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165-485: He is best known for his contributions to algebra and the theory of abelian categories . From 1964 to 2007 he collaborated with Pierre Gabriel on the characterization of abelian categories; their best-known result is the Gabriel–Popescu theorem , published in 1964. His areas of expertise were category theory , abelian categories with applications to rings and modules , adjoint functors , limits and colimits ,

330-563: A {\displaystyle a} to object b {\displaystyle b} , and another morphism from object b {\displaystyle b} to object c {\displaystyle c} , then there must also exist one from object a {\displaystyle a} to object c {\displaystyle c} . Composition of morphisms is required to be associative, and there must be an "identity morphism" for every object. Categories are widely used in contemporary mathematics since they provide

495-403: A {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are usually used for constants and coefficients . The expression 5 x + 3 {\displaystyle 5x+3} is an algebraic expression created by multiplying the number 5 with the variable x {\displaystyle x} and adding

660-746: A 2 x 2 + . . . + a n x n = b {\displaystyle a_{1}x_{1}+a_{2}x_{2}+...+a_{n}x_{n}=b} where a 1 {\displaystyle a_{1}} , a 2 {\displaystyle a_{2}} , ..., a n {\displaystyle a_{n}} and b {\displaystyle b} are constants. Examples are x 1 − 7 x 2 + 3 x 3 = 0 {\displaystyle x_{1}-7x_{2}+3x_{3}=0} and 1 4 x − y = 4 {\textstyle {\frac {1}{4}}x-y=4} . A system of linear equations

825-429: A ∘ a − 1 = a − 1 ∘ a = e {\displaystyle a\circ a^{-1}=a^{-1}\circ a=e} . Every algebraic structure that fulfills these requirements is a group. For example, ⟨ Z , + ⟩ {\displaystyle \langle \mathbb {Z} ,+\rangle } is a group formed by the set of integers together with

990-433: A ∘ b ) ∘ c {\displaystyle (a\circ b)\circ c} is the same as a ∘ ( b ∘ c ) {\displaystyle a\circ (b\circ c)} for all elements. An operation has an identity element or a neutral element if one element e exists that does not change the value of any other element, i.e., if a ∘ e = e ∘

1155-402: A + c a . {\displaystyle (b+c)a=ba+ca.} Moreover, multiplication is associative and has an identity element generally denoted as 1 . Multiplication needs not to be commutative; if it is commutative, one has a commutative ring . The ring of integers ( Z {\displaystyle \mathbb {Z} } ) is one of the simplest commutative rings. A field

1320-437: A = a {\displaystyle a\circ e=e\circ a=a} . An operation has inverse elements if for any element a {\displaystyle a} there exists a reciprocal element a − 1 {\displaystyle a^{-1}} that undoes a {\displaystyle a} . If an element operates on its inverse then the result is the neutral element e , expressed formally as

1485-661: A Lie algebra or an associative algebra . The word algebra comes from the Arabic term الجبر ( al-jabr ), which originally referred to the surgical treatment of bonesetting . In the 9th century, the term received a mathematical meaning when the Persian mathematician Muhammad ibn Musa al-Khwarizmi employed it to describe a method of solving equations and used it in the title of a treatise on algebra, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah [ The Compendious Book on Calculation by Completion and Balancing ] which

1650-500: A Protestant German family in Hamburg and worked as a journalist. As teenagers, both of his parents had broken away from their early backgrounds. At the time of his birth, Grothendieck's mother was married to the journalist Johannes Raddatz and initially, his birth name was recorded as "Alexander Raddatz." That marriage was dissolved in 1929 and Schapiro acknowledged his paternity, but never married Hanka Grothendieck. Grothendieck had

1815-414: A commune in the early 1970s. Grothendieck's early mathematical work was in functional analysis . Between 1949 and 1953 he worked on his doctoral thesis in this subject at Nancy , supervised by Jean Dieudonné and Laurent Schwartz . His key contributions include topological tensor products of topological vector spaces , the theory of nuclear spaces as foundational for Schwartz distributions , and

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1980-453: A 600-page manuscript entitled Pursuing Stacks . It began with a letter addressed to Daniel Quillen . This letter and successive parts were distributed from Bangor (see External links below). Within these, in an informal, diary-like manner, Grothendieck explained and developed his ideas on the relationship between algebraic homotopy theory and algebraic geometry and prospects for a noncommutative theory of stacks . The manuscript, which

2145-479: A complete variety is finite-dimensional; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent; this reduces to Serre's theorem over a one-point space. In 1956, he applied the same thinking to the Riemann–Roch theorem , which recently had been generalized to any dimension by Hirzebruch . The Grothendieck–Riemann–Roch theorem was announced by Grothendieck at

2310-429: A conference in his memory. Algebra Algebra is the branch of mathematics that studies certain abstract systems , known as algebraic structures , and the manipulation of statements within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations such as addition and multiplication . Elementary algebra

2475-429: A connection between the topological characteristics of a variety and its diophantine (number theoretic) properties. For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers . Weil had realized that to prove such a connection, one needed a new cohomology theory, but neither he nor any other expert saw how to accomplish this until such

2640-503: A continuous stream of high-quality mathematical publications in international, peer-reviewed mathematics journals by several members participating in his Seminar series. Popescu died in Bucharest on July 29, 2010. He is survived by his wife, Professor Dr. Elena Liliana Popescu  [ ro ] (a mathematician, poet, literary translator and editor), and their three children, one of whom, Dan Cristian Popescu  [ ro ] ,

2805-498: A freshman. Popescu earned his M.S. degree in mathematics in 1964, and his Ph.D. degree in mathematics in 1967, with thesis Krull–Remak–Schmidt Theorem and Theory of Decomposition written under the direction of Gheorghe Galbură  [ ro ] . He was awarded a D. Phil. degree (Doctor Docent) in 1972, also by the University of Bucharest. While still a student, Popescu focused on category theory . He first approached

2970-418: A higher level of abstraction and turned them into a key organising principle of his theory. He shifted attention from the study of individual varieties to his relative point of view (pairs of varieties related by a morphism ), allowing a broad generalization of many classical theorems. The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on

3135-465: A key turning point in the history of algebra and consider what came before it as the prehistory of algebra because it lacked the abstract nature based on symbolic manipulation. In the 17th and 18th centuries, many attempts were made to find general solutions to polynomials of degree five and higher. All of them failed. At the end of the 18th century, the German mathematician Carl Friedrich Gauss proved

3300-467: A large part of linear algebra. A vector space is an algebraic structure formed by a set with an addition that makes it an abelian group and a scalar multiplication that is compatible with addition (see vector space for details). A linear map is a function between vector spaces that is compatible with addition and scalar multiplication. In the case of finite-dimensional vector spaces , vectors and linear maps can be represented by matrices. It follows that

3465-405: A list of 14 open questions, relevant for locally convex spaces . Grothendieck introduced new mathematical methods that enabled him to solve all of these problems within a few months. In Nancy, he wrote his dissertation under those two professors on functional analysis , from 1950 to 1953. At this time he was a leading expert in the theory of topological vector spaces. In 1953 he moved to

