [[Euclid]]
'''Publication data:''' c. 300 BC
 
'''Online version:''' [http://aleph0.clarku.edu/~djoyce/java/elements/elements.html Interactive Java version]
 
'''Description:''' 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 [[geometry]], [[number theory]] and the first algorithm as well.  More than any specific result in the publication, it seems that the major achievement of this publication is the popularization of [[logic]] and mathematical proof as a method of solving problems.
 
===''[[The Nine Chapters on the Mathematical Art]]''===
 
* Unknown author
'''Description:''' This was a Chinese [[mathematics]] book, mostly geometric, composed during the [[Han Dynasty]], perhaps as early as 200 BC. It remained the most important textbook in [[China]] 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]]''. Problems with several unknowns, solved by a principle similar to [[Gaussian elimination]]. Problems involving the principle known in the West as the [[Pythagorean theorem]]. The earliest solution of a [[matrix (mathematics)|matrix]] using a method equivalent to the modern method.
 
===''[[On Conic Sections|The Conics]]''===
 
* [[Apollonius of Perga]]
'''Description:''' The Conics was written by Apollonius of Perga, a [[Greek people|Greek]] mathematician. His innovative methodology and terminology, especially in the field of [[conics]], influenced many later scholars including [[Ptolemy]], [[Francesco Maurolico]], [[Isaac Newton]], and [[René Descartes]]. It was Apollonius who gave the [[ellipse]], the [[parabola]], and the [[hyperbola]] the names by which we know them.
 
===''[[Surya Siddhanta]]''===
 
* Unknown (400 CE)
'''Description:''' Contains the roots of modern trigonometry. It describes the archeo-astronomy theories, principles and methods of the ancient Hindus. This siddhanta is supposed to be the knowledge that the Sun god gave to an Asura called Maya. It uses sine (jya), cosine (kojya or "perpendicular sine") and inverse sine (otkram jya) for the first time, and also contains the earliest use of the tangent and secant. Later Indian mathematicians such as Aryabhata made references to this text, while later Arabic and Latin translations were very influential in Europe and the Middle East.
 
===''[[Aryabhatiya]]''===
 
*[[Aryabhata]] (499 CE)
'''Description:''' This was a highly influential text during the Golden Age of mathematics in India. The text was highly concise and therefore elaborated upon in commentaries by later mathematicians. It made significant contributions to geometry and astronomy, including introduction of sine/ cosine, determination of the approximate value of pi and accurate calculation of the earth's circumference.
 
===''[[La Géométrie]]''===
 
* [[René Descartes]]
'''Description:''' La Géométrie was [[publishing|published]] in 1637 and [[writing|written]] by [[René Descartes]]. The book was influential in developing the [[Cartesian coordinate system]] and specifically discussed the representation of [[Point (geometry)|point]]s of a [[plane (mathematics)|plane]], via [[real number]]s; and the representation of [[curve]]s, via [[equation]]s.
 
===''Grundlagen der Geometrie''===
 
*[[David Hilbert]]
'''Online version:''' [http://www.gutenberg.org/ebooks/17384 English]
 
'''Publication data:'''  {{Cite book|first=David|last=Hilbert|year=1899|title=Grundlagen der Geometrie
|publisher=Teubner-Verlag Leipzig|isbn=1-4020-2777-X}}
 
'''Description:''' 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 (book)|Regular Polytopes]]''===
 
* [[H.S.M. Coxeter]]
'''Description:''' ''Regular Polytopes'' is a comprehensive survey of the geometry of [[regular polytope]]s, the generalisation of regular [[polygon]]s and regular [[polyhedron|polyhedra]] to higher dimensions. Originating with an essay entitled ''Dimensional Analogy'' written in 1923, the first edition of the book took Coxeter 24 years to complete. Originally written in 1947, the book was updated and republished in 1963 and 1973.
 
===[[Differential geometry]]===
 
====''Recherches sur la courbure des surfaces''====
 
* [[Leonhard Euler]] (1760)
'''Publication data:''' Mémoires de l'académie des sciences de Berlin '''16''' (1760) pp.&nbsp;119–143; published 1767. ([http://math.dartmouth.edu/~euler/pages/E333.html Full text] and an English translation available from the Dartmouth Euler archive.)
 
