Jean-Pierre Kahane was a French mathematician with contributions to harmonic analysis. Kahane attended the École normale supérieure and obtained the agrégation of mathematics in 1949, he worked for the CNRS from 1949 to 1954, first as an intern and as a research assistant. He defended his PhD in 1954, he was assistant professor professor of mathematics in Montpellier from 1954 to 1961. Since he has been professor until his retirement in 1994 professor emeritus at the Université de Paris-Sud in Orsay, he was a Plenary Speaker at the International Congress of Mathematicians in 1962 in Stockholm and an Invited Speaker at the 1986 ICM meeting in Berkeley, California. He was elected corresponding member of the French Academy of Sciences in 1982 and full member in 1998, he was president of the Société mathématique de France, the French Mathematical Society from 1971 to 1973. In 2000 Kahane received an honorary doctorate from the Faculty of Science and Technology at Uppsala University, Sweden In 2002 he was elevated to the rank of commander in the order of the Légion d'Honneur.
In 2012 he became a fellow of the American Mathematical Society. Kahane was known for his lifelong activism as part of the French Communist Party. Lectures on mean periodic functions. Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 50, Springer Verlag 1970 with Raphaël Salem: Ensembles parfaits et séries trigonométriques, Hermann, 1963, 1994 Some random series of functions, Massachusetts, D. C. Heath 1968, 2nd edition Cambridge University Press 1985 Séries de Fourier aléatoires, Presse de l’ Université de Montreal 1967 editor with A. G. Howson: The Popularization of Mathematics, Cambridge University Press 1990 Des series de Taylor au movement brownien, avec un apercu sur le retour, in Jean-Paul Pier, Development of mathematics 1900–1950, Birkhäuser 1994 Jean-Pierre Kahane at the Mathematics Genealogy Project Bio from the French Academy of Sciences
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise and in mathematics and physics as infinite-dimensional function spaces; the earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis, ergodic theory. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications.
The success of Hilbert space methods ushered in a fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, Hardy spaces of holomorphic functions. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes, in analogy with Cartesian coordinates in the plane; when that set of axes is countably infinite, the Hilbert space can be usefully thought of in terms of the space of infinite sequences that are square-summable. The latter space is in the older literature referred to as the Hilbert space.
Linear operators on a Hilbert space are fairly concrete objects: in good cases, they are transformations that stretch the space by different factors in mutually perpendicular directions in a sense, made precise by the study of their spectrum. One of the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ3, equipped with the dot product; the dot product takes two vectors x and y, produces a real number x · y. If x and y are represented in Cartesian coordinates the dot product is defined by ⋅ = x 1 y 1 + x 2 y 2 + x 3 y 3; the dot product satisfies the properties: It is symmetric in x and y: x · y = y · x. It is linear in its first argument: · y = ax1 · y + bx2 · y for any scalars a, b, vectors x1, x2, y, it is positive definite: for all vectors x, x · x ≥ 0, with equality if and only if x = 0. An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a inner product. A vector space equipped with such an inner product is known as a inner product space.
Every finite-dimensional inner product space is a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length of a vector, denoted ||x||, to the angle θ between two vectors x and y by means of the formula x ⋅ y = ‖ x ‖ ‖ y ‖ cos θ. Multivariable calculus in Euclidean space relies on the ability to compute limits, to have useful criteria for concluding that limits exist. A mathematical series ∑ n = 0 ∞ x n consisting of vectors in ℝ3 is convergent provided that the sum of the lengths converges as an ordinary series of real numbers: ∑ k = 0 ∞ ‖ x k ‖ < ∞. Just as with a series of scalars, a series of vectors that converges also converges to some limit vector L in the Euclidean space, in the sense that ‖ L − ∑ k = 0 N x k ‖ → 0 as N → ∞; this property expresses the completeness of
In probability theory and related fields, a stochastic or random process is a mathematical object defined as a collection of random variables. The random variables were associated with or indexed by a set of numbers viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes are used as mathematical models of systems and phenomena that appear to vary in a random manner, they have applications in many disciplines including sciences such as biology, ecology and physics as well as technology and engineering fields such as image processing, signal processing, information theory, computer science and telecommunications. Furthermore random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes.
Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time; these two stochastic processes are considered the most important and central in the theory of stochastic processes, were discovered and independently, both before and after Bachelier and Erlang, in different settings and countries. The term random function is used to refer to a stochastic or random process, because a stochastic process can be interpreted as a random element in a function space; the terms stochastic process and random process are used interchangeably with no specific mathematical space for the set that indexes the random variables. But these two terms are used when the random variables are indexed by the integers or an interval of the real line. If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space the collection of random variables is called a random field instead.
The values of a stochastic process are not always numbers and can be vectors or other mathematical objects. Based on their mathematical properties, stochastic processes can be divided into various categories, which include random walks, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, branching processes; the study of stochastic processes uses mathematical knowledge and techniques from probability, linear algebra, set theory, topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, functional analysis. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications. A stochastic or random process can be defined as a collection of random variables, indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set.
The set used to index. The index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time; each random variable in the collection takes values from the same mathematical space known as the state space. This state space can be, for example, the integers, the real n - dimensional Euclidean space. An increment is the amount that a stochastic process changes between two index values interpreted as two points in time. A stochastic process can have many outcomes, due to its randomness, a single outcome of a stochastic process is called, among other names, a sample function or realization. A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the state space; when interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers the stochastic process is said to be in discrete time.
If the index set is some interval of the real line time is said to be continuous. The two types of stochastic processes are referred to as discrete-time and continuous-time stochastic processes. Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge due to the index set being uncountable. If the index set is the integers, or some subset of them the stochastic process can be called a random sequence. If the state space is the integers or natural numbers the stochastic process is called a discrete or integer-valued stochastic process. If the state space is the real line the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space. If the state space is n -dimensional Euclidean space the stochastic process is called a n -dimensional vector process or n -vector process; the word stochastic in English was used as an adjective with the definition "pertaining to conjecturing", stemming from a Greek word meaning "to aim at a mark, guess", the Oxford English Dictionary gives the year 16
Michel Demazure is a French mathematician. He made contributions in the fields of abstract algebra, algebraic geometry, computer vision, participated in the Nicolas Bourbaki collective, he has been president of the French Mathematical Society and directed two French science museums. In the 1960s, Demazure was a student of Alexandre Grothendieck, together with Grothendieck, he ran and edited the Séminaire de Géométrie Algébrique du Bois Marie on group schemes at the Institut des Hautes Études Scientifiques near Paris from 1962 to 1964. Demazure obtained his doctorate from the Université de Paris in 1965 under Grothendieck's supervision, with a dissertation entitled Schémas en groupes reductifs, he was maître de conférence at Strasbourg University, university professor at Paris-Sud in Orsay and the École Polytechnique in Palaiseau. From 1965 to 1985, he was one of the core members of the Bourbaki group, a group of French mathematicians writing under the collective pseudonym Nicolas Bourbaki. In 1988 Demazure was the president of the Société Mathématique de France.
From 1991 to 1998, he was the director of the Palais de la Découverte in Paris and, from 1998 to 2002, the chairman of the Cité des Sciences et de l'Industrie in La Villette, two major science museums in France. Demazure chairs the regional advisory committee of research for Languedoc-Roussillon. In SGA3, Demazure introduced the definition of a root datum, a generalization of root systems for reductive groups, central to the notion of Langlands duality. A 1970 paper of Demazure on subgroups of the Cremona group has been recognized as the beginning of the study of toric varieties; the Demazure character formula and Demazure modules and Demazure conjecture are named after Demazure, who wrote about them in 1974. Demazure modules are submodules of a finite-dimensional representation of a semisimple Lie algebra, the Demazure character formula is an extension of the Weyl character formula to these modules. Demazure's work in this area was marred by a dependence on a false lemma in an earlier paper.