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3630-605: A maternal sibling, his half sister Maidi. Grothendieck lived with his parents in Berlin until the end of 1933, when his father moved to Paris to evade Nazism . His mother followed soon thereafter. Grothendieck was left in the care of Wilhelm Heydorn, a Lutheran pastor and teacher in Hamburg . According to Winfried Scharlau , during this time, his parents took part in the Spanish Civil War as non-combatant auxiliaries. However, others state that Schapiro fought in

3795-420: A new address that he did not share with his previous contacts in the mathematical community. Very few people visited him afterward. Local villagers helped sustain him with a more varied diet after he tried to live on a staple of dandelion soup. At some point, Leila Schneps and Pierre Lochak  [ fr ] located him, then carried on a brief correspondence. Thus they became among "the last members of

3960-412: A new privately funded research institute that, in effect, had been created for Jean Dieudonné and Grothendieck. Grothendieck attracted attention by an intense and highly productive activity of seminars there ( de facto working groups drafting into foundational work some of the ablest French and other mathematicians of the younger generation). Grothendieck practically ceased publication of papers through

4125-487: A political group entitled Survivre —the name later changed to Survivre et vivre . The group published a bulletin and was dedicated to antimilitary and ecological issues. It also developed strong criticism of the indiscriminate use of science and technology. Grothendieck devoted the next three years to this group and served as the main editor of its bulletin. Although Grothendieck continued with mathematical enquiries, his standard mathematical career mostly ended when he left

4290-412: A positive degree can be factorized into linear polynomials. This theorem was proved at the beginning of the 19th century, but this does not close the problem since the theorem does not provide any way for computing the solutions. Linear algebra starts with the study systems of linear equations . An equation is linear if it can be expressed in the form a 1 x 1 +

4455-462: A refugee. Records of his nationality were destroyed in the fall of Nazi Germany in 1945 and he did not apply for French citizenship after the war. Thus, he became a stateless person for at least the majority of his working life and he traveled on a Nansen passport . Part of his reluctance to hold French nationality is attributed to not wishing to serve in the French military, particularly due to

4620-428: A second-degree polynomial equation of the form a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} is given by the quadratic formula x = − b ± b 2 − 4 a c   2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.} Solutions for

4785-447: A similar way, if one knows the value of one variable one may be able to use it to determine the value of other variables. Algebraic equations can be interpreted geometrically to describe spatial figures in the form of a graph . To do so, the different variables in the equation are understood as coordinates and the values that solve the equation are interpreted as points of a graph. For example, if x {\displaystyle x}

4950-435: A statement formed by comparing two expressions, saying that they are equal. This can be expressed using the equals sign ( = {\displaystyle =} ), as in 5 x 2 + 6 x = 3 y + 4 {\displaystyle 5x^{2}+6x=3y+4} . Inequations involve a different type of comparison, saying that the two sides are different. This can be expressed using symbols such as

5115-532: A unifying framework to describe and analyze many fundamental mathematical concepts. For example, sets can be described with the category of sets , and any group can be regarded as the morphisms of a category with just one object. The origin of algebra lies in attempts to solve mathematical problems involving arithmetic calculations and unknown quantities. These developments happened in the ancient period in Babylonia , Egypt , Greece , China , and India . One of

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5280-479: A website devoted to his work as "an abomination". His dictate may have been reversed in 2010. In September 2014, almost totally deaf and blind, he asked a neighbour to buy him a revolver so he could kill himself. On 13 November 2014, aged 86, Grothendieck died in the hospital of Saint-Lizier or Saint-Girons, Ariège . Grothendieck was born in Weimar Germany . In 1938, aged ten, he moved to France as

5445-400: A whole is zero if and only if one of its factors is zero, i.e., if x {\displaystyle x} is either −2 or 5. Before the 19th century, much of algebra was devoted to polynomial equations , that is equations obtained by equating a polynomial to zero. The first attempts for solving polynomial equations was to express the solutions in terms of n th roots . The solution of

5610-530: Is Grothendieck's account of how his consideration of the source of dreams led him to conclude that a deity exists. As part of the notes to this manuscript, Grothendieck described the life and the work of 18 "mutants", people whom he admired as visionaries far ahead of their time and heralding a new age. The only mathematician on his list was Bernhard Riemann . Influenced by the Catholic mystic Marthe Robin who

5775-397: Is a commutative group under addition: the addition of the ring is associative, commutative, and has an identity element and inverse elements. The multiplication is associative and distributive with respect to addition; that is, a ( b + c ) = a b + a c {\displaystyle a(b+c)=ab+ac} and ( b + c ) a = b

5940-438: Is a commutative ring such that ⁠ 1 ≠ 0 {\displaystyle 1\neq 0} ⁠ and each nonzero element has a multiplicative inverse . The ring of integers does not form a field because it lacks multiplicative inverses. For example, the multiplicative inverse of 7 {\displaystyle 7} is 1 7 {\displaystyle {\tfrac {1}{7}}} , which

6105-475: Is a function from the underlying set of one algebraic structure to the underlying set of another algebraic structure that preserves certain structural characteristics. If the two algebraic structures use binary operations and have the form ⟨ A , ∘ ⟩ {\displaystyle \langle A,\circ \rangle } and ⟨ B , ⋆ ⟩ {\displaystyle \langle B,\star \rangle } then

6270-504: Is a method used to simplify polynomials, making it easier to analyze them and determine the values for which they evaluate to zero . Factorization consists in rewriting a polynomial as a product of several factors. For example, the polynomial x 2 − 3 x − 10 {\displaystyle x^{2}-3x-10} can be factorized as ( x + 2 ) ( x − 5 ) {\displaystyle (x+2)(x-5)} . The polynomial as

6435-814: Is a politician. In 1971 Popescu received the Simion Stoilow Prize in Mathematics of the Romanian Academy. He was elected President of the Romanian Mathematical Society in 1990 and corresponding Member of the Romanian Academy in 1997. On the 80th anniversary of his birthday, the Faculty of Mathematics and Informatics at the University of Bucharest and the Institute of Mathematics of the Romanian Academy organized

6600-487: Is a polynomial with one term while two- and three-term polynomials are called binomials and trinomials. The degree of a polynomial is the maximal value (among its terms) of the sum of the exponents of the variables (4 in the above example). Polynomials of degree one are called linear polynomials . Linear algebra studies systems of linear polynomials. A polynomial is said to be univariate or multivariate , depending on whether it uses one or more variables. Factorization

6765-941: Is a set of linear equations for which one is interested in common solutions. Matrices are rectangular arrays of values that have been originally introduced for having a compact and synthetic notation for systems of linear equations For example, the system of equations 9 x 1 + 3 x 2 − 13 x 3 = 0 2.3 x 1 + 7 x 3 = 9 − 5 x 1 − 17 x 2 = − 3 {\displaystyle {\begin{aligned}9x_{1}+3x_{2}-13x_{3}&=0\\2.3x_{1}+7x_{3}&=9\\-5x_{1}-17x_{2}&=-3\end{aligned}}} can be written as A X = B , {\displaystyle AX=B,} where A , B {\displaystyle A,B} and C {\displaystyle C} are

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6930-532: Is all the more incredible; quite unique in the history of mathematics. His first works on topological vector spaces in 1953 have been successfully applied to physics and computer science, culminating in a relation between Grothendieck inequality and the Einstein–Podolsky–Rosen paradox in quantum physics . In 1958, Grothendieck was installed at the Institut des hautes études scientifiques (IHÉS),