'''Description:''' Established the theory of [[surface]]s, and introduced the idea of [[principal curvatures]], laying the foundation for subsequent developments in the [[differential geometry of surfaces]].
 
====''Disquisitiones generales circa superficies curvas''====
 
* [[Carl Friedrich Gauss]] (1827)
'''Publication data:''' [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN35283028X_0006_2NS "Disquisitiones generales circa superficies curvas"], ''Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores'' Vol. '''VI''' (1827), pp.&nbsp;99–146; "[http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABR1255 General Investigations of Curved Surfaces]" (published 1965) Raven Press, New York, translated by A.M.Hiltebeitel and J.C.Morehead.
 
'''Description:''' Groundbreaking work in [[differential geometry]], introducing the notion of [[Gaussian curvature]] and Gauss' celebrated [[Theorema Egregium]].
 
====''Über die Hypothesen, welche der Geometrie zu Grunde Liegen''====
 
* Bernhard Riemann (1854)
'''Publication data:''' [http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/ "Über die Hypothesen, welche der Geometrie zu Grunde Liegen"], ''Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen'',  Vol. 13, 1867.[http://www.emis.de/classics/Riemann/WKCGeom.pdf English translate]
 
'''Description:''' Riemann's famous Habiltationsvortrag, in which he introduced the notions of a [[manifold]], [[Riemannian metric]], and [[Riemann curvature tensor|curvature tensor]].
 
====''Leçons sur la théorie génerale des surfaces et les applications géométriques du calcul infinitésimal''====
 
*[[Gaston Darboux]]
'''Publication data:'''  {{Cite book|first=Gaston|last=Darboux|year=1887,1889,1896|title=Leçons sur la théorie génerale des surfaces: [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0001.001 Volume I], [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0002.001 Volume II], [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0003.001 Volume III], [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0003.001 Volume IV]
|publisher=Gauthier-Villars}}
 
'''Description:''' Leçons sur la théorie génerale des surfaces et les applications géométriques du calcul infinitésimal (on the General Theory of Surfaces and the Geometric Applications of Infinitesimal Calculus).  A treatise covering virtually every aspect of the 19th century [[differential geometry]] of [[surface]]s.
 
==[[Topology]]==
 
===''Analysis situs''===
 
* [[Henri Poincaré]] (1895, 1899–1905)
'''Description:''' Poincaré's [[Analysis Situs (paper)|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 (mathematics)|homology]] and the [[fundamental group]], provided an early formulation of [[Poincaré duality]], gave the [[Euler–Poincaré characteristic]] for [[chain complexes]], and mentioned several important conjectures including the [[Poincaré conjecture]].
 
===''L'anneau d'homologie d'une représentation'', ''Structure de l'anneau d'homologie d'une représentation''===
 
* [[Jean Leray]] (1946)
'''Description:''' These two [[Comptes Rendus]] notes of Leray from 1946 introduced the novel concepts of [[Sheaf (mathematics)|sheafs]], [[sheaf cohomology]], and [[spectral sequences]], which he had developed during his years of captivity as a prisoner of war.  Leray's announcements and applications (published in other Comptes Rendus notes from 1946) drew immediate attention from other mathematicians.  Subsequent clarification, development, and generalization by [[Henri Cartan]], [[Jean-Louis Koszul]], [[Armand Borel]], [[Jean-Pierre Serre]], and Leray himself allowed these concepts to be understood and applied to many other areas of mathematics.<ref name=HaynesMiller>{{Cite web| last = Miller | first = Haynes | title = Leray in Oflag XVIIA: The origins of sheaf theory, sheaf cohomology, and spectral sequences | work = | publisher = | year = 2000 | url = http://www-math.mit.edu/~hrm/papers/ss.ps | format = [[PostScript|ps]]}}</ref>  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''".<ref name=Dieudonné123>{{Cite book| last = Dieudonné | first = Jean | title = A history of algebraic and differential topology 1900–1960 | publisher = Birkhäuser | year = 1989 | pages = 123–141 | isbn = 0-8176-3388-X}}</ref>
 