In his career, Demazure's research emphasis shifted from pure mathematics to more computational problems, involving the application of algebraic geometry to image reconstruction problems in computer vision. The Kruppa–Demazure theorem, stemming from this work, shows that if a scene consisting of five points is viewed from two cameras with unknown positions but known focal lengths in general, there will be ten different scenes that could have generated the same two images. Austrian mathematician Erwin Kruppa had many years earlier narrowed the number of possible scenes to eleven, Demazure provided the first complete solution to the problem. Schémas en groupes. I: Propriétés générales des schémas en groupes. Lecture Notes in Mathematics 151, Berlin: Springer-Verlag, 1970. MR0274458. Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux. Lecture Notes in Mathematics 152, Berlin: Springer-Verlag, 1970. MR0274459. Schémas en groupes. III: Structure des schémas en groupes réductifs.
Lecture Notes in Mathematics 153, Berlin: Springer-Verlag, 1970. MR0274460. Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. Masson, Amsterdam: North Holland, 1970. MR0302656. Translated into English by J. Bell as Introduction to Algebraic Geometry and Algebraic Groups, Volume 39 of North-Holland Mathematics Studies, Elsevier, 1980, MR0563524. Lectures on p-divisible groups. Lecture Notes in Mathematics 302, Berlin: Springer-Verlag, 1972, 1986, ISBN 3-540-06092-8. MR0344261, MR0883960. Bifurcations and catastrophes: Geometry of solutions to nonlinear problems. Universitext, Berlin: Springer-Verlag, 2000. Translated from the French by David Chillingworth. MR1739190. Cours d'Algèbre: Primalité. Divisibilité. Codes. Paris: Cassini, 1997, 2008. MR1466448
Serge Lang was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the influential Algebra, he was a member of the Bourbaki group. As an activist, he campaigned against the nomination of the political scientist Samuel P. Huntington to the National Academies of Science, descended into AIDS denialism, claiming that HIV had not been proven to cause AIDS and protesting Yale's research into HIV/AIDS. Lang was born in Saint-Germain-en-Laye, close to Paris, in 1927, he had a twin brother who became a sister who became an actress. Lang moved with his family to California as a teenager, where he graduated in 1943 from Beverly Hills High School, he subsequently graduated from the California Institute of Technology in 1946, received a doctorate from Princeton University in 1951. He held faculty positions at the University of Chicago, Columbia University, Yale University. Lang studied under Emil Artin at Princeton University, writing his thesis on quasi-algebraic closure, worked on the geometric analogues of class field theory and diophantine geometry.
He moved into diophantine approximation and transcendental number theory, proving the Schneider–Lang theorem. A break in research while he was involved in trying to meet 1960s student activism halfway caused him difficulties in picking up the threads afterwards, he wrote on modular forms and modular units, the idea of a'distribution' on a profinite group, value distribution theory. He made a number of conjectures in diophantine geometry: Mordell–Lang conjecture, Bombieri–Lang conjecture, Lang–Trotter conjecture, the Lang conjecture on analytically hyperbolic varieties, he introduced the Lang map, the Katz–Lang finiteness theorem, the Lang–Steinberg theorem in algebraic groups. Lang was a prolific writer of mathematical texts completing one on his summer vacation. Most are at the graduate level, he wrote calculus texts and prepared a book on group cohomology for Bourbaki. Lang's Algebra, a graduate-level introduction to abstract algebra, was a influential text that ran through numerous updated editions.
His Steele prize citation stated, "Lang's Algebra changed the way graduate algebra is taught... It has affected all subsequent graduate-level algebra books." It contained ideas of Artin. Lang was noted for his eagerness for contact with students, he was described as a passionate teacher who would throw chalk at students who he believed were not paying attention. One of his colleagues recalled: "He would rave in front of his students, he would say,'Our two aims are truth and clarity, to achieve these I will shout in class.'" He won a Leroy P. Steele Prize for Mathematical Exposition from the American Mathematical Society. In 1960, he won the sixth Frank Nelson Cole Prize in Algebra for his paper Unramified class field theory over function fields in several variables. Lang spent much of his professional time engaged in political activism, he was a staunch socialist and active in opposition to the Vietnam War, volunteering for the 1966 anti-war campaign of Robert Scheer. Lang quit his position at Columbia in 1971 in protest over the university's treatment of anti-war protesters.