7095-559: Is also available on the Internet. An English translation by Leila Schneps will be published by MIT Press in 2025. A partial English translation can be found on the Internet. A Japanese translation of the whole book in four volumes was completed by Tsuji Yuichi (1938–2002), a friend of Grothendieck from the Survivre period. The first three volumes (corresponding to Parts 0 to III of the book) were published between 1989 and 1993, while

7260-414: Is an expression consisting of one or more terms that are added or subtracted from each other, like x 4 + 3 x y 2 + 5 x 3 − 1 {\displaystyle x^{4}+3xy^{2}+5x^{3}-1} . Each term is either a constant, a variable, or a product of a constant and variables. Each variable can be raised to a positive-integer power. A monomial

7425-629: Is applied to one side of an equation also needs to be done to the other side. For example, if one subtracts 5 from the left side of an equation one also needs to subtract 5 from the right side to balance both sides. The goal of these steps is usually to isolate the variable one is interested in on one side, a process known as solving the equation for that variable. For example, the equation x − 7 = 4 {\displaystyle x-7=4} can be solved for x {\displaystyle x} by adding 7 to both sides, which isolates x {\displaystyle x} on

7590-497: Is being edited for publication by G. Maltsiniotis, later led to another of his monumental works, Les Dérivateurs . Written in 1991, this latter opus of approximately 2000 pages, further developed the homotopical ideas begun in Pursuing Stacks . Much of this work anticipated the subsequent development during the mid-1990s of the motivic homotopy theory of Fabien Morel and Vladimir Voevodsky . In 1984, Grothendieck wrote

7755-536: Is collected in the monumental, yet incomplete, Éléments de géométrie algébrique ( EGA ) and Séminaire de géométrie algébrique ( SGA ). The collection Fondements de la Géometrie Algébrique ( FGA ), which gathers together talks given in the Séminaire Bourbaki , also contains important material. Grothendieck's work includes the invention of the étale and l-adic cohomology theories, which explain an observation made by André Weil that argued for

7920-414: Is considered by many to be the greatest mathematician of the twentieth century. Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the Institut des hautes études scientifiques (IHÉS) and remained there until 1970, when, driven by personal and political convictions, he left following a dispute over military funding. He received

8085-408: Is no solution since they never intersect. If two equations are not independent then they describe the same line, meaning that every solution of one equation is also a solution of the other equation. These relations make it possible to seek solutions graphically by plotting the equations and determining where they intersect. The same principles also apply to systems of equations with more variables, with

8250-482: Is not an integer. The rational numbers , the real numbers , and the complex numbers each form a field with the operations of addition and multiplication. Ring theory is the study of rings, exploring concepts such as subrings , quotient rings , polynomial rings , and ideals as well as theorems such as Hilbert's basis theorem . Field theory is concerned with fields, examining field extensions , algebraic closures , and finite fields . Galois theory explores

8415-400: Is set to zero in the equation y = 0.5 x − 1 {\displaystyle y=0.5x-1} , then y {\displaystyle y} must be −1 for the equation to be true. This means that the ( x , y ) {\displaystyle (x,y)} -pair ( 0 , − 1 ) {\displaystyle (0,-1)} is part of

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8580-403: Is the identity matrix . Then, multiplying on the left both members of the above matrix equation by A − 1 , {\displaystyle A^{-1},} one gets the solution of the system of linear equations as X = A − 1 B . {\displaystyle X=A^{-1}B.} Methods of solving systems of linear equations range from

8745-414: Is the application of group theory to analyze graphs and symmetries. The insights of algebra are also relevant to calculus, which uses mathematical expressions to examine rates of change and accumulation . It relies on algebra, for instance, to understand how these expressions can be transformed and what role variables play in them. Algebraic logic employs the methods of algebra to describe and analyze

8910-423: Is the branch of mathematics that studies algebraic structures and the operations they use. An algebraic structure is a non-empty set of mathematical objects , such as the integers , together with algebraic operations defined on that set, like addition and multiplication . Algebra explores the laws, general characteristics, and types of algebraic structures. Within certain algebraic structures, it examines

9075-425: Is the case because the sum of two even numbers is again an even number. But the set of odd integers together with addition is not a subalgebra because it is not closed: adding two odd numbers produces an even number, which is not part of the chosen subset. Universal algebra is the study of algebraic structures in general. As part of its general perspective, it is not concerned with the specific elements that make up

9240-443: Is the main form of algebra taught in school and examines mathematical statements using variables for unspecified values. It seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called systems of linear equations . It provides methods to find

9405-421: Is the process of applying algebraic methods and principles to other branches of mathematics , such as geometry , topology , number theory , and calculus . It happens by employing symbols in the form of variables to express mathematical insights on a more general level, allowing mathematicians to develop formal models describing how objects interact and relate to each other. One application, found in geometry,

9570-472: Is the study of numerical operations and investigates how numbers are combined and transformed using the arithmetic operations of addition , subtraction , multiplication , division , exponentiation , extraction of roots , and logarithm . For example, the operation of addition combines two numbers, called the addends, into a third number, called the sum, as in 2 + 5 = 7 {\displaystyle 2+5=7} . Elementary algebra relies on

9735-718: Is the use of algebraic statements to describe geometric figures. For example, the equation y = 3 x − 7 {\displaystyle y=3x-7} describes a line in two-dimensional space while the equation x 2 + y 2 + z 2 = 1 {\displaystyle x^{2}+y^{2}+z^{2}=1} corresponds to a sphere in three-dimensional space. Of special interest to algebraic geometry are algebraic varieties , which are solutions to systems of polynomial equations that can be used to describe more complex geometric figures. Algebraic reasoning can also solve geometric problems. For example, one can determine whether and where

9900-461: Is there that he completed his last major work on that topic (on "metric" theory of Banach spaces ). Grothendieck moved to Lawrence, Kansas at the beginning of 1955, and there he set his old subject aside in order to work in algebraic topology and homological algebra , and increasingly in algebraic geometry. It was in Lawrence that Grothendieck developed his theory of abelian categories and

10065-466: Is true for all elements of the underlying set. For example, commutativity is a universal equation that states that a ∘ b {\displaystyle a\circ b} is identical to b ∘ a {\displaystyle b\circ a} for all elements. A variety is a class of all algebraic structures that satisfy certain identities. For example, if two algebraic structures satisfy commutativity then they are both part of

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10230-446: Is true if x {\displaystyle x} is either 2 or −2 and false otherwise. Equations with variables can be divided into identity equations and conditional equations. Identity equations are true for all values that can be assigned to the variables, such as the equation 2 x + 5 x = 7 x {\displaystyle 2x+5x=7x} . Conditional equations are only true for some values. For example,

10395-472: The École Normale Supérieure -trained students at that time ( Pierre Samuel , Roger Godement , René Thom , Jacques Dixmier , Jean Cerf , Yvonne Bruhat , Jean-Pierre Serre , and Bernard Malgrange ), Leila Schneps said: He was so completely unknown to this group and to their professors, came from such a deprived and chaotic background, and was, compared to them, so ignorant at the start of his research career, that his fulgurating ascent to sudden stardom