===Quelques propriétés globales des variétés differentiables===
 
*[[René Thom]] (1954)
'''Description:''' In this paper, Thom proved the Thom transversality theorem, introduced the notions of [[cobordism#Oriented cobordism|oriented]] and [[List of cohomology theories#Unoriented cobordism|unoriented cobordism]], and demonstrated that cobordism groups could be computed as the homotopy groups of certain [[Thom space]]s.  Thom completely characterized the unoriented cobordism ring and achieved strong results for several problems, including [[Steenrod problem|Steenrod's problem]] on the realization of cycles.<ref name=Dieudonné556>{{Cite book| last = Dieudonné | first = Jean | title = A history of algebraic and differential topology 1900–1960 | publisher = Birkhäuser | year = 1989 | pages = 556–575 | isbn = 0-8176-3388-X}}</ref><ref name=Sullivan>{{Cite journal| last = Sullivan  | first = Dennis  |date=April 2004 | title = René Thom's Work On Geometric Homology And Bordism | journal = [[Bulletin of the American Mathematical Society]]  | volume = 41  | issue = 3  | pages = 341–350  | url = http://www.ams.org/bull/2004-41-03/S0273-0979-04-01026-2/S0273-0979-04-01026-2.pdf | format = PDF  | doi = 10.1090/S0273-0979-04-01026-2}}</ref>
 
==[[Category theory]]==
 
===General theory of natural equivalences===
 
* [[Samuel Eilenberg]] and [[Saunders Mac Lane]] (1945)
'''Description:''' The first paper on category theory. Mac Lane later wrote in ''Categories for the Working Mathematician'' that he and Eilenberg introduced categories so that they could introduce functors, and they introduced functors so that they could introduce natural equivalences. Prior to this paper, "natural" was used in an informal and imprecise way to designate constructions that could be made without making any choices. Afterwards, "natural" had a precise meaning which occurred in a wide variety of contexts and had powerful and important consequences.
 
===[[Categories for the Working Mathematician]]===
 
* [[Saunders Mac Lane]] (1971, second edition 1998)
'''Description:''' Saunders Mac Lane, one of the founders of category theory, wrote this exposition to bring categories to the masses.  Mac Lane brings to the fore the important concepts that make category theory useful, such as [[adjoint functor]]s and [[universal properties]].
 
==[[Set theory]]==
 
===Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen===
 
* [[Georg Cantor]] (1874)
'''Online version:''' [http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002155583 Online version]
 
'''Description''': Contains the first proof that the set of all real numbers is uncountable; also contains a proof that the set of algebraic numbers is denumerable. (For history and controversies about this article, see [[Cantor's first uncountability proof]].)
 
===[[Grundzüge der Mengenlehre]]===
 
* [[Felix Hausdorff]]
'''Description:''' First published in 1914, this was the first comprehensive introduction to set theory. Besides the systematic treatment of known results in set theory, the book also contains chapters on [[measure theory]] and topology, which were then still considered parts of set theory. Here Hausdorff presents and develops highly original material which was later to become the basis for those areas.
 
===The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory===
 
* [[Kurt Gödel]] (1938)
'''Description:''' Gödel proves the results of the title.  Also, in the process, introduces the [[constructible universe|class L of constructible sets]], a major influence in the development of axiomatic set theory.
 
===The Independence of the Continuum Hypothesis===
 
* [[Paul Cohen (mathematician)|Paul J. Cohen]] (1963, 1964)
'''Description:''' 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 (mathematics)|forcing]]'' which led to many other major results in axiomatic set theory.
 
==[[Logic]]==
 
===[[Begriffsschrift]]===
 
* [[Gottlob Frege]] (1879)
'''Description''': Published in 1879, the title '''''Begriffsschrift''''' is usually translated as ''concept writing'' or ''concept notation''; the full title of the book identifies it as "''a [[formula]] [[language]], modelled on that of [[arithmetic]], of pure [[thought]]''". Frege's motivation for developing his [[formal logical system]] was similar to [[Gottfried Wilhelm Leibniz|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)
'''Description''': First published in 1895, the '''Formulario mathematico''' was the first mathematical book written entirely in a [[formal language|formalized language]]. It contained a description of [[mathematical logic]] and many important theorems in other branches of mathematics. Many of the notations introduced in the book are now in common use.
 
