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This is a list of important publications in mathematics, organized by field. Euclid, detail from The School of Athens by Raphael. ...
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
- Introduction – A publication that is a good introduction or survey of a topic
- Influence – A publication which has significantly influenced the world
- Latest and greatest – The current most advanced result in a topic
Early manuscripts
These are publications that are not necessarily relevant to a mathematician nowadays, but are nonetheless important publications in the history of mathematics. The word mathematics comes from the Greek μάθημα (máthema) which means science, knowledge, or learning; μαθηματικός (mathematikós) means fond of learning. Today, the term refers to a specific body of knowledge - the rigorous, deductive study of quantity, structure, space, and change. ...
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 papyrus. It laid the foundations of Egyptian mathematics and in turn, later influenced Greek and Hellenistic mathematics. Besides describing how to obtain an approximation of π only missing the mark by under 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 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. The Moscow and Rhind Mathematical Papyri are two of the oldest mathematical texts discovered. ...
Ahmes (more accurately Ahmose) was an Egyptian scribe who lived during the Second Intermediate Period. ...
Illustration of a 15th century scribe This is about scribe, the profession. ...
The Second Intermediate Period marks a period when Ancient Egypt once again fell into disarray between the end of the Middle Kingdom, and the start of the New Kingdom. ...
Ancient Egypt was an African civilization located along the upper Nile, reaching from the Nile Delta in the north to as far south as Jebel Barkal at the Fourth Cataract of the Nile at the time of its greatest extension (15th century BC). ...
Ahmes (more accurately Ahmose) was an Egyptian scribe who lived during the Second Intermediate Period. ...
This name may refer to (amongst others): Ahmose I, a pharaoh of ancient Egypt and founder of the Eighteenth dynasty. ...
The Middle Kingdom is: a old name for China a period in the History of Ancient Egypt, the Middle Kingdom of Egypt This is a disambiguation page — a navigational aid which lists pages that might otherwise share the same title. ...
Papyrus plant Cyperus papyrus at Kew Gardens, London Papyrus is an early form of paper made from the pith of the papyrus plant, Cyperus papyrus, a wetland sedge that grows to 5 meters (15 ft) in height and was once abundant in the Nile Delta of Egypt. ...
Egyptian mathematics refers to the style and methods of mathematics performed by scribes in Ancient Egypt, deriving in large part from the rare discoveries of ancient papyri: in particular, the Rhind Mathematical Papyrus, dating from the Second Intermediate Period (though it is a copy of a now lost Middle Kingdom...
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BCE to the 5th century CE around the Eastern shores of the Mediterranean. ...
This square and circle have the same area. ...
Geometric shape created by connecting a polygonal base to an apex For other versions including architectural Pyramids, see Pyramid (disambiguation). ...
Trigonometry In trigonometry, the cotangent is a function (see trigonometric function) defined as: or An interpretation of the cotangent of an angle x is as follows. ...
Importance: Topic creator, Breakthrough, Influence
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 infinitesimals 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. Contrary to historically ignorant statements found in some 20th-century calculus textbooks, he did not use anything like Riemann sums, either in the work embodied in this palimpsest or in any of his other works. For explicit details of the method used, see how Archimedes used infinitesimals. The Archimedes Palimpsest is a palimpsest on parchment in the form of a codex which originally was a copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes of Syracuse. ...
Archimedes of Syracuse. ...
Table of Geometry, from the 1728 Cyclopaedia. ...
In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ...
This article or section may contain original research or unverified claims. ...
A parabola The parabola (from the Greek: παÏαβολή) is a conic section generated by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. ...
In mathematics, a Riemann sum is a method for approximating the values of integrals. ...
The ancient Greek mathematician, physicist, and engineer Archimedes of Syracuse was the first mathematician to make explicit use of infinitesimals. ...
Online version: Online version The Sand Reckoner is probably the most accessible work of Archimedes, in some sense, it is the first research-expository paper. ...
Archimedes of Syracuse. ...
Description: The first known (European) system of number-naming that can be expanded beyond the needs of everyday life. A numeral is a symbol or group of symbols that represents a number. ...
Table of Geometry, from the 1728 Cyclopaedia. ...
Description: Written around the 8th century BC, this is one of the oldest geometrical texts. It laid the foundations of Indian mathematics and was influential in South Asia 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, the earliest geometrical proof 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 goemetrical text, it also contained some important algebraic developments, including the earliest use of quadratic equations of the forms ax2 = c and ax2 + bx = c, and integral solutions of simultaneous Diophantine equations with upto four unknowns. Baudhayana, (circa 800 BC), was a Vedic Indian mathematician/scribe. ...
The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ...
Baudhayana, (circa 800 BC), was a Vedic Indian mathematician/scribe. ...
(2nd millennium BCE - 1st millennium BCE - 1st millennium) // Overview Events Assyria conquers Damascus and Samaria Nineveh destroyed (789 BCE) First recorded Olympic Games held in Greece (776 BCE) Zhou Dynasty moved its capital to Luoyang (771 BC); The Spring and Autumn Period (771-481 BCE) began. ...
The chronology of Indian mathematics spans from the Indus Valley civilization (3300-1500 BC) and Vedic civilization (1500-500 BC) to modern India (21st century CE). ...
South Asia or Southern Asia is a southern geopolitical region of the Asian continent comprising territories on and in proximity to the Indian subcontinent. ...
The chronology of Indian mathematics spans from the Indus Valley civilization (3300-1500 BC) and Vedic civilization (1500-500 BC) to modern India (21st century CE). ...
Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek πεÏιφÎÏεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ...
In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. ...
Importance: Topic creator, Breakthrough, Influence
Publication data: c. 300 BC The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Egypt during the early 3rd century BC. It comprises a collection of definitions, postulates...
Euclid Euclid of Alexandria (Greek: ) (ca. ...
Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 350s BC 340s BC 330s BC 320s BC 310s BC - 300s BC - 290s BC 280s BC 270s BC 260s BC 250s BC Years: 305 BC 304 BC 303 BC 302 BC 301 BC - 300 BC - 299 BC 298 BC...
Online version: 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. The Elements is still a valuable resource and a good introduction to algorithm. 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. Table of Geometry, from the 1728 Cyclopaedia. ...
Table of Geometry, from the 1728 Cyclopaedia. ...
To meet Wikipedias quality standards, this article or section may require cleanup. ...
Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ...
Importance: Topic creator, Breakthrough, Influence, Introduction, Latest and greatest (though it is one of the first, some of the results are still the latest)
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 using a method equivalent to the modern method. The Nine Chapters on the Mathematical Art (ä¹ç« ç®—è¡“) is a Chinese mathematics book, probably composed in the 1st century AD, but perhaps as early as 200 BC. This book is the earliest surviving mathematical text from China that has come down to us by being copied by scribes and (centuries later...
Euclid, detail from The School of Athens by Raphael. ...
The Han Dynasty (Traditional Chinese: æ¼¢æœ; Simplified Chinese: 汉æœ; Hanyu Pinyin: ; Wade-Giles: Han Chau; 206 BC–AD 220) followed the Qin Dynasty and preceded the Three Kingdoms in China. ...
Centuries: 3rd century BC - 2nd century BC - 1st century BC Decades: 250s BC 240s BC 230s BC 220s BC 210s BC - 200s BC - 190s BC 180s BC 170s BC 160s BC 150s BC Years: 205 BC 204 BC 203 BC 202 BC 201 BC - 200 BC - 199 BC 198 BC...
Geographic scope of East Asia East Asia is a subregion of Asia that can be defined in either geographical or cultural terms. ...
In numerical analysis, the false position method or regula falsi method is a root-finding algorithm that combines features from the bisection method and the secant method. ...
In mathematics, Gaussian elimination or Gauss–Jordan elimination, named after Carl Friedrich Gauss and Wilhelm Jordan (for many, Gaussian elimination is regarded as the front half of the complete Gauss–Jordan elimination), is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining...
The Pythagorean theorem: The sum of the areas of the two squares on the legs (blue and red) equals the area of the square on the hypotenuse (purple). ...
In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ...
Importance: Topic creator, Breakthrough, Influence
Description: La Géométrie was published in 1637 and written by René Descartes. The book was influential in developing the Cartesian coordinate system and specifically discussed the representation of points of a plane, via real numbers; and the representation of curves, via equations. La Géométrie was published in 1637 and written by René Descartes. ...
For other things named Descartes, see Descartes (disambiguation). ...
This article is concerned with the production of books, magazines, and other literary material (whether in printed or electronic formats). ...
Events February 3 - Tulipmania collapses in Netherlands by government order February 15 - Ferdinand III becomes Holy Roman Emperor December 17 - Shimabara Rebellion erupts in Japan Pierre de Fermat makes a marginal claim to have proof of what would become known as Fermats last theorem. ...
see also Creative Writing Writing may refer to two activities: the inscribing of characters on a medium, with the intention of forming words and other constructs that represent language or record information, and the creation of material to be conveyed through written language. ...
For other things named Descartes, see Descartes (disambiguation). ...
Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
Point can refer to: Look up Point in Wiktionary, the free dictionary // Mathematics In mathematics: Point (geometry), an entity that has a location in space but no extent Fixed point (mathematics), a point that is mapped to itself by a mathematical function Point at infinity Point group Point charge, an...
Two intersecting planes in R3 In mathematics, a plane is a fundamental two-dimensional object. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...
In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ...
Importance: Topic creator, Breakthrough, Influence
Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ...
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 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. Begriffsschrift is the title of a short book on logic by Gottlob Frege, published in 1879, and is also the name of the formal system set out in that book. ...
Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, Bad Kleinen) was a German mathematician who evolved into a logician and philosopher. ...
1879 (MDCCCLXXIX) was a common year starting on Wednesday (see link for calendar). ...
In mathematics and in the sciences, a formula is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ...
Arithmetic or arithmetics (from the Greek word αÏιθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though professional mathematicians often treat arithmetic as a synonym for number theory. ...
Thought or thinking is a mental process which allows beings to model the world, and so to deal with it effectively according to their goals, plans, ends and desires. ...
Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ...
The Calculus Ratiocinator is a concept appearing in the writings of Gottfried Leibniz, usually paired with his characteristica universalis, which he mentioned much more frequently. ...
Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ...
Importance: Arguably the most significant publication in logic since Aristotle. Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ...
Aristotle (Ancient Greek: Aristotelēs 384–March 7 322 BCE) was an ancient Greek philosopher, who studied with Plato and taught Alexander the Great. ...
Description: 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. Formulario Mathematico (interlingua: Formulation of mathematics) is a book by Giuseppe Peano which expresses fundamental theorems of mathematics in a symbolic language developed by Peano. ...
Giuseppe Peano Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher best known for his contributions to set theory. ...
1895 (MDCCCXCV) was a common year starting on Tuesday (see link for calendar) of the Gregorian calendar (or a common year starting on Thursday of the 12-day-slower Julian calendar). ...