10560-707: The Algerian War (1954–62). He eventually applied for French citizenship in the early 1980s, after he was well past the age that would have required him to do military service. Grothendieck was very close to his mother, to whom he dedicated his dissertation. She died in 1957 from tuberculosis that she contracted in camps for displaced persons . He had five children: a son with his landlady during his time in Nancy; three children, Johanna (1959), Alexander (1961), and Mathieu (1965) with his wife Mireille Dufour; and one child with Justine Skalba, with whom he lived in

10725-477: The Crafoord Prize with an open letter to the media. He wrote that he and other established mathematicians had no need for additional financial support and criticized what he saw as the declining ethics of the scientific community that was characterized by outright scientific theft that he believed had become commonplace and tolerated. The letter also expressed his belief that totally unforeseen events before

10890-523: The Fields Medal in 1966 for advances in algebraic geometry , homological algebra , and K-theory . He later became professor at the University of Montpellier and, while still producing relevant mathematical work, he withdrew from the mathematical community and devoted himself to political and religious pursuits (first Buddhism and later, a more Catholic Christian vision). In 1991, he moved to

11055-576: The Le Collège-Lycée Cévenol International ), a unique secondary school founded in 1938 by local Protestant pacifists and anti-war activists. Many of the refugee children hidden in Le Chambon attended Collège Cévenol, and it was at this school that Grothendieck apparently first became fascinated with mathematics. In 1990, for risking their lives to rescue Jews, the entire village was recognized as " Righteous Among

11220-504: The Romanian Revolution . Between 1962 and 2008 Popescu published more than 102 papers in peer-reviewed mathematics journals, several monographs on the theory of sheaves, and several books on abelian category theory and abstract algebra , including In a Grothendieck -like, energetic style, he initiated and provided scientific leadership to several seminars on category theory, sheaves and abstract algebra which resulted in

11385-407: The University of Iași . In his third year of studies he was expelled from the university, having been deemed "hostile to the regime" for remarking that "the achievements of American scientists are also worth of consideration." He then went back home to Strehaia, where he worked for a year in a collective farm, after which he was admitted in 1959 at the University of Bucharest , only to start anew as

11550-603: The University of São Paulo in Brazil, where he immigrated by means of a Nansen passport , given that he had refused to take French nationality (as that would have entailed military service against his convictions). He stayed in São Paulo (apart from a lengthy visit in France from October 1953 to March 1954) until the end of 1954. His published work from the time spent in Brazil is still in the theory of topological vector spaces; it

11715-509: The Weil conjectures , as well as crystalline cohomology and algebraic de Rham cohomology to complement it. Closely linked to these cohomology theories, he originated topos theory as a generalisation of topology (relevant also in categorical logic ). He also provided, by means of a categorical Galois theory , an algebraic definition of fundamental groups of schemes giving birth to the now famous étale fundamental group and he then conjectured

11880-551: The difference of two squares method and later in Euclid's Elements . In the 3rd century CE, Diophantus provided a detailed treatment of how to solve algebraic equations in a series of books called Arithmetica . He was the first to experiment with symbolic notation to express polynomials. Diophantus's work influenced Arab development of algebra with many of his methods reflected in the concepts and techniques used in medieval Arabic algebra. In ancient China, The Nine Chapters on

12045-535: The fundamental theorem of algebra , which describes the existence of zeros of polynomials of any degree without providing a general solution. At the beginning of the 19th century, the Italian mathematician Paolo Ruffini and the Norwegian mathematician Niels Henrik Abel were able to show that no general solution exists for polynomials of degree five and higher. In response to and shortly after their findings,

12210-602: The fundamental theorem of finite abelian groups and the Feit–Thompson theorem . The latter was a key early step in one of the most important mathematical achievements of the 20th century: the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete classification of finite simple groups . A ring is an algebraic structure with two operations that work similarly to addition and multiplication of numbers and are named and generally denoted similarly. A ring

12375-461: The less-than sign ( < {\displaystyle <} ), the greater-than sign ( > {\displaystyle >} ), and the inequality sign ( ≠ {\displaystyle \neq } ). Unlike other expressions, statements can be true or false and their truth value usually depends on the values of the variables. For example, the statement x 2 = 4 {\displaystyle x^{2}=4}

12540-679: The 12th century further refined Brahmagupta's methods and concepts. In 1247, the Chinese mathematician Qin Jiushao wrote the Mathematical Treatise in Nine Sections , which includes an algorithm for the numerical evaluation of polynomials , including polynomials of higher degrees. The Italian mathematician Fibonacci brought al-Khwarizmi's ideas and techniques to Europe in books including his Liber Abaci . In 1545,

12705-426: The 16th and 17th centuries, when a rigorous symbolic formalism was developed. In the mid-19th century, the scope of algebra broadened beyond a theory of equations to cover diverse types of algebraic operations and structures. Algebra is relevant to many branches of mathematics, such as geometry, topology , number theory , and calculus , and other fields of inquiry, like logic and the empirical sciences . Algebra

12870-533: The 1930s, the American mathematician Garrett Birkhoff expanded these ideas and developed many of the foundational concepts of this field. The invention of universal algebra led to the emergence of various new areas focused on the algebraization of mathematics—that is, the application of algebraic methods to other branches of mathematics. Topological algebra arose in the early 20th century, studying algebraic structures such as topological groups and Lie groups . In

13035-464: The 1940s and 50s, homological algebra emerged, employing algebraic techniques to study homology . Around the same time, category theory was developed and has since played a key role in the foundations of mathematics . Other developments were the formulation of model theory and the study of free algebras . The influence of algebra is wide-reaching, both within mathematics and in its applications to other fields. The algebraization of mathematics

13200-405: The 1958 International Congress of Mathematicians , he introduced the theory of schemes , developing it in detail in his Éléments de géométrie algébrique ( EGA ) and providing the new more flexible and general foundations for algebraic geometry that has been adopted in the field since that time. He went on to introduce the étale cohomology theory of schemes, providing the key tools for proving

13365-484: The 1958 International Congress of Mathematicians . His foundational work on algebraic geometry is at a higher level of abstraction than all prior versions. He adapted the use of non-closed generic points , which led to the theory of schemes . Grothendieck also pioneered the systematic use of nilpotents . As 'functions' these can take only the value 0, but they carry infinitesimal information, in purely algebraic settings. His theory of schemes has become established as

13530-589: The 1980s, he produced several influential manuscripts with limited distribution, with both mathematical and biographical content. Produced during 1980 and 1981, La Longue Marche à travers la théorie de Galois ( The Long March Through Galois Theory ) is a 1600-page handwritten manuscript containing many of the ideas that led to the Esquisse d'un programme . It also includes a study of Teichmüller theory . In 1983, stimulated by correspondence with Ronald Brown and Tim Porter at Bangor University , Grothendieck wrote

13695-402: The 9th century and the Persian mathematician Omar Khayyam in the 11th and 12th centuries. In India, Brahmagupta investigated how to solve quadratic equations and systems of equations with several variables in the 7th century CE. Among his innovations were the use of zero and negative numbers in algebraic equations. The Indian mathematicians Mahāvīra in the 9th century and Bhāskara II in