===[[Principia Mathematica]]===
 
* [[Bertrand Russell]] and [[Alfred North Whitehead]] (1910–1913)
'''Description:''' The '''''Principia Mathematica''''' is a three-volume work on the foundations of [[mathematics]], written by [[Bertrand Russell]] and [[Alfred North Whitehead]] and published in 1910–1913. It is an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in [[Mathematical logic|symbolic logic]]. The questions remained whether a contradiction could be derived from the Principia's axioms, and whether there exists a mathematical statement which could neither be proven nor disproven in the system. These questions were settled, in a rather surprising way, by [[Gödel's incompleteness theorem]] in 1931.
 
===[[Systems of Logic Based on Ordinals]]===
 
* [[Alan Turing]]'s Ph.D. thesis
 
===Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I===
 
([[On Formally Undecidable Propositions of Principia Mathematica and Related Systems]])
*  [[Kurt Gödel]] (1931)
'''Online version:''' [http://www.springerlink.com/content/p03501kn35215860/ Online version]
 
'''Description:''' In [[mathematical logic]], '''[[Gödel's incompleteness theorems]]''' are two celebrated theorems proved by [[Kurt Gödel]] in 1931.
The first incompleteness theorem states:
 
<blockquote>
For any formal system such that (1) it is <math>\omega</math>-consistent ([[omega-consistent]]), (2) it has a [[recursively definable]] set of [[axioms]] and [[rules of derivation]], and (3) every [[computable function#Computable sets and relations|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.
</blockquote>
 
==[[Combinatorics]]==
 
===On sets of integers containing no k elements in arithmetic progression===
 
*[[Endre Szemerédi]] (1975)
'''Description:'''  Settled a conjecture of [[Paul Erdős]] and [[Paul 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"<ref name=Steele2008>{{Cite journal| year = 2008  | month = April | title = 2008 Steele Prizes; Seminal Contribution to Research: Endre Szemerédi  | journal = [[Notices of the American Mathematical Society]]  | volume = 55  | issue = 4  | page = 488  | url = http://www.ams.org/notices/200804/tx080400486p.pdf  | format = PDF  | accessdate = 19 July 2008}}</ref> and it introduced new ideas and tools to the field including a weak form of the [[Szemerédi regularity lemma]].<ref name=interview>{{Cite journal| year = 2013  | month = April | title = Interview with Endre Szemerédi  | journal = [[Notices of the American Mathematical Society]]  | volume = 60  | issue = 2  | page = 226  | url = http://www.ams.org/notices/201302/rnoti-p221.pdf  | format = PDF  | accessdate = 27 January 2013}}</ref>
 
===[[Graph theory]]===
 
====Solutio problematis ad geometriam situs pertinentis====
 
* [[Leonhard Euler]] (1741)
* [http://math.dartmouth.edu/~euler/docs/originals/E053.pdf Euler's original publication] (in Latin)
'''Description:''' Euler's solution of the [[Königsberg bridge problem]] in ''Solutio problematis ad geometriam situs pertinentis'' (''The solution of a problem relating to the geometry of position'') is considered to be the first theorem of [[graph theory]].
 
====On the evolution of random graphs====
 
*[[Paul Erdős]] and [[Alfréd Rényi]] (1960)
'''Description:''' Provides a detailed discussion of sparse [[random graphs]], including distribution of components, occurrence of small subgraphs, and phase transitions.<ref name=Bollobás>{{Cite book| last = Bollobás | first = Béla | title = Modern Graph Theory | publisher = Springer | year = 2002 | page = 252 | isbn = 978-0-387-98488-9}}</ref>
 
====Network Flows and General Matchings====
 
* [[L. R. Ford, Jr.|Ford, L.]], & [[D. R. Fulkerson|Fulkerson, D.]]
* Flows in Networks. Prentice-Hall, 1962.
'''Description:''' 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#Computational complexity theory|List of important publications in theoretical computer science]].''
 
