École normale supérieure (Paris)
The École normale supérieure is one of the French grandes écoles and a school of PSL University since 2010. It was conceived during the French Revolution and was intended to provide the Republic with a new body of professors, trained in the critical spirit and secular values of the Enlightenment, it has since developed into an institution which has become a platform for a select few of France's students to pursue careers in government and academia. Founded in 1794 and reorganised by Napoleon, ENS has two main sections and a competitive selection process consisting of written and oral examinations. During their studies, ENS students hold the status of paid civil servants; the principal goal of ENS is the training of professors and public administrators. Among its alumni there are 13 Nobel Prize laureates including 8 in Physics, 12 Fields Medalists, more than half the recipients of the CNRS's Gold Medal, several hundred members of the Institut de France, scores of politicians and statesmen; the school has achieved particular recognition in the fields of mathematics and physics as one of France's foremost scientific training grounds, along with notability in the human sciences as the spiritual birthplace of authors such as Julien Gracq, Jean Giraudoux, Assia Djebar, Charles Péguy, philosophers such as Henri Bergson, Jean-Paul Sartre, Louis Althusser, Simone Weil, Maurice Merleau-Ponty and Alain Badiou, social scientists such as Émile Durkheim, Raymond Aron, Pierre Bourdieu, "French theorists" such as Michel Foucault and Jacques Derrida.
The school's students are referred to as normaliens. The ENS is a grande école and, as such, is not part of the mainstream university system, although it maintains extensive connections with it; the vast majority of the academic staff hosted at ENS belong to external academic institutions such as the CNRS, the EHESS and the University of Paris. This mechanism for constant scientific turnover allows ENS to benefit from a continuous stream of researchers in all fields. ENS full professorships are competitive. Generalistic in its recruitment and organisation, the ENS is the only grande école in France to have departments of research in all the natural and human sciences, its status as one of the foremost centres of French research has led to its model being replicated elsewhere, in France, in Italy, in Romania, in China and in former French colonies such as Morocco, Mali and Cameroon. The current institution finds its roots in the creation of the Ecole normale de l'an III by the post-revolutionary National Convention led by Robespierre in 1794.
The school was created based on a recommendation by Joseph Lakanal and Dominique-Joseph Garat, who were part of the commission on public education. The Ecole normale was intended as the core of a planned centralised national education system; the project was conceived as a way to reestablish trust between the Republic and the country's elites, alienated to some degree by the Reign of Terror. The decree establishing the school, issued on 9 brumaire, states in its first article that "There will be established in Paris an Ecole normale, from all the parts of the Republic, citizens educated in the useful sciences shall be called upon to learn, from the best professors in all the disciplines, the art of teaching." The inaugural course was given on 20 January 1795 and the last on 19 May of the same year at the Museum of Natural History. The goal of these courses was to train a body of teachers for all the secondary schools in the country and thereby to ensure a homogenous education for all; these courses covered all the existing sciences and humanities and were given by scholars such as: scientists Monge, Daubenton and philosophers Bernardin de Saint-Pierre and Volney were some of the teachers.
The school was closed as a result of the arrival of the Consulate but this Ecole normale was to serve as a basis when the school was founded for the second time by Napoleon I in 1808. On 17 March 1808, Napoleon created by decree a pensionnat normal within the imperial University of France charged with "training in the art of teaching the sciences and the humanities"; the establishment was opened in its strict code including a mandatory uniform. By a sister establishment had been created by Napoleon in Pisa under the name of Scuola normale superiore, which continues to exist today and still has close ties to the Paris school. Up to 1818, the students are handpicked by the academy inspectors based on their results in the secondary school. However, the "pensionnat" created by Napoleon came to be perceived under the Restoration as a nexus of liberal thought and was suppressed by then-minister of public instruction Denis-Luc Frayssinous in 1824. An École préparatoire was created on 9 March 1826 at the site of collège Louis-le-Grand.