13860-401: The Faculty of Mathematics in Bucharest in 1968. Like Grothendieck, he had a long-standing interest in category theory and number theory , and supported promising young mathematicians in his fields of interest. He also promoted the early developments of category theory applications in relational biology and mathematical biophysics/ mathematical biology . Popescu was appointed as a Lecturer at

14025-432: The French mathematician Évariste Galois developed what came later to be known as Galois theory , which offered a more in-depth analysis of the solutions of polynomials while also laying the foundation of group theory . Mathematicians soon realized the relevance of group theory to other fields and applied it to disciplines like geometry and number theory. Starting in the mid-19th century, interest in algebra shifted from

14190-633: The French village of Lasserre in the Pyrenees , where he lived in seclusion, still working on mathematics and his philosophical and religious thoughts until his death in 2014. Grothendieck was born in Berlin to anarchist parents. His father, Alexander "Sascha" Schapiro (also known as Alexander Tanaroff), had Hasidic Jewish roots and had been imprisoned in Russia before moving to Germany in 1922, while his mother, Johanna "Hanka" Grothendieck , came from

14355-608: The German mathematicians David Hilbert , Ernst Steinitz , and Emmy Noether as well as the Austrian mathematician Emil Artin . They researched different forms of algebraic structures and categorized them based on their underlying axioms into types, like groups, rings, and fields. The idea of the even more general approach associated with universal algebra was conceived by the English mathematician Alfred North Whitehead in his 1898 book A Treatise on Universal Algebra . Starting in

14520-629: The IHÉS established several unifying themes in algebraic geometry , number theory , topology , category theory , and complex analysis . His first (pre-IHÉS) discovery in algebraic geometry was the Grothendieck–Hirzebruch–Riemann–Roch theorem , a generalisation of the Hirzebruch–Riemann–Roch theorem proved algebraically; in this context he also introduced K-theory . Then, following the programme he outlined in his talk at

14685-670: The IHÉS. Grothendieck's political views were radical and pacifistic . He strongly opposed both United States intervention in Vietnam and Soviet military expansionism . To protest against the Vietnam War , he gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed. In 1966, he had declined to attend the International Congress of Mathematicians (ICM) in Moscow, where he

14850-472: The IHÉS. After leaving the IHÉS, Grothendieck became a temporary professor at Collège de France for two years. He then became a professor at the University of Montpellier, where he became increasingly estranged from the mathematical community. He formally retired in 1988, a few years after having accepted a research position at the CNRS . While not publishing mathematical research in conventional ways during

15015-592: The Italian polymath Gerolamo Cardano published his book Ars Magna , which covered many topics in algebra, discussed imaginary numbers , and was the first to present general methods for solving cubic and quartic equations . In the 16th and 17th centuries, the French mathematicians François Viète and René Descartes introduced letters and symbols to denote variables and operations, making it possible to express equations in an abstract and concise manner. Their predecessors had relied on verbal descriptions of problems and solutions. Some historians see this development as

15180-569: The Mathematical Art , a book composed over the period spanning from the 10th century BCE to the 2nd century CE, explored various techniques for solving algebraic equations, including the use of matrix-like constructs. There is no unanimity as to whether these early developments are part of algebra or only precursors. They offered solutions to algebraic problems but did not conceive them in an abstract and general manner, focusing instead on specific cases and applications. This changed with

15345-648: The Nations ". After the war, the young Grothendieck studied mathematics in France, initially at the University of Montpellier where at first he did not perform well, failing such classes as astronomy. Working on his own, he rediscovered the Lebesgue measure . After three years of increasingly independent studies there, he went to continue his studies in Paris in 1948. Initially, Grothendieck attended Henri Cartan 's Seminar at École Normale Supérieure , but he lacked

15510-458: The Persian mathematician al-Khwarizmi , who published his The Compendious Book on Calculation by Completion and Balancing in 825 CE. It presents the first detailed treatment of general methods that can be used to manipulate linear and quadratic equations by "reducing" and "balancing" both sides. Other influential contributions to algebra came from the Arab mathematician Thābit ibn Qurra also in

15675-528: The SGA projects also included Michael Artin ( étale cohomology ), Nick Katz ( monodromy theory , and Lefschetz pencils ). Jean Giraud worked out torsor theory extensions of nonabelian cohomology there as well. Many others such as David Mumford , Robin Hartshorne , Barry Mazur and C.P. Ramanujam were also involved. Alexander Grothendieck's work during what is described as the "Golden Age" period at

15840-456: The University of Bucharest in 1968 where he taught graduate students until 1972. Starting in 1964 he also held a research appointment at the Institute of Mathematics of the Romanian Academy . The institute was closed in 1976 by order of Nicolae Ceaușescu (for reasons related to his daughter Zoia Ceaușescu , who had been hired at the institute two years before), but was reopened in 1990, after

16005-401: The addition of numbers. While elementary algebra and linear algebra work within the confines of particular algebraic structures, abstract algebra takes a more general approach that compares how algebraic structures differ from each other and what types of algebraic structures there are, such as groups , rings , and fields . The key difference between these types of algebraic structures lies in

16170-613: The anarchist militia. In May 1939, Grothendieck was put on a train in Hamburg for France. Shortly afterward his father was interned in Le Vernet . He and his mother were then interned in various camps from 1940 to 1942 as "undesirable dangerous foreigners." The first camp was the Rieucros Camp , where his mother contracted the tuberculosis that would eventually cause her death in 1957. While there, Grothendieck managed to attend

16335-483: The application of L spaces in studying linear maps between topological vector spaces. In a few years, he had become a leading authority on this area of functional analysis—to the extent that Dieudonné compares his impact in this field to that of Banach . It is, however, in algebraic geometry and related fields where Grothendieck did his most important and influential work. From approximately 1955 he started to work on sheaf theory and homological algebra , producing

16500-506: The best universal foundation for this field, because of its expressiveness as well as its technical depth. In that setting one can use birational geometry , techniques from number theory , Galois theory , commutative algebra , and close analogues of the methods of algebraic topology , all in an integrated way. Grothendieck is noted for his mastery of abstract approaches to mathematics and his perfectionism in matters of formulation and presentation. Relatively little of his work after 1960

16665-443: The characteristics of algebraic structures in general. The term "algebra" is sometimes used in a more narrow sense to refer only to elementary algebra or only to abstract algebra. When used as a countable noun , an algebra is a specific type of algebraic structure that involves a vector space equipped with a certain type of binary operation . Depending on the context, "algebra" can also refer to other algebraic structures, like

16830-428: The conventional, learned journal route. However, he was able to play a dominant role in mathematics for approximately a decade, gathering a strong school. Officially during this time, he had as students Michel Demazure (who worked on SGA3, on group schemes ), Luc Illusie (cotangent complex), Michel Raynaud , Jean-Louis Verdier (co-founder of the derived category theory), and Pierre Deligne . Collaborators on

16995-416: The corresponding variety. Category theory examines how mathematical objects are related to each other using the concept of categories . A category is a collection of objects together with a collection of so-called morphisms or "arrows" between those objects. These two collections must satisfy certain conditions. For example, morphisms can be joined, or composed : if there exists a morphism from object