==Probability theory==
''See [[list of important publications in statistics]].''
 
==[[Game theory]]==
 
===Zur Theorie der Gesellschaftsspiele===
 
* [[John von Neumann]] (1928)
'''Description:''' Went well beyond [[Émile Borel]]'s initial investigations into strategic two-person game theory by proving the [[minimax theorem]] for two-person, zero-sum games.
 
===''[[Theory of Games and Economic Behavior]]''===
 
* [[Oskar Morgenstern]], [[John von Neumann]] (1944)
'''Description:''' This book led to the investigation of modern game theory as a prominent branch of mathematics. This profound work contained the method for finding optimal solutions for two-person zero-sum games.
 
===''Equilibrium Points in N-person Games''===
 
* [[John Forbes Nash]]
* ''[[Proceedings of the National Academy of Sciences]]'' 36 (1950), 48–49.  {{MR|0031701}}
*  [http://www.pnas.org/cgi/reprint/36/1/48 "Equilibrium Points in N-person Games"]
'''Description:'''[[Nash equilibrium]]
 
===''[[On Numbers and Games]]''===
 
*[[John Horton Conway]]
'''Description:''' The book is in two, {0,1<nowiki>|}</nowiki>, parts. The zeroth part is about numbers, the first part about games – both the values of games and also some real games that can be played such as [[Nim]], [[Hackenbush]], [[map-coloring games#Col and Snort|Col and Snort]] amongst the  many described.
 
===''[[Winning Ways for your Mathematical Plays]]''===
 
*[[Elwyn Berlekamp]], [[John Horton Conway|John Conway]] and [[Richard K. Guy]]
'''Description:''' 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]]s==
 
===''[[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension]]''===
 
*[[Benoît Mandelbrot]]
'''Description:''' 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 (mathematics)|Optimization]]===
 
====Method of Fluxions====
 
* [[Isaac Newton]]
'''Description:''' ''[[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's method|Newton–Raphson method]]) for finding the real zeroes of a [[function (mathematics)|function]].
 
====''Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies''====
 
* [[Joseph Louis Lagrange]] (1761)
'''Description:''' Major early work on the [[calculus of variations]], building upon some of Lagrange's prior investigations as well as those of [[Euler]].  Contains investigations of minimal surface determination as well as the initial appearance of [[Lagrange multipliers]].
 
====Математические методы организации и планирования производства====
 
* [[Leonid Kantorovich]] (1939) "[The Mathematical Method of Production Planning and Organization]" (in Russian).
'''Description:''' Kantorovich wrote the first paper on production planning, which used Linear Programs as the model. He proposed the simplex algorithm as a systematic procedure to solve these Linear Programs. 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.
'''Description:''' Dantzig's is considered the father of [[linear programming]] in the western world. He independently invented the [[simplex algorithm]]. Dantzig and Wolfe worked on decomposition algorithms for large-scale linear programs in factory and production planning.
 
====How good is the simplex algorithm?====
 
* [[Victor Klee]] and George J. Minty
* {{cite book|title=Inequalities&nbsp;III (Proceedings of the Third Symposium on Inequalities held at the University of California, Los Angeles, Calif., September&nbsp;1–9,&nbsp;1969, dedicated to the memory of Theodore&nbsp;S. Motzkin)|editor-first=Oved|editor-last=Shisha|publisher=Academic Press|location=New York-London|year=1972|mr=332165|last1=Klee|first1=Victor|authorlink1=Victor Klee|last2=Minty|first2= George&nbsp;J.|authorlink2=George J. Minty|chapter=How good is the simplex algorithm?|pages=159–175|ref=harv}}
'''Description:''' 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====
 
* {{Cite journal| last=Khachiyan | first=Leonid Genrikhovich | authorlink=Leonid Khachiyan | year=1979 | title=Полиномиальный алгоритм в линейном программировании | trans_title=A polynomial algorithm for linear programming | journal=[[Doklady Akademii Nauk SSSR]] | volume=244 | pages=1093–1096 | language=Russian}}.
'''Description:''' Khachiyan's work on Ellipsoid method. This was the first polynomial time algorithm for linear programming.
 