In mathematics, logic and computer science, a formal language is a set of finite-length words (i. ...
Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...
Importance:Influence
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 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 disappointing way, by Gödel's incompleteness theorem in 1931. The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ...
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970), was a famous and influential British philosopher, logician, and mathematician, working mostly in the 20th century. ...
Alfred North Whitehead, OM (February 15, 1861 – December 30, 1947) was a British mathematician who became a philosopher. ...
Euclid, detail from The School of Athens by Raphael. ...
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970), was a famous and influential British philosopher, logician, and mathematician, working mostly in the 20th century. ...
Alfred North Whitehead, OM (February 15, 1861 – December 30, 1947) was a British mathematician who became a philosopher. ...
-1...
1913 (MCMXIII) was a common year starting on Wednesday. ...
Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...
In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ...
1931 (MCMXXXI) was a common year starting on Thursday (link is to a full 1931 calendar). ...
Importance: Influence
(Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931).) In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ...
Online version: Online version Kurt Gödel (IPA: ) (April 28, 1906 Brno, then Austria-Hungary, now Czech Republic – January 14, 1978 Princeton, New Jersey) was a logician, mathematician, and philosopher of mathematics. ...
Description: In mathematical logic, Gödel's incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1930. The first incompleteness theorem states: Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...
Kurt Gödel (IPA: ) (April 28, 1906 Brno, then Austria-Hungary, now Czech Republic – January 14, 1978 Princeton, New Jersey) was a logician, mathematician, and philosopher of mathematics. ...
1930 (MCMXXX) is a common year starting on Wednesday. ...
For any formal system such that (1) it is ω-consistent (omega-consistent), (2) it has a recursively definable set of axioms and rules of derivation, and (3) every recursive relation of natural numbers is definable in it, there exists a formula of the system such that, according to the intended interpretation of the system, it expresses a truth about natural numbers and yet it is not a theorem of the system. In mathematical logic, an omega-consistent (or ω-consistent) theory is a theory (collection of sentences) that is not only consistent (that is, does not prove a contradiction), but also avoids proving certain infinite combinations of sentences that are intuitively contradictory. ...
In computability theory a countable set is called recursive, computable or decidable if we can construct an algorithm which terminates after a finite amount of time and decides whether a given element belongs to the set or not. ...
For the algebra software named Axiom, see Axiom computer algebra system. ...
In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ...
See: Recursion Recursively enumerable language Recursively enumerable set Recursive filter Recursive function Recursive set Primitive recursive function This is a disambiguation page — a list of pages that otherwise might share the same title. ...
In mathematics, a mathematical object X of some type T is definable, if there exists some predicate P(x) which is expressible using a finite string of mathematical symbols drawn from a finite language, such that P(X) is true and P(Y) is false for all Y of type...
A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...
Importance: Breakthrough, Influence
See the list of publications in information theory. To meet Wikipedias quality standards, this article or section may require cleanup. ...
This is a list of important publications in computer science, organized by field. ...
To meet Wikipedias quality standards, this article or section may require cleanup. ...
Description: The Disquisitiones Arithmeticae is a textbook of number theory written by 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, Lagrange and Legendre and adds important new results of his own. The Disquisitiones Arithmeticae is a textbook of number theory written by German mathematician Carl Friedrich Gauss and first published in 1801 when Gauss was 24. ...
(help· info) (30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ...
The Disquisitiones Arithmeticae is a textbook of number theory written by German mathematician Carl Friedrich Gauss and first published in 1801 when Gauss was 24. ...
To meet Wikipedias quality standards, this article or section may require cleanup. ...
A mathematician is a person whose primary area of study and research is mathematics. ...
(help· info) (30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ...
The Union Jack, flag of the newly formed United Kingdom of Great Britain and Ireland. ...
Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parliament of Toulouse and a mathematician who is given credit for the development of modern calculus. ...
Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ...
Joseph Louis Lagrange Joseph Louis Lagrange (January 25, 1736 – April 10, 1813; born Giuseppe Luigi Lagrangia in Turin, Lagrange moved to Paris (1787) and became a French citizen, adopting the French translation of his name, Joseph Louis Lagrange) was an Italian-French mathematician and astronomer who made important contributions to...
Adrien-Marie Legendre (September 18, 1752–January 10, 1833) was a French mathematician. ...
Importance: Breakthrough, Influence
Description: On the Number of Primes Less Than a Given Magnitude (or Über die Anzahl der Primzahlen unter einer gegebenen Grösse) 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. On the Number of Primes Less Than a Given Magnitude (or Über die Anzahl der Primzahlen unter einer gegebenen Grösse) is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monthly Reports of the Berlin Academy. ...
Bernhard Riemann. ...
Bernhard Riemann. ...
1859 is a common year starting on Saturday. ...
Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ...
Analytic number theory is the branch of number theory that uses methods from mathematical analysis. ...
Importance: Breakthrough, Influence
Description: Vorlesungen über Zahlentheorie (Lectures on Number Theory) is a textbook of number theory written by German mathematicians P.G.L. Dirichlet and Richard Dedekind, and published in 1863. The Vorlesungen can be seen as a watershed between the classical number theory of Fermat, Jacobi 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 Vorlesungen über Zahlentheorie (Lectures on Number Theory) is a textbook of number theory written by German mathematicians P.G.L. Dirichlet and Richard Dedekind, and published in 1863. ...
Peter Gustav Lejeune Dirichlet. ...
Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ...
Vorlesungen über Zahlentheorie (Lectures on Number Theory) is a textbook of number theory written by German mathematicians P.G.L. Dirichlet and Richard Dedekind, and published in 1863. ...