17160-593: The degrees 3 and 4 are given by the cubic and quartic formulas. There are no general solutions for higher degrees, as proven in the 19th century by the so-called Abel–Ruffini theorem . Even when general solutions do not exist, approximate solutions can be found by numerical tools like the Newton–Raphson method . The fundamental theorem of algebra asserts that every univariate polynomial equation of positive degree with real or complex coefficients has at least one complex solution. Consequently, every polynomial of

17325-455: The difference being that the equations do not describe lines but higher dimensional figures. For instance, equations with three variables correspond to planes in three-dimensional space , and the points where all planes intersect solve the system of equations. Abstract algebra, also called modern algebra, is the study of algebraic structures . An algebraic structure is a framework for understanding operations on mathematical objects , like

17490-469: The distributive property. For statements with several variables, substitution is a common technique to replace one variable with an equivalent expression that does not use this variable. For example, if one knows that y = 3 x {\displaystyle y=3x} then one can simplify the expression 7 x y {\displaystyle 7xy} to arrive at 21 x 2 {\displaystyle 21x^{2}} . In

17655-446: The downtrodden. As Cartier puts it, Grothendieck came to find Bures-sur-Yvette as " une cage dorée " ("a gilded cage"). While Grothendieck was at the IHÉS, opposition to the Vietnam War was heating up, and Cartier suggests that this also reinforced Grothendieck's distaste at having become a mandarin of the scientific world. In addition, after several years at the IHÉS, Grothendieck seemed to cast about for new intellectual interests. By

17820-507: The earliest documents on algebraic problems is the Rhind Mathematical Papyrus from ancient Egypt, which was written around 1650 BCE. It discusses solutions to linear equations , as expressed in problems like "A quantity; its fourth is added to it. It becomes fifteen. What is the quantity?" Babylonian clay tablets from around the same time explain methods to solve linear and quadratic polynomial equations , such as

17985-403: The elements of the two algebraic structures. This implies that every element of the first algebraic structure is mapped to one unique element in the second structure without any unmapped elements in the second structure. Another tool of comparison is the relation between an algebraic structure and its subalgebra . The algebraic structure and its subalgebra use the same operations, which follow

18150-537: The end of the century would lead to an unprecedented collapse of civilization. Grothendieck added however that his views were "in no way meant as a criticism of the Royal Academy's aims in the administration of its funds" and he added, "I regret the inconvenience that my refusal to accept the Crafoord prize may have caused you and the Royal Academy." La Clef des Songes , a 315-page manuscript written in 1987,

18315-404: The equation x + 4 = 9 {\displaystyle x+4=9} is only true if x {\displaystyle x} is 5. The main goal of elementary algebra is to determine the values for which a statement is true. This can be achieved by transforming and manipulating statements according to certain rules. A key principle guiding this process is that whatever operation

18480-612: The existence of loops or holes in them. Number theory is concerned with the properties of and relations between integers. Algebraic number theory applies algebraic methods and principles to this field of inquiry. Examples are the use of algebraic expressions to describe general laws, like Fermat's Last Theorem , and of algebraic structures to analyze the behavior of numbers, such as the ring of integers . The related field of combinatorics uses algebraic techniques to solve problems related to counting, arrangement, and combination of discrete objects. An example in algebraic combinatorics

18645-530: The existence of a further generalization of it, which is now known as the fundamental group scheme . As a framework for his coherent duality theory, he also introduced derived categories , which were further developed by Verdier. The results of his work on these and other topics were published in the EGA and in less polished form in the notes of the Séminaire de géométrie algébrique ( SGA ) that he directed at

18810-429: The form of variables in addition to numbers. A higher level of abstraction is found in abstract algebra , which is not limited to a particular domain and examines algebraic structures such as groups and rings . It extends beyond typical arithmetic operations by also covering other types of operations. Universal algebra is still more abstract in that it is not interested in specific algebraic structures but investigates

18975-422: The fourth volume (Part IV) was completed and, although unpublished, copies of it as a typed manuscript are circulated. Grothendieck helped with the translation and wrote a preface for it, in which he called Tsuji his "first true collaborator". Parts of Récoltes et Semailles have been translated into Spanish, as well as into a Russian translation that was published in Moscow. In 1988, Grothendieck declined

19140-438: The function h : A → B {\displaystyle h:A\to B} is a homomorphism if it fulfills the following requirement: h ( x ∘ y ) = h ( x ) ⋆ h ( y ) {\displaystyle h(x\circ y)=h(x)\star h(y)} . The existence of a homomorphism reveals that the operation ⋆ {\displaystyle \star } in

19305-594: The general theory, with its connections to homological algebra and algebraic topology, then shifted his focus on theory of Abelian categories , being one of the main promoters of this theory in Romania. He carried out mathematics studies at the Institute of Mathematics of the Romanian Academy in the Algebra research group, and also had international collaborations on three continents. He shared many moral, ethical, and religious values with Alexander Grothendieck , who visited

19470-413: The graph of the equation. The ( x , y ) {\displaystyle (x,y)} -pair ( 0 , 7 ) {\displaystyle (0,7)} , by contrast, does not solve the equation and is therefore not part of the graph. The graph encompasses the totality of ( x , y ) {\displaystyle (x,y)} -pairs that solve the equation. A polynomial

19635-574: The influential " Tôhoku paper " ( Sur quelques points d'algèbre homologique , published in the Tohoku Mathematical Journal in 1957) where he introduced abelian categories and applied their theory to show that sheaf cohomology may be defined as certain derived functors in this context. Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre and others, after sheaves had been defined by Jean Leray . Grothendieck took them to

19800-546: The initial Mathematische Arbeitstagung in Bonn , in 1957. It appeared in print in a paper written by Armand Borel with Serre. This result was his first work in algebraic geometry. Grothendieck went on to plan and execute a programme for rebuilding the foundations of algebraic geometry, which at the time were in a state of flux and under discussion in Claude Chevalley 's seminar. He outlined his programme in his talk at

19965-500: The introductory, like substitution and elimination, to more advanced techniques using matrices, such as Cramer's rule , the Gaussian elimination , and LU decomposition . Some systems of equations are inconsistent , meaning that no solutions exist because the equations contradict each other. Consistent systems have either one unique solution or an infinite number of solutions. The study of vector spaces and linear maps form

20130-446: The late 1960s, he had started to become interested in scientific areas outside mathematics. David Ruelle , a physicist who joined the IHÉS faculty in 1964, said that Grothendieck came to talk to him a few times about physics . Biology interested Grothendieck much more than physics, and he organized some seminars on biological topics. In 1970, Grothendieck, with two other mathematicians, Claude Chevalley and Pierre Samuel , created

20295-607: The left side and results in the equation x = 11 {\displaystyle x=11} . There are many other techniques used to solve equations. Simplification is employed to replace a complicated expression with an equivalent simpler one. For example, the expression 7 x − 3 x {\displaystyle 7x-3x} can be replaced with the expression 4 x {\displaystyle 4x} since 7 x − 3 x = ( 7 − 3 ) x = 4 x {\displaystyle 7x-3x=(7-3)x=4x} by