Lang engaged in several efforts to challenge anyone he believed was spreading misinformation or misusing science or mathematics to further their own goals. He attacked the 1977 Survey of the American Professoriate, an opinion questionnaire that Seymour Martin Lipset and E. C. Ladd had sent to thousands of college professors in the United States, accusing it of containing numerous biased and loaded questions; this led to a public and acrimonious conflict. In 1986, Lang mounted what the New York Times described as a "one-man challenge" against the nomination of political scientist Samuel P. Huntington to the National Academy of Sciences. Lang described Huntington's research, in particular his use of mathematical equations to demonstrate that South Africa was a "satisfied society", as "pseudoscience", arguing that it gave "the illusion of science without any of its substance." Despite support for Huntington from the Academy's social and behavioral scientists, Lang's challenge was successful, Huntington was twice rejected for Academy membership.
Huntington's supporters argued that Lang's opposition was political rather than scientific in nature. Lang kept his political correspondence and related documentation in extensive "files", he would send letters or publish articles, wait for responses, engage the writers in further correspondence, collect all these writings together and point out what he considered contradictions. He mailed these files to people he considered important, his extensive file criticizing Nobel laureate David Baltimore was published in the journal Ethics and Behaviour in January 1993. Lang fought the decision by Yale University to hire Daniel Kevles, a historian of science, because Lang disagreed with Kevles' analysis in The Baltimore Case. Lang's most controversial political stance was as an AIDS denialist.
In mathematics, a differentiable manifold is a type of manifold, locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts known as an atlas. One may apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their composition on chart intersections in the atlas must be differentiable functions on the corresponding linear space. In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas.
The maps that relate the coordinates defined by the various charts to one another are called transition maps. Differentiability means different things in different contexts including: continuously differentiable, k times differentiable and holomorphic. Furthermore, the ability to induce such a differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A differential structure allows one to define the globally differentiable tangent space, differentiable functions, differentiable tensor and vector fields. Differentiable manifolds are important in physics. Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics, general relativity, Yang–Mills theory, it is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the exterior calculus; the study of calculus on differentiable manifolds is known as differential geometry.
The emergence of differential geometry as a distinct discipline is credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture before the faculty at Göttingen, he motivated the idea of a manifold by an intuitive process of varying a given object in a new direction, presciently described the role of coordinate systems and charts in subsequent formal developments: Having constructed the notion of a manifoldness of n dimensions, found that its true character consists in the property that the determination of position in it may be reduced to n determinations of magnitude... – B. RiemannThe works of physicists such as James Clerk Maxwell, mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita led to the development of tensor analysis and the notion of covariance, which identifies an intrinsic geometric property as one, invariant with respect to coordinate transformations; these ideas found a key application in Einstein's theory of general relativity and its underlying equivalence principle.
A modern definition of a 2-dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surfaces. The accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney. A presentation of a topological manifold is a second countable Hausdorff space, locally homeomorphic to a vector space, by a collection of homeomorphisms called charts; the composition of one chart with the inverse of another chart is a function called a transition map, defines a homeomorphism of an open subset of the linear space onto another open subset of the linear space. This formalizes the notion of "patching together pieces of a space to make a manifold" – the manifold produced contains the data of how it has been patched together. However, different atlases may produce "the same" manifold. Thus, one defines a topological manifold to be a space as above with an equivalence class of atlases, where one defines equivalence of atlases below. There are a number of different types of differentiable manifolds, depending on the precise differentiability requirements on the transition functions.
Some common examples include the following: A differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition maps are all differentiable. More a Ck-manifold is a topological manifold with an atlas whose transition maps are all k-times continuously differentiable. A smooth manifold or C∞-manifold is a differentiable manifold for which all the transition maps are smooth; that is, derivatives of all orders exist. An equivalence class of such atlases is said to be a smooth structure. An analytic manifold, or Cω-manifold is a smooth manifold with the additional condition that each transition map is analytic: the Taylor expansion is convergent and equals the function on some open ball. A complex manifold is a topological space modeled on a Euclidean space over the complex field and for which all the transition maps are holomorphic. While there is a meaningful notion of a Ck atlas, there is no distinct notion of a Ck manifold other than C0 and C∞, because for every Ck-structure with k > 0, there is a unique Ck-equivalent C∞-structure – a result of Whitney.