==Early manuscripts==
<!-- This section is linked from [[History of mathematics]] -->
{{Globalize|date=November 2009}}
These are publications that are not necessarily relevant to a mathematician nowadays, but are nonetheless important publications in the [[history of mathematics]].
 
===''[[Rhind Mathematical Papyrus]]''===
 
* [[Ahmes]] ([[scribe]])
 
'''Description:''' It is one of the oldest mathematical texts, dating to the [[Second Intermediate Period]] of [[ancient Egypt]]. It was copied by the scribe [[Ahmes]] (properly ''Ahmose'') from an older [[Middle Kingdom of Egypt|Middle Kingdom]] [[papyrus]]. It laid the foundations of [[Egyptian mathematics]] and in turn, later influenced [[Greek mathematics|Greek and Hellenistic mathematics]]. Besides describing how to obtain an approximation of π only missing the mark by less than one per cent, it is describes one of the earliest attempts at [[squaring the circle]] and in the process provides persuasive evidence against the theory that the [[Egyptians]] deliberately built their [[pyramid]]s to enshrine the value of π in the proportions. Even though it would be a strong overstatement to suggest that the papyrus represents even rudimentary attempts at analytical geometry, Ahmes did make use of a kind of an analogue of the [[cotangent]].
 
===''[[Archimedes Palimpsest]]''===
 
*[[Archimedes|Archimedes of Syracuse]]
'''Description:''' 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]]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|Archimedes of Syracuse]]
 
'''Online version:''' [http://web.fccj.org/~ethall/archmede/sandreck.htm Online version]
 
'''Description:''' The first known (European) [[numeral system|system of number-naming]] that can be expanded beyond the needs of everyday life.
 
==Textbooks==
 
===''[[Synopsis of Pure Mathematics]]''===
 
* [[G. S. Carr]]
'''Description''': Contains over 6000 theorems of mathematics, assembled by George Shoobridge Carr for the purpose of training students in the art of mathematics, studied extensively by [[Ramanujan]]. [http://books.google.com/books?id=FTgAAAAAQAAJ&pg=PA1&dq=george+shoobridge+carr#PPR7,M1 (first half here)] It was one of the few books that attempts to summarize the entirety of known mathematics.
 
===''[[Arithmetick: or, The Grounde of Arts]]''===
 
* [[Robert Recorde]]
'''Description''': Written in 1542, it was the first really popular arithmetic book written in the English Language.
 
===''[[Cocker's Arithmetick]]''===
 
* [[Edward Cocker]] (authorship disputed)
'''Description''': 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]]
'''Description''': An early and popular English arithmetic textbook published in [[United States of America|America]] in the 18th century. The book reached from the introductory topics to the advanced in five sections.
 
===''Geometry''===
 
*[[Andrei Petrovich Kiselyov|Andrei Kiselyov]]
'''Publication data:''' 1892
 
'''Description:''' The most widely used and influential textbook in Russian mathematics. (See Kiselyov page and [http://www.maa.org/reviews/KiselevGeomI.html MAA review].)
 
===''[[A Course of Pure Mathematics]]''===
 
* [[G. H. Hardy]]
 
'''Description:''' A classic textbook in introductory [[mathematical analysis]], written by [[G. H. Hardy]]. 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]]===
 
* [[Bartel Leendert van der Waerden|B. L. van der Waerden]]
'''Description:''' The first introductory textbook (graduate level) expounding the abstract approach to algebra developed by Emil Artin and Emmy Noether.  First published in German in 1931 by Springer Verlag.  A later English translation was published in 1949 by [[Frederick Ungar Publishing Company]].
 
===[[Algebra (book)|Algebra]]===
 
* [[Saunders Mac Lane]] and [[Garrett Birkhoff]]
'''Description:''' A definitive introductory text for abstract algebra using a [[Category theory|category theoretic]] approach. Both a rigorous introduction from first principles, and a reasonably comprehensive survey of the field.
 