To meet Wikipedias quality standards, this article or section may require cleanup. ...
Peter Gustav Lejeune Dirichlet. ...
Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ...
Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parliament of Toulouse and a mathematician who is given credit for the development of modern calculus. ...
Karl Gustav Jacob Jacobi (Potsdam December 10, 1804 - Berlin February 18, 1851), was not only a great German mathematician but also considered by many as the most inspiring teacher of his time (Bell, p. ...
(help· info) (30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ...
Bernhard Riemann. ...
David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ...
Group theory is that branch of mathematics concerned with the study of groups. ...
Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ...
Importance: Breakthrough, Influence
Number Theory, An approach through history from Hammurapi to Legendre 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. André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ...
Importance:
Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ...
Yuktibhasa Description: Written in India in 1501, this was the world's first calculus text. "This work laid the foundation for a complete system of fluxions" (Charles Whish, 1835) 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'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 differentiation and integration, 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. Some of its important developments in analysis include: the infinite series expansion of a function, the power series, the Taylor series, the trigonometric series of sine, cosine, tangent and arctangent, the second and third order Taylor series approximations of sine and cosine, the power series of π, π/4, θ, the radius, diamater and circumference, and tests of convergence. Jyestadeva (1500-1610), was an astronomer of the Kerala school founded by Madhava of Sangamagrama and a student of Damodara. ...
1501 was a common year starting on Tuesday (see link for calendar) of the Gregorian calendar. ...
| Come and take it, slogan of the Texas Revolution 1835 was a common year starting on Thursday (see link for calendar). ...
The Kerala School was a school of mathematics and astronomy founded by Madhava in Kerala (in South India) which included as its prominent members Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ...
Wikibooks has more about this subject: Trigonometry Table of Trigonometry, 1728 Cyclopaedia Trigonometry (from the Greek trigonon = three angles and metro = measure) is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine, cosine and tangent. ...
Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ...
This 14th-century statue from south India depicts the gods Shiva (on the left) and Uma (on the right). ...
Madhava (माधव) of Sangamagrama (1350-1425) was a major mathematician from Kerala, in South India. ...
The Kerala School was a school of mathematics and astronomy founded by Madhava in Kerala (in South India) which included as its prominent members Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ...
Differentiation can mean the following: In biology: cellular differentiation; evolutionary differentiation; In mathematics: see: derivative In cosmogony: planetary differentiation Differentiation (geology); Differentiation (logic); Differentiation (marketing). ...
Integration may be any of the following: In the most general sense, integration may be any bringing together of things: the integration of two or more economies, cultures, religions (usually called syncretism), etc. ...
In mathematics, the derivative is defined to be the instantaneous rate of change of a function. ...
Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...
For any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c. ...
In mathematics, a series is a sum of a sequence of terms. ...
In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...
As the degree of the Taylor series rises, it approaches the correct function. ...
In mathematics, a Fourier series of a periodic function, named in honor of Joseph Fourier (1768-1830), represents the function as a sum of periodic functions of the form where e is Eulers number and i the imaginary unit. ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
This article is about the mathematical concept of tangent. For other meanings, see tangent (disambiguation). ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek πεÏιφÎÏεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ...
Note: A theta probe is a device for measuring soil moisture. ...
The integral test for convergence is a method used to test infinite series of nonnegative terms for convergence. ...
Importance: Topic creator, Breakthrough, Influence
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 July 5, 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. He derives Kepler's laws for the motion of the planets (which were first obtained empirically). In formulating his physical theories, Newton had developed a field of mathematics known as calculus. Newtons own copy of his Principia, with hand written corrections for the second edition. ...
Sir Isaac Newton, PRS, (4 January 1643 – 31 March 1727) [OS: 25 December 1642 – 20 March 1727] was an English physicist, mathematician, astronomer, alchemist, inventor, and natural philosopher who is generally regarded as one of the most influential scientists in history. ...
Latin is an ancient Indo-European language originally spoken in the region around Rome called Latium. ...
Sir Isaac Newton, PRS, (4 January 1643 – 31 March 1727) [OS: 25 December 1642 – 20 March 1727] was an English physicist, mathematician, astronomer, alchemist, inventor, and natural philosopher who is generally regarded as one of the most influential scientists in history. ...
July 5 is the 186th day of the year (187th in leap years) in the Gregorian Calendar, with 179 days remaining. ...
Events March 19 - The men under explorer Robert Cavelier de La Salle murder him while searching for the mouth of the Mississippi River. ...
Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...
Classical mechanics is a branch of physics which studies the deterministic motion of objects. ...
Gravity is a force of attraction that acts between bodies that have mass. ...
Johannes Keplers primary contributions to astronomy/astrophysics were the three laws of planetary motion. ...
A planet is generally considered to be a relatively large mass of accreted matter in orbit around a star that is not a star itself. ...
Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ...
Up to the publication of this book, mathematics was only used to describe nature. This is the first instance when mathematics is used to explain nature. 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 other words, the greatness of the Principia is not only in developing a number of fundamental theories in physics and mathematics but first and foremost (amply demonstrated in the title!) in the very linking of science and mathematics. The influence of this book is so deep that nowadays we find this link obvious and cannot imagine doing science in any other way.