This date can be taken as the definitive date of creation of the current school. After the July Revolution, the school regained its original name of École normale and in 1845 was renamed École normale supérieure. During the 1830s, under the direction of philosopher Victor Cousin, the school enhanced its status as an institution to prepare the agrégation by expanding the duration of study to three years, was divided into its present-day "
Kunihiko Kodaira was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954. Kodaira was born in Tokyo, he graduated from the University of Tokyo in 1938 with a degree in mathematics and graduated from the physics department at the University of Tokyo in 1941. During the war years he worked in isolation, but was able to master Hodge theory as it stood, he obtained his Ph. D. from the University of Tokyo in 1949, with a thesis entitled Harmonic fields in Riemannian manifolds. He was involved in cryptographic work from about 1944, while holding an academic post in Tokyo. In 1949 he travelled to the Institute for Advanced Study in Princeton, New Jersey at the invitation of Hermann Weyl, he was subsequently appointed Associate Professor at Princeton University in 1952 and promoted to Professor in 1955. At this time the foundations of Hodge theory were being brought in line with contemporary technique in operator theory.
Kodaira became involved in exploiting the tools it opened up in algebraic geometry, adding sheaf theory as it became available. This work was influential, for example on Friedrich Hirzebruch. In a second research phase, Kodaira wrote a long series of papers in collaboration with Donald C. Spencer, founding the deformation theory of complex structures on manifolds; this gave the possibility of constructions of moduli spaces, since in general such structures depend continuously on parameters. It identified the sheaf cohomology groups, for the sheaf associated with the holomorphic tangent bundle, that carried the basic data about the dimension of the moduli space, obstructions to deformations; this theory is still foundational, had an influence on the scheme theory of Grothendieck. Spencer continued this work, applying the techniques to structures other than complex ones, such as G-structures. In a third major part of his work, Kodaira worked again from around 1960 through the classification of algebraic surfaces from the point of view of birational geometry of complex manifolds.
This resulted in a typology of seven kinds of two-dimensional compact complex manifolds, recovering the five algebraic types known classically. He provided detailed studies of elliptic fibrations of surfaces over a curve, or in other language elliptic curves over algebraic function fields, a theory whose arithmetic analogue proved important soon afterwards; this work included a characterisation of K3 surfaces as deformations of quartic surfaces in P4, the theorem that they form a single diffeomorphism class. Again, this work has proved foundational.. Kodaira left Princeton University and the Institute for Advanced Study in 1961, served as chair at the Johns Hopkins University and Stanford University. In 1967, returned to the University of Tokyo, he was awarded a Wolf Prize in 1984/5. He died in Kofu on 26 July 1997. Morrow, James. J. ISBN 978-0-691-08158-8, MR 0366598 Kodaira, Baily, Walter L. ed. Kunihiko Kodaira: collected works, II, Iwanami Shoten, Tokyo. J. ISBN 978-0-691-08163-2, MR 0366599 Kodaira, Baily, Walter L. ed. Kunihiko Kodaira: collected works, III, Iwanami Shoten, Tokyo.
J. ISBN 978-0-691-08164-9, MR 0366600 Kodaira, Complex manifolds and deformation of complex structures, Classics in Mathematics, New York: Springer-Verlag, ISBN 978-3-540-22614-7, MR 0815922, review by Andrew J. Sommese Kodaira, Complex analysis, Cambridge Studies in Advanced Mathematics, 107, Cambridge University Press, ISBN 978-0-521-80937-5, MR 2343868 Bochner–Kodaira–Nakano identity Spectral theory of ordinary differential equations Kodaira vanishing theorem Kodaira–Spencer mapping Kodaira dimension Kodaira embedding theorem Enriques–Kodaira classification Kodaira's classification of singular fibers O'Connor, John J.. Donald C. Spencer, "Kunihiko Kodaira", Notices of the AMS, 45: 388–389. Friedrich Hirzebruch, "Kunihiko Kodaira: Mathematician and Teacher", Notices of the American Mathematical Society, 45: 1456–1462. "Special Issue to Honor Professor Kunihiko Kodaira on his 85th birthday", Asian Journal of Mathematics, 4, 2000
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, measure, infinite series, analytic functions. These theories are studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary techniques of analysis. Analysis may be distinguished from geometry. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids; the explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century.
In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would be called Cavalieri's principle to find the volume of a sphere in the 5th century; the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine and arctangent. Alongside his development of the Taylor series of the trigonometric functions, he estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series, his followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century. The modern foundations of mathematical analysis were established in 17th century Europe. Descartes and Fermat independently developed analytic geometry, a few decades Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations and partial differential equations, Fourier analysis, generating functions.