===''Calculus, Vol. 1''===
 
* [[Tom M. Apostol]]
 
===''[[Algebraic Geometry (book)|Algebraic Geometry]]''===
 
* [[Robin Hartshorne]]
'''Description:''' 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 [[hom functor|functor of points]].
 
===''[[Naive Set Theory (book)|Naive Set Theory]]''===
 
* [[Paul Halmos]]
'''Description:''' 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]]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]]
'''Description:'''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]]
'''Description:''' This book is not really for beginners, but graduate students with some minimal experience in set theory and formal logic will find it a valuable self-teaching tool, particularly in regard to [[forcing (mathematics)|forcing]]. It is far easier to read than a true reference work such as Jech, ''Set Theory''. It may be the best textbook from which to learn forcing, though it has the disadvantage that the exposition of forcing relies somewhat on the earlier presentation of Martin's axiom.
 
===Topologie===
 
* [[Pavel Sergeevich Alexandrov]]
* [[Heinz Hopf]]
'''Description:''' 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]]
'''Description:'''First published in 1955,for many years the only introductory graduate level textbook in the U.S.A. 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]]
'''Description:''' This short book introduces the main concepts of differential topology in Milnor's lucid and concise style.  While the book does not cover very much, its topics are explained beautifully in a way that illuminates all their details.
 
===''[[Number Theory, An approach through history from Hammurapi to Legendre]]''===
 
* [[André Weil]]
'''Description:''' An historical study of number theory, written by one of the 20th century's greatest researchers in the field.  The book covers some thirty six centuries of arithmetical work but the bulk of it is devoted to a detailed study and exposition of the work of Fermat, Euler, Lagrange, and Legendre.  The author wishes to take the reader into the workshop of his subjects to share their successes and failures. A rare opportunity to see the historical development of a subject through the mind of one of its greatest practitioners.
 
===An Introduction to the Theory of Numbers===
 
* [[G. H. Hardy]] and [[E. M. Wright]]
'''Description:''' ''[[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]]
 
===''Hodge Theory and Complex Algebraic Geometry I''===
 
===''Hodge Theory and Complex Algebraic Geometry II''===
 
* [[Claire Voisin]]
 
==Popular writing==
 
===Gödel, Escher, Bach===
 
* [[Douglas Hofstadter]]
'''Description:''' ''[[Gödel, Escher, Bach]]: an Eternal Golden Braid'' is a Pulitzer Prize-winning book, first published in 1979 by Basic Books.
It is a book about how the creative achievements of logician Kurt Gödel, artist M. C. Escher and composer Johann Sebastian Bach interweave. As the author states: "I realized that to me, Gödel and Escher and Bach were only shadows cast in different directions by some central solid essence. I tried to reconstruct the central object, and came up with this book."
 
===The World of Mathematics===
 
*  [[James R. Newman]]
'''Description:'''  ''[[The World of Mathematics]]'' was specially designed to make mathematics more accessible to the inexperienced. It comprises nontechnical essays on every aspect of the vast subject, including articles by and about scores of eminent mathematicians, as well as literary figures, economists, biologists, and many other eminent thinkers. Includes the work of Archimedes, Galileo, Descartes, Newton, Gregor Mendel, Edmund Halley, Jonathan Swift, John Maynard Keynes, Henri Poincaré, Lewis Carroll, George Boole, Bertrand Russell, Alfred North Whitehead, John von Neumann, and many others. In addition, an informative commentary by distinguished scholar James R. Newman precedes each essay or group of essays, explaining their relevance and context in the history and development of mathematics. Originally published in 1956, it does not include many of the exciting discoveries of the later years of the 20th century but it has no equal as a general historical survey of important topics and applications.
 
==References==
{{Reflist|colwidth=30em}}
 
{{Important publications|state=expanded}}
{{Mathematics-footer}}
 
{{DEFAULTSORT:List Of Important Publications In Mathematics}}
[[Category:Mathematics-related lists|Publications]]
[[Category:Lists of publications in science|Mathematics]]
[[Category:Mathematics books|*]]
[[Category:Mathematics literature|*]]

Latest revision as of 21:35, 8 September 2014

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