20460-620: The line described by y = x + 1 {\displaystyle y=x+1} intersects with the circle described by x 2 + y 2 = 25 {\displaystyle x^{2}+y^{2}=25} by solving the system of equations made up of these two equations. Topology studies the properties of geometric figures or topological spaces that are preserved under operations of continuous deformation . Algebraic topology relies on algebraic theories such as group theory to classify topological spaces. For example, homotopy groups classify topological spaces based on

20625-426: The linear map to the basis vectors. Systems of equations can be interpreted as geometric figures. For systems with two variables, each equation represents a line in two-dimensional space . The point where the two lines intersect is the solution of the full system because this is the only point that solves both the first and the second equation. For inconsistent systems, the two lines run parallel, meaning that there

20790-490: The local school, at Mendel. Once, he managed to escape from the camp, intending to assassinate Hitler . Later, his mother Hanka was transferred to the Gurs internment camp for the remainder of World War II . Grothendieck was permitted to live separated from his mother. In the village of Le Chambon-sur-Lignon , he was sheltered and hidden in local boarding houses or pensions , although he occasionally had to seek refuge in

20955-472: The lowercase letters x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} represent variables. In some cases, subscripts are added to distinguish variables, as in x 1 {\displaystyle x_{1}} , x 2 {\displaystyle x_{2}} , and x 3 {\displaystyle x_{3}} . The lowercase letters

21120-504: The mathematical community, a community that initially accepted him in an open and welcoming manner, but which he progressively perceived to be governed by competition and status. He complains about what he saw as the "burial" of his work and betrayal by his former students and colleagues after he had left the community. Récoltes et Semailles was finally published in 2022 by Gallimard and, thanks to French science historian Alain Herreman,

21285-609: The mathematical establishment to come into contact with him". After his death, it was revealed that he lived alone in a house in Lasserre, Ariège , a small village at the foot of the Pyrenees . In January 2010, Grothendieck wrote the letter entitled "Déclaration d'intention de non-publication" to Luc Illusie , claiming that all materials published in his absence had been published without his permission. He asked that none of his work be reproduced in whole or in part and that copies of this work be removed from libraries. He characterized

21450-647: The matrices A = [ 9 3 − 13 2.3 0 7 − 5 − 17 0 ] , X = [ x 1 x 2 x 3 ] , B = [ 0 9 − 3 ] . {\displaystyle A={\begin{bmatrix}9&3&-13\\2.3&0&7\\-5&-17&0\end{bmatrix}},\quad X={\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}},\quad B={\begin{bmatrix}0\\9\\-3\end{bmatrix}}.} Under some conditions on

21615-481: The method of completing the square . Many of these insights found their way to the ancient Greeks. Starting in the 6th century BCE, their main interest was geometry rather than algebra, but they employed algebraic methods to solve geometric problems. For example, they studied geometric figures while taking their lengths and areas as unknown quantities to be determined, as exemplified in Pythagoras ' formulation of

21780-455: The necessary background to follow the high-powered seminar. On the advice of Cartan and André Weil , he moved to the University of Nancy where two leading experts were working on Grothendieck's area of interest, topological vector spaces : Jean Dieudonné and Laurent Schwartz . The latter had recently won a Fields Medal. Dieudonné and Schwartz showed the new student their latest paper La dualité dans les espaces ( F ) et ( LF ) ; it ended with

21945-399: The number 3 to the result. Other examples of algebraic expressions are 32 x y z {\displaystyle 32xyz} and 64 x 1 2 + 7 x 2 − c {\displaystyle 64x_{1}^{2}+7x_{2}-c} . Some algebraic expressions take the form of statements that relate two expressions to one another. An equation is

22110-539: The number of operations they use and the laws they follow . Universal algebra and category theory provide general frameworks to investigate abstract patterns that characterize different classes of algebraic structures. Algebraic methods were first studied in the ancient period to solve specific problems in fields like geometry . Subsequent mathematicians examined general techniques to solve equations independent of their specific applications. They described equations and their solutions using words and abbreviations until

22275-470: The number of operations they use and the laws they obey. In mathematics education , abstract algebra refers to an advanced undergraduate course that mathematics majors take after completing courses in linear algebra. On a formal level, an algebraic structure is a set of mathematical objects, called the underlying set, together with one or several operations. Abstract algebra is primarily interested in binary operations , which take any two objects from

22440-511: The number of rows and columns, matrices can be added , multiplied , and sometimes inverted . All methods for solving linear systems may be expressed as matrix manipulations using these operations. For example, solving the above system consists of computing an inverted matrix A − 1 {\displaystyle A^{-1}} such that A − 1 A = I , {\displaystyle A^{-1}A=I,} where I {\displaystyle I}

22605-436: The numbers with variables, it is possible to express a general law that applies to any possible combination of numbers, like the commutative property of multiplication , which is expressed in the equation a × b = b × a {\displaystyle a\times b=b\times a} . Algebraic expressions are formed by using arithmetic operations to combine variables and numbers. By convention,

22770-425: The operation of addition. The neutral element is 0 and the inverse element of any number a {\displaystyle a} is − a {\displaystyle -a} . The natural numbers with addition, by contrast, do not form a group since they contain only positive integers and therefore lack inverse elements. Group theory examines the nature of groups, with basic theorems such as

22935-432: The operations are not restricted to regular arithmetic operations. For instance, the underlying set of the symmetry group of a geometric object is made up of geometric transformations , such as rotations , under which the object remains unchanged . Its binary operation is function composition , which takes two transformations as input and has the transformation resulting from applying the first transformation followed by

23100-493: The proposal Esquisse d'un Programme ("Sketch of a Programme") for a position at the Centre National de la Recherche Scientifique (CNRS). It describes new ideas for studying the moduli space of complex curves. Although Grothendieck never published his work in this area, the proposal inspired other mathematicians to work in the area by becoming the source of dessin d'enfant theory and anabelian geometry . Later, it

23265-570: The reformulation of sheaf cohomology based on them, leading to the very influential " Tôhoku paper ". In 1957 he was invited to visit Harvard University by Oscar Zariski , but the offer fell through when he refused to sign a pledge promising not to work to overthrow the United States government—a refusal which, he was warned, threatened to land him in prison. The prospect of prison did not worry him, so long as he could have access to books. Comparing Grothendieck during his Nancy years to

23430-449: The relation between field theory and group theory, relying on the fundamental theorem of Galois theory . Besides groups, rings, and fields, there are many other algebraic structures studied by algebra. They include magmas , semigroups , monoids , abelian groups , commutative rings , modules , lattices , vector spaces , algebras over a field , and associative and non-associative algebras . They differ from each other in regard to

23595-411: The rupture ran more deeply. Pierre Cartier , a visiteur de longue durée ("long-term guest") at the IHÉS, wrote a piece about Grothendieck for a special volume published on the occasion of the IHÉS's fortieth anniversary. In that publication, Cartier notes that as the son of an antimilitary anarchist and one who grew up among the disenfranchised, Grothendieck always had a deep compassion for the poor and

23760-430: The same axioms. The only difference is that the underlying set of the subalgebra is a subset of the underlying set of the algebraic structure. All operations in the subalgebra are required to be closed in its underlying set, meaning that they only produce elements that belong to this set. For example, the set of even integers together with addition is a subalgebra of the full set of integers together with addition. This