Importance: Topic creator, Breakthrough, Influence
Newton's Principia for the Common Reader Description: An exposition, using modern notation and language, of a large part of Newton's above-cited masterwork. Mathematical and physical language and notation have evolved considerably since Newton's time, making it difficult for a modern reader to read Newton's original work even in translation from the original Latin. Chandrasekhar's labor of love makes it possible for a modern reader, familiar with the modern treatment of algebra, geometry and calculus to appreciate Newton's genius through following his work as he originally conceived it. Subrahmanyan Chandrasekhar (October 19, 1910 – August 21, 1995) was an Indian-American physicist, astrophysicist and mathematician. ...
Importance: Interpretation for the modern reader of a great classic of mathematics and science
Description: Introductions to differential and integral calculus in a single and many variables respectively. Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ...
Michael Spivaks Calculus on Manifolds is a text treating analysis in several variables in Euclidean spaces and on differentiable manifolds. ...
Michael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. ...
Importance: Introduction
Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ...
Description: Method of Fluxions was a book written by Isaac Newton. The book was completed in 1671, and published in 1736. Method of Fluxions was a book by Isaac Newton. ...
Sir Isaac Newton, PRS, (4 January 1643 – 31 March 1727) [OS: 25 December 1642 – 20 March 1727] was an English physicist, mathematician, astronomer, alchemist, inventor, and natural philosopher who is generally regarded as one of the most influential scientists in history. ...
Sir Isaac Newton, PRS, (4 January 1643 – 31 March 1727) [OS: 25 December 1642 – 20 March 1727] was an English physicist, mathematician, astronomer, alchemist, inventor, and natural philosopher who is generally regarded as one of the most influential scientists in history. ...
Events May 9 - Thomas Blood, disguised as a clergyman, attempts to steal the Crown Jewels from the Tower of London. ...
Events January 26 - Stanislaus I of Poland abdicates his throne. ...
Within this book, Newton describes a method (the Newton-Raphson method) for finding the real zeroes of a function. In numerical analysis, Newtons method (or the Newton-Raphson method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ...
Partial plot of a function f. ...
Importance: Topic creator, Breakthrough, Influence
Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. ...
John Maynard Smith Book cover Evolution and the Theory of Games is a 1982 book by the British evolutionary biologist John Maynard Smith on evolutionary game theory. ...
John Maynard Smith Professor John Maynard Smith, F.R.S. (6 January 1920 – 19 April 2004) was a British evolutionary biologist and geneticist. ...
(Theory of Games and Economic Behavior, 3rd ed., Princeton University Press 1953) In 1944 Princeton University Press published Theory of Games and Economic Behavior, a book by the mathematician John von Neumann and economist Oskar Morgenstern. ...
Description: This book led to the investigation of modern game theory as a prominent branch of mathematics. This profound work contained the method -- alluded to above -- for finding optimal solutions for two-person zero-sum games. Oskar Morgenstern (January 24, 1902 - July 26, 1977) was an German- American economist who, working with John von Neumann, helped found the mathematical field of game theory. ...
John von Neumann in the 1940s. ...
Importance: Influence, Topic creator, Breakthrough
Description: The book is in two, {0,1|}, parts. The zeroth part is about numbers, the first part about games - both the values of games and also some real games that can be played such as Nim, Hackenbush, Col and Snort amongst the many described. On Numbers and Games is a mathematics book by John Conway, published by Academic Press Inc in 1976, ISBN 0121863506, and re-released by AK Peters in 2000 (ISBN 1568811276). ...
See John B. Conway for the functional analyst. ...
Nim is a two-player mathematical game of strategy in which players take turns removing objects from heaps, one or more objects at a time but only from a single heap. ...
Hackenbush is a two-player partisan mathematical game that consists of several colored line segments connected to the ground. ...
Col may refer to: the French word for mountain pass a common abbreviation for the military rank colonel This is a disambiguation page, a list of pages that otherwise might share the same title. ...
Snort is an open source network intrusion detection system, capable of performing real-time traffic analysis and packet logging on IP networks. ...
Importance:
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. Winning Ways for your Mathematical Plays (ISBN 1568811306) by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy is a compendium of information on mathematical games. ...
Elwyn Ralph Berlekamp is professor of mathematics at University of California, Berkeley. ...
See John B. Conway for the functional analyst. ...
Richard K. Guy is a Professor Emeritus in the Department of Mathematics at the University of Calgary. ...
Mathematical games include many topics which are a part of recreational mathematics, but can also cover topics such as the mathematics of games, and playing games with mathematics. ...
1982 (MCMLXXXII) was a common year starting on Friday of the Gregorian calendar. ...
Combinatorial game theory (CGT) is a mathematical theory that studies a certain kind of game. ...
In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similiar to superreal numbers and hyperreal numbers. ...
Importance:
The boundary of the Mandelbrot set is a famous example of a fractal. ...
How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension 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. How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension is a paper by mathematician Benoît Mandelbrot, first published in Science in 1967. ...
Benoît Mandelbrot Benoît B. Mandelbrot (born November 20, 1924) is a Polish-born French mathematician and leading proponent of fractal geometry. ...
Importance:
Textbooks
Online version: Online version A Course of Pure Mathematics is a classic textbook in introductory mathematical analysis, written by G. H. Hardy. ...
G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. ...
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. Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ...
G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. ...
1908 (MCMVIII) was a leap year starting on Wednesday (link will take you to calendar). ...
The University of Cambridge (often called Cambridge University), located in Cambridge, England, is the second-oldest university in the English-speaking world. ...
Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ...
In mathematics, a series is a sum of a sequence of terms. ...
Importance:
- Richard Rusczyk and Sandor Lehoczky
Description: The Art of Problem Solving began as a set of two books coauthored by Richard Rusczyk and Sandor Lehoczky. The books, which are about 750 pages together, are for students who are interested in math and/or compete in math competitions. The Art of Problem Solving began as a set of two books coauthored by Richard Rusczyk and Sandor Lehoczky. ...