During this period, calculus techniques were applied to approximate discrete problems by continuous ones. In the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra used in earlier work by Euler. Instead, Cauchy formulated calculus in terms of geometric infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y, he introduced the concept of the Cauchy sequence, started the formal theory of complex analysis. Poisson, Liouville and others studied partial differential equations and harmonic analysis; the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis.
In the middle of the 19th century Riemann introduced his theory of integration. The last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, introduced the "epsilon-delta" definition of limit. Mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions. "monsters" began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, Baire proved the Baire category theorem.
In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue solved the problem of measure, Hilbert introduced Hilbert spaces to solve integral equations; the idea of normed vector space was in the air, in the 1920s Banach created functional analysis. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. Much of analysis happens in some metric space. Examples of analysis without a metric include functional analysis. Formally, a metric space is an ordered pair where M is a set
Vladimir Alexandrovich Voevodsky was a Russian-American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002, he is known for the proof of the Milnor conjecture and motivic Bloch-Kato conjectures and for the univalent foundations of mathematics and homotopy type theory. Vladimir Voevodsky's father, Aleksander Voevodsky, was head of the Laboratory of High Energy Leptons in the Institute for Nuclear Research at the Russian Academy of Sciences, his mother Tatyana was a chemist. Voevodsky attended Moscow State University for a while, but was forced to leave without a diploma for refusing to attend classes and failing academically, he received his Ph. D. in mathematics from Harvard University in 1992 after being recommended without applying, following several independent publications. While he was a first year undergraduate, he was given a copy of Esquisse d'un Programme by his advisor George Shabat.
He learned the French language "with the sole purpose of being able to read this text" and started his research on some of the themes mentioned there. Voevodsky's work was in the intersection of algebraic geometry with algebraic topology. Along with Fabien Morel, Voevodsky introduced a homotopy theory for schemes, he formulated what is now believed to be the correct form of motivic cohomology, used this new tool to prove Milnor's conjecture relating the Milnor K-theory of a field to its étale cohomology. For the above, he received the Fields Medal at the 24th International Congress of Mathematicians held in Beijing, China. In 1998 he gave a plenary lecture at the International Congress of Mathematicians in Berlin, he coauthored Cycles and Motivic Homology Theories, which develops the theory of motivic cohomology in some detail. From 2002, Voevodsky was a professor at the Institute for Advanced Study in New Jersey. In January 2009, at an anniversary conference in honor of Alexander Grothendieck, held at the Institut des Hautes Études Scientifiques, Voevodsky announced a proof of the full Bloch-Kato conjectures.
In 2009, he constructed the univalent model of Martin-Löf type theory in simplicial sets. This led to important advances in type theory and in the development of new Univalent foundations of mathematics that Voevodsky worked on in his final years, he worked on a Coq library UniMath using univalent ideas. In April 2016, the University of Gothenburg awarded an honorary doctorate to Voevodsky. Voevodsky died on 30 September 2017 at his home from an aneurysm, he is survived by Natalia Dalia Shalaby. Voevodsky, Suslin and Friedlander, Eric M.. Cycles and motivic homology theories. Annals of Mathematics Studies Vol. 143. Princeton University Press. Mazza, Voevodsky and Weibel, Charles A. Lecture notes on motivic cohomology. Clay Mathematics Monographs, Vol. 2. American Mathematical Soc. 2011 Friedlander Eric M. Rapoport Michael, Suslin Andrei. "The mathematical work of the 2002 Fields medalists". Notices Amer. Math. Soc. 50: 212–217. CS1 maint: Multiple names: authors list More information about his work can be found on his website Vladimir Voevodsky on GitHub Contains the slides of many of his recent lectures.
По большому филдсовскому счету Интервью с Владимиром Воеводским и Лораном Лаффоргом Julie Rehmeyer, Vladimir Voevodsky, Revolutionary Mathematician, Dies at 51, New York Times, 6 October 2017 O'Connor, John J.. Vladimir Voevodsky at the Mathematics Genealogy Project
France the French Republic, is a country whose territory consists of metropolitan France in Western Europe and several overseas regions and territories. The metropolitan area of France extends from the Mediterranean Sea to the English Channel and the North Sea, from the Rhine to the Atlantic Ocean, it is bordered by Belgium and