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Alexander Grothendieck was a 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 is considered by many to be the greatest mathematician of the 20th century. Born in Germany, Grothendieck was raised and lived in France. For much of his working life, however, he was, in effect, stateless; as he spelled his first name "Alexander" rather than "Alexandre" and his surname, taken from his mother, was the Dutch-like Low German "Grothendieck", he was sometimes mistakenly believed to be of Dutch origin. 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 and remained there until 1970, driven by personal and political convictions, he left following a dispute over military funding.
He 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 causes. Soon after his formal retirement in 1988, he moved to the Pyrenees, where he lived in isolation until his death in 2014. Grothendieck was born in Berlin to anarchist parents, his father, Alexander "Sascha" Schapiro, had Hasidic Jewish roots and had been imprisoned in Russia before moving to Germany in 1922, while his mother, Johanna "Hanka" Grothendieck, came from a Protestant family in Hamburg and worked as a journalist. Both had broken away from their early backgrounds in their teens. At the time of his birth, Grothendieck's mother was married to the journalist Johannes Raddatz and his birthname was recorded as "Alexander Raddatz." The marriage was dissolved in 1929 and Schapiro/Tanaroff acknowledged his paternity, but never married Hanka. Grothendieck lived with his parents in Berlin until the end of 1933, when his father moved to Paris to evade Nazism, followed soon thereafter by his mother.
They left Grothendieck in the care of a Lutheran pastor and teacher in Hamburg. During this time, his parents took part in the Spanish Civil War, according to Winfried Scharlau, as non-combatant auxiliaries, though others state that Sascha fought in the anarchist militia. In May 1939, Grothendieck was put on a train in Hamburg for France. Shortly afterwards his father was interned in Le Vernet, he and his mother were interned in various camps from 1940 to 1942 as "undesirable dangerous foreigners". The first was the Rieucros Camp, where his mother contracted the tuberculosis which caused her death and where Alexander managed to attend the local school, at Mende. Once Alexander managed to escape from the camp, intending to assassinate Hitler, his mother Hanka was transferred to the Gurs internment camp for the remainder of World War II. Alexander was permitted to live, separated from his mother, in the village of Le Chambon-sur-Lignon and hidden in local boarding houses or pensions, though he had to seek refuge in the woods during Nazis raids, surviving at times without food or water for several days.
His father was arrested under the Vichy anti-Jewish legislation, sent to the Drancy, handed over by the French Vichy government to the Germans to be sent to be murdered at the Auschwitz concentration camp in 1942. In Chambon, Grothendieck attended the Collège Cévenol, a unique secondary school founded in 1938 by local Protestant pacifists and anti-war activists. Many of the refugee children hidden in Chambon attended Cévenol, it was at this school that Grothendieck first became fascinated with mathematics. After the war, the young Grothendieck studied mathematics in France at the University of Montpellier where he did not perform well, failing such classes as astronomy. Working on his own, he rediscovered the Lebesgue measure. After three years of independent studies there, he went to continue his studies in Paris in 1948. Grothendieck attended Henri Cartan's Seminar at École Normale Supérieure, but he lacked 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 he wrote his dissertation under Laurent Schwartz and Jean Dieudonné on functional analysis, from 1950 to 1953.
At this time he was a leading expert in the theory of topological vector spaces. From 1953 to 1955 he moved to the University of São Paulo in Brazil, where he immigrated by means of a Nansen passport, given that he refused to take French Nationality. By 1957, he set this subject aside in order to work in algebraic homological algebra; the same year he was invited to visit Harvard 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 position that, he was warned, might have landed him in prison. The prospect did not worry him. Comparing Grothendieck during his Nancy years to the École Normale Supérieure trained students at that time: Pierre Samuel, Roger Godement, René Thom, Jacques Dixmier, Jean Cerf, Yvonne Bruhat, Jean-Pierre Serre, Bernard Malgrange, Leila Schneps says: He was so unknown to this group and to their professors, came from such a deprived and chaotic background, was, compared to them, so ignorant at the start of his research career