Importance:
Metalogic: an Introduction to the Metatheory of Standard First Order Logic Description: An excellent introduction to the mathematical theory of logical formal systems, covering completeness-proofs, consistency-proofs, and so on and even set-theory. In logic, mathematics, and computer science, a formal system is a formal grammar used for modelling purposes. ...
In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ...
Consistency has three technical meanings: In mathematics and logic, as well as in theoretical physics, it refers to the proposition that a formal theory or a physical theory contains no contradictions. ...
Importance:
Popular writing
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." GEB cover Gödel, Escher, Bach: an Eternal Golden Braid (commonly GEB) is a Pulitzer Prize-winning book by Douglas Hofstadter, published in 1979 by Basic Books. ...
Douglas Richard Hofstadter (born February 15, 1945) is an American academic. ...
Importance:
The World of Mathematics 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. James Roy Newman was a mathematician and mathematical historian. ...
Importance:
Arithmetic or arithmetics (from the Greek word αÏιθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though professional mathematicians often treat arithmetic as a synonym for number theory. ...
Arithmetick: or, The Grounde of Arts Description: Written in 1542, it was the first really popular arithmetic book written in the English Language. Robert Recorde (c. ...
Importance:
Description: An early and popular English arithmetic textbook published in America in the eighteenth century. The book reached from the introductory topics to the advanced in five sections. The Schoolmasters Assistant, Being a Compendium of Arithmetic both Practical and Theoretical was an early and popular English arithmetic textbook, written by Thomas Dilworth and published in America in the eighteenth century. ...
The Reverend Mr. ...
The United States of America — also referred to as the United States, the U.S.A., the U.S., America, the States, or (archaically) Columbia—is a federal republic of 50 states located primarily in central North America (with the exception of two states: Alaska and Hawaii). ...
(17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ...
Importance:
Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ...
Moderne Algebra 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. Bartel Leendert van der Waerden (February 2, 1903 – January 12, 1996) was a Dutch mathematician who born in Amsterdam, Netherlands and died in Zürich, Switzerland. ...
Importance: Influence
Algebra Description: A definitive introductory text for abstract algebra using a category theoretic approach. Both a rigorous introduction from first principles, and a reasonably comprehensive survey of the field. Saunders Mac Lane (4 August 1909, Taftville, Connecticut - 14 April 2005, San Francisco) was an American mathematician who cofounded category theory with Samuel Eilenberg. ...
Garrett Birkhoff (January 19, 1911, Princeton, New Jersey, USA - November 22, 1996, Water Mill, New York, USA) was an American mathematician. ...
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
Importance: Introduction
Abstract Algebra - David Dummit and Richard Foote
Description: A clear and concise introduction to abstract algebra, beginning with material suitable for undergraduates, and ranging through to cover graduate level topics in some detail.
Importance: Introduction
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ...
Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
Faisceaux Algébriques Cohérents Publication data: Annals of Mathematics, 1955 Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ...
Description: FAC, as it is usually called, first introduced the use of sheaves into algebraic geometry. Serre introduced Cech cohomology of sheaves in this paper, and, despite its technical deficiencies, revolutionized 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. Before FAC, this was next to impossible. While Grothendieck's derived functor cohomology has replaced Cech cohomology for technical reasons, actual calculations, such as of the cohomology of projective space, are usually carried out by Cech techniques, and for this reason Serre's paper remains important even today. In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and...
Čech cohomology is a particular type of cohomology in mathematics. ...
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ...
Importance: Topic creator, Breakthrough, Influence
Description: In mathematics, algebraic geometry and analytic geometry are closely related subjects, where analytic geometry is the theory of complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. A (mathematical) theory of the relationship between the two was put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory. (NB While analytic geometry as use of Cartesian coordinates is also in a sense included in the scope of algebraic geometry, that is not the topic being discussed in this article.) The major paper consolidating the theory was Géometrie Algébrique et Géométrie Analytique by Serre, now usually referred to as GAGA. A GAGA-style result would now mean any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, and a well-defined subcategory of analytic geometry objects and holomorphic mappings. In mathematics, algebraic geometry and analytic geometry are two closely related subjects. ...
Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ...
Euclid, detail from The School of Athens by Raphael. ...
Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. ...
In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ...
Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. ...
In mathematics, an analytic function is a function that is locally given by a convergent power series. ...
The theory of functions of several complex variables is the branch of mathematics dealing with functions f(z1, z2, ... , zn) on the space Cn of n-tuples of complex numbers. ...
In mathematics, Hodge theory is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric...
Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. ...
Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ...
Importance: Topic creator, Breakthrough, Influence
Written with the assistance of Jean Dieudonne, 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. The Éléments de géométrie algébrique (Elements of Algebraic Geometry) by Alexander Grothendieck (assisted by Jean Dieudonné), or EGA for short, are an unfinished 1500-page treatise, in French, on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the Institut...
Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century. ...
Importance: Seminal work which revolutionized the field
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 1967–1969. 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 Weil conjectures in the 1970s. Other authors who worked on one or several volumes of SGA include Michel Raynaud, Michael Artin, Jean-Pierre Serre, Jean Verdier, Pierre Deligne, and Nicholas Katz. In mathematics, Alexander Grothendiecks Séminaire de géométrie algébrique was a unique phenomenon of research and publication outside of the main mathematical journals, reporting on work done starting from 1960 and centred on the IHÉS near Paris (the official title was the seminar of Bois...
Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century. ...
IHÉS main building The Institut des Hautes Études Scientifiques (I.H.É.S.) is a French institute supporting advanced research in mathematics and theoretical physics. ...
Pierre Deligne, March 2005 Pierre Deligne (born 3 October 1944) is a Belgian mathematician. ...
In mathematics, the Weil conjectures, which had become theorems by 1975, were some highly-influential proposals from the late 1940s by Andre Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. ...
Michel Raynaud (June 16, 1938) is a French mathematician working in algebraic geometry. ...
Michael Artin is an American mathematician, known for his contributions to algebraic geometry. ...
Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ...
Pierre Deligne, March 2005 Pierre Deligne (born 3 October 1944) is a Belgian mathematician. ...
Nick Katz (Nicholas M. Katz) is an American mathematician, working in the fields of algebraic geometry, particularly on p-adic methods, monodromy and moduli problems, and number theory. ...
Importance: Seminal work which revolutionized the field
Description: The first comprehensive introductory (graduate level) text in algebraic geometry that used the language of schemes and cohomology. Published in 1977, it remains, in 2005, a good introduction to its subject. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
Importance: Breakthrough textbook, influence
Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ...
Universal algebra - Wolfgang Wechler.
- Springer-Verlag.
Description:
Importance:
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...
Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...
Topologie 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. Pavel Sergeevich Alexandrov (Па́вел Серге́евич Алекса́ндров, sometimes romanized Alexandroff or Aleksandrov) (born May 7, 1896 - died November 16, 1982) was a Russian mathematician. ...
Heinz Hopf (November 19, 1894 – June 3, 1971) was a mathematician born in Gräbschen, Germany. ...
Importance: Influence
General Topology Description:First published in the mid-1950's,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.
Importance: Pioneering text. Influence.
A labeled graph with 6 vertices and 7 edges. ...
Elementary graph theory - Chartrand, Gary, Introductory Graph Theory, Dover. ISBN 0-486-24775-9.
Frank Harary (March 11, 1921 - January 4, 2005) was a prolific American mathematician, who specialized in graph theory. ...
Enumerative graph theory - Harary, Frank, and Palmer, Edgar M., Graphical Enumeration, Academic Press, New York, NY, 1973.
Asymptotic and random graph theory - Palmer, Edgar M., Graphical Evolution: An Introduction to the Theory of Random Graphs, John Wiley & Sons, New York, NY, 1985.
Algorithmic graph theory - Even, Shimon, Graph Algorithms, Computer Science Press, Rockville, MD, 1979.
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
Description: Saunders Mac Lane, one of the founders of category theory, wrote this exposition to bring categories to the masses. Mac Lane does not get lost in pointless abstraction, but instead brings to the fore the important concepts that make category theory useful, such as adjoint functors and universal objects. His text is more comprehensive than most mathematicians will ever need, and consequently is also an excellent reference. Categories for the Working Mathematician is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. ...
Saunders Mac Lane (4 August 1909, Taftville, Connecticut - 14 April 2005, San Francisco) was an American mathematician who cofounded category theory with Samuel Eilenberg. ...
The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...
In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...
Importance: Introduction
Category Theory for Computing Science - Michael Barr and Charles Wells
Description: Slower-paced introduction than Mac Lane's, assuming much less math background. Suitable for budding computer-scientists, logicians, linguists, etc. 1999 edition contains extensive exercises and solutions.
Importance: Introduction
Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...
Wikibooks has more about this subject: Trigonometry Table of Trigonometry, 1728 Cyclopaedia Trigonometry (from the Greek trigonon = three angles and metro = measure) is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine, cosine and tangent. ...
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...
Topology from the Differentiable Viewpoint 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. John Willard Milnor (b. ...
Importance: Influence
Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...
Analysis situs - Henri Poincaré's major contribution to algebraic topology, published in 1895, was the first real systematic look at topology.
Henri Poincaré, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri Poincaré (April 29, 1854 – July 17, 1912), generally known as Henri Poincaré, was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ...
Algebraic Topology Publication data: Cambridge University Press, 2002.
Online version: http://www.math.cornell.edu/~hatcher/AT/ATpage.html
Description: This is the first in a series of three textbooks in algebraic topology having the goal of covering all the basics while remaining readable by newcomers seeing the subject for the first time. The first book contains the basic core material along with a number of optional topics of a relatively elementary nature.
Importance: Introduction
A fractal is a geometric object which can be divided into parts, each of which is similar to the original object. ...
Discrete mathematics, sometimes called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. ...
Dividing a circle into areas. ...
Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...
Grundzüge der Mengenlehre 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. Grundzüge der Mengenlehre (German: Basics of set theory) is an influential book on set theory written by Felix Hausdorff. ...
Felix Hausdorff Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory and functional analysis. ...
In mathematics, a measure is a function that assigns a number, e. ...
Importance: Influence, Introduction
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 advanced topics like large cardinals. Instead it tried, and succeeds, in being intelligible to someone who has never thought about set theory before. Naive Set Theory is a mathematics textbook by Paul Halmos originally published in 1960. ...
Paul Halmos Paul Richard Halmos (born March 3, 1916) is a Hungarian-born American mathematician who has done research in the fields of logarithm theory, probability theory, statistics, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). ...
Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ...
In mathematics, a cardinal is called a large cardinal if it belongs to a class of cardinals, the existence of which provably cannot be proved within the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent. ...