23925-543: The same operations while allowing variables in addition to regular numbers. Variables are symbols for unspecified or unknown quantities. They make it possible to state relationships for which one does not know the exact values and to express general laws that are true, independent of which numbers are used. For example, the equation 2 × 3 = 3 × 2 {\displaystyle 2\times 3=3\times 2} belongs to arithmetic and expresses an equality only for these specific numbers. By replacing

24090-401: The second algebraic structure plays the same role as the operation ∘ {\displaystyle \circ } does in the first algebraic structure. Isomorphisms are a special type of homomorphism that indicates a high degree of similarity between two algebraic structures. An isomorphism is a bijective homomorphism, meaning that it establishes a one-to-one relationship between

24255-442: The second as its output. Abstract algebra classifies algebraic structures based on the laws or axioms that its operations obey and the number of operations it uses. One of the most basic types is a group, which has one operation and requires that this operation is associative and has an identity element and inverse elements . An operation is associative if the order of several applications does not matter, i.e., if (

24420-656: The structures and patterns that underlie logical reasoning , exploring both the relevant mathematical structures themselves and their application to concrete problems of logic. It includes the study of Boolean algebra to describe propositional logic as well as the formulation and analysis of algebraic structures corresponding to more complex systems of logic . Alexander Grothendieck Alexander Grothendieck , later Alexandre Grothendieck in French ( / ˈ ɡ r oʊ t ən d iː k / ; German: [ˌalɛˈksandɐ ˈɡʁoːtn̩ˌdiːk] ; French: [ɡʁɔtɛndik] ; 28 March 1928 – 13 November 2014),

24585-410: The study of diverse types of algebraic operations and structures together with their underlying axioms , the laws they follow. Elementary algebra, also called school algebra, college algebra, and classical algebra, is the oldest and most basic form of algebra. It is a generalization of arithmetic that relies on variables and examines how mathematical statements may be transformed. Arithmetic

24750-485: The study of polynomials associated with elementary algebra towards a more general inquiry into algebraic structures, marking the emergence of abstract algebra . This approach explored the axiomatic basis of arbitrary algebraic operations. The invention of new algebraic systems based on different operations and elements accompanied this development, such as Boolean algebra , vector algebra , and matrix algebra . Influential early developments in abstract algebra were made by

24915-406: The theories of matrices and finite-dimensional vector spaces are essentially the same. In particular, vector spaces provide a third way for expressing and manipulating systems of linear equations. From this perspective, a matrix is a representation of a linear map: if one chooses a particular basis to describe the vectors being transformed, then the entries in the matrix give the results of applying

25080-638: The theory of sheaves, the theory of rings , fields and polynomials , and valuation theory . He also had interests and published in algebraic topology , algebraic geometry , commutative algebra , K-theory , class field theory , and algebraic function theory . Popescu was born on September 22, 1937, in Strehaia-Comanda , Mehedinți County , Romania . In 1954 he graduated from the Carol I High School in Craiova and went on to study mathematics at

25245-413: The types of objects they describe and the requirements that their operations fulfill. Many are related to each other in that a basic structure can be turned into a more advanced structure by adding additional requirements. For example, a magma becomes a semigroup if its operation is associative. Homomorphisms are tools to examine structural features by comparing two algebraic structures. A homomorphism

25410-510: The underlying set as inputs and map them to another object from this set as output. For example, the algebraic structure ⟨ N , + ⟩ {\displaystyle \langle \mathbb {N} ,+\rangle } has the natural numbers ( N {\displaystyle \mathbb {N} } ) as the underlying set and addition ( + {\displaystyle +} ) as its binary operation. The underlying set can contain mathematical objects other than numbers and

25575-400: The underlying sets and considers operations with more than two inputs, such as ternary operations . It provides a framework for investigating what structural features different algebraic structures have in common. One of those structural features concerns the identities that are true in different algebraic structures. In this context, an identity is a universal equation or an equation that

25740-516: The use of variables in equations and how to manipulate these equations. Algebra is often understood as a generalization of arithmetic . Arithmetic studies operations like addition, subtraction , multiplication, and division , in a particular domain of numbers, such as the real numbers. Elementary algebra constitutes the first level of abstraction. Like arithmetic, it restricts itself to specific types of numbers and operations. It generalizes these operations by allowing indefinite quantities in

25905-533: The values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathematical objects together with one or several operations defined on that set. It is a generalization of elementary and linear algebra, since it allows mathematical objects other than numbers and non-arithmetic operations. It distinguishes between different types of algebraic structures, such as groups , rings , and fields , based on

26070-758: The woods during Nazi raids, surviving at times without food or water for several days. His father was arrested under the Vichy anti-Jewish legislation , and sent to the Drancy internment camp , and then handed over by the French Vichy government to the Germans to be sent to be murdered at the Auschwitz concentration camp in 1942. In Le Chambon, Grothendieck attended the Collège Cévenol (now known as

26235-409: Was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry . His research extended the scope of the field and added elements of commutative algebra , homological algebra , sheaf theory , and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics . He

26400-521: Was a three-volume collection of research papers to mark his sixtieth birthday in 1988. More than 20,000 pages of Grothendieck's mathematical and other writings are held at the University of Montpellier and remain unpublished. They have been digitized for preservation and are freely available in open access through the Institut Montpelliérain Alexander Grothendieck portal. In 1991, Grothendieck moved to

26565-553: Was claimed to have survived on the Holy Eucharist alone, Grothendieck almost starved himself to death in 1988. His growing preoccupation with spiritual matters was also evident in a letter entitled Lettre de la Bonne Nouvelle sent to 250 friends in January 1990. In it, he described his encounters with a deity and announced that a "New Age" would commence on 14 October 1996. The Grothendieck Festschrift , published in 1990,

26730-565: Was published by the conventional route of the learned journal , circulating initially in duplicated volumes of seminar notes; his influence was to a considerable extent personal. His influence spilled over into many other branches of mathematics, for example the contemporary theory of D-modules . Although lauded as "the Einstein of mathematics", his work also provoked adverse reactions, with many mathematicians seeking out more concrete areas and problems. The bulk of Grothendieck's published work

26895-666: Was published in two-volumes and entitled Geometric Galois Actions (Cambridge University Press, 1997). During this period, Grothendieck also gave his consent to publishing some of his drafts for EGA on Bertini-type theorems ( EGA  V, published in Ulam Quarterly in 1992–1993 and later made available on the Grothendieck Circle web site in 2004). In the extensive autobiographical work, Récoltes et Semailles ('Harvests and Sowings', 1986), Grothendieck describes his approach to mathematics and his experiences in

27060-465: Was to receive the Fields Medal. He retired from scientific life around 1970 after he had found out that IHÉS was partly funded by the military. He returned to academia a few years later as a professor at the University of Montpellier . While the issue of military funding was perhaps the most obvious explanation for Grothendieck's departure from the IHÉS, those who knew him say that the causes of

27225-403: Was translated into Latin as Liber Algebrae et Almucabola . The word entered the English language in the 16th century from Italian , Spanish , and medieval Latin . Initially, its meaning was restricted to the theory of equations , that is, to the art of manipulating polynomial equations in view of solving them. This changed in the 19th century when the scope of algebra broadened to cover

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