Importance: Influence, Introduction
Cardinal and Ordinal Numbers Description:The ne 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 1950's but based on the author's lectures on the subject over the preceding 40 years. Wacław Franciszek Sierpiński, was born on March 14, 1882 in Warsaw and died on October 21, 1969 in Warsaw. ...
Importance: Influence, unique reference
The Consistency of the Continuum Hypothesis Description:Gödel proves the result of the title and also the consistency of the axiom of choice. Also, in the process, introduces the class L of constructible sets, a major influence in the development of axiomatic set theory. Kurt Gödel (IPA: ) (April 28, 1906 Brno, then Austria-Hungary, now Czech Republic – January 14, 1978 Princeton, New Jersey) was a logician, mathematician, and philosopher of mathematics. ...
In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...
In mathematics, the constructible universe (or Gödels constructible universe), denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. ...
Importance: Breakthrough, influence
Set Theory and the Continuum Hypothesis Description:Published in 1966, these lecture notes from a course at Stanford University made accessible to the general mathematical community Cohen's breakthrough work proving the independence of the continuum hypothesis. In proving this Cohen introduced the concept of forcing which led to many other major results in axiomatic set theory. Paul Joseph Cohen (born April 2, 1934) is an American mathematician. ...
Importance: Breakthrough, influence
Set Theory: An Introduction to Independence Proofs 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. 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. In axiomatic set theory, forcing is a technique, invented by Paul Cohen, for proving consistency and independence results with respect to the Zermelo-Fraenkel axioms. ...
Importance: Textbook, reference
In mathematics, the term optimization refers to the study of problems that have the form Given: a function f : A R from some set A to the real numbers Sought: an element x0 in A such that f(x0) ≤ f(x) for all x in A (minimization) or such that...
The New Variational Method 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. Leonid Vitaliyevich Kantorovich (January 19, 1912 in Petersburg – April 7, 1986 in Moscow) was a Soviet/Russian mathematician and economist. ...
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Decomposition Principle for Linear Programs. 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. George Bernard Dantzig (8 November 1914 – 13 May 2005) was a mathematician who introduced the simplex algorithm and is considered the Father of linear programming. He was the recipient of many honors, including the National Medal of Science in 1975, the John von Neumann Theory Prize in 1974. ...
In mathematical optimization theory, the simplex algorithm of George Dantzig is the fundamental technique for numerical solution of the linear programming problem. ...
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Network Flows and General Matchings - Ford, L., & Fulkerson, D.
- Flows in Networks. Prentice-Hall, 1962.
Description: Ford and Fulkerson paper on Network Flows. The algorithm along with many ideas on flow-based models can be found in their book. This book is supposedly very well written. 1962 (MCMLXII) was a common year starting on Monday (the link is to a full 1962 calendar). ...
Importance:
Paths, trees and Flowers - J. Edmonds.
- Canadian Journal of Mathematics, 17:449–467, 1965.
Description:
Importance:
The complexity of theorem proving procedures - S. A. Cook
- Proceedings of the 3rd Annual ACM Symposium on Theory of Computing (1971), pp. 151--158.
Description: This paper introduced the concept of NP-Completeness and proved that Boolean satisfiability problem (SAT) is NP-Complete. Stephen A. Cook is a noted computer scientist. ...
In complexity theory, the NP-complete problems are the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. The reason is that if you could find a way to solve an NP-complete problem quickly, then you could use...
The Boolean satisfiability problem (SAT) is a decision problem considered in complexity theory. ...
In complexity theory, the NP-complete problems are the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. The reason is that if you could find a way to solve an NP-complete problem quickly, then you could use...
Importance: Topic creator, Breakthrough, Influence
Reducibility among combinatorial problems - R. M. Karp
- In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations, pages 85-103. Plenum Press, New York, NY, 1972.
Description: This paper showed that 21 different problems are NP-Complete and showed the importance of the concept. Richard M. Karp (born 1935) is a computer scientist, notable for research in the theory of algorithms, for which he received a Turing Award in 1985. ...
In complexity theory, the NP-complete problems are the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. The reason is that if you could find a way to solve an NP-complete problem quickly, then you could use...
Importance: Influence
How good is the simplex algorithm? - V. Klee and G. J. Minty
- In: O. Shisha (ed.) Inequalities III, Academic Press (1972) 159–175.
Description: Klee and Minty gave example showing that simplex method can take exponentially many steps to solve a linear program if it chooses the greedy ascent rule.
Importance:
Linear Programming and Polynomial time algorithms - L. Khachiyan
- Doklady Akademii Nauk SSSR 244 (1979) pp. 1093–1096 (Russian).
Description:' Khachiyan's work on Ellipsoid method. This was the first polynomial time algorithm for Linear programming. Leonid Khachiyan Leonid Khachiyan (May 3, 1952 - April 29, 2005) was a Russian-born mathematician who taught Computer Science at Rutgers University. ...
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New polynomial-time algorithm for linear programming - Karmarkar, N.
- Combinatorica 4, 373–395, 1984.
Description: Karmarkars path-breaking work on Interior-Point algorithms for Linear Programming. 1984 (MCMLXXXIV) was a leap year starting on Sunday of the Gregorian calendar. ...
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Interior Point Polynomial Algorithms in Convex Programming - Yurii NESTEROV and A. NEMIROVSKY.
- Philadelphia : Society for Industrial and Applied Mathematics, 1994. (SIAM Studies in Applied Mathematics).
Description: Nesterov and Nemirovski's work on Self-concordant barriers and Interior-Point Methods for general convex programming. All their series of papers (both individual and combined) is compiled more coherently in the following "bible" of convex optimization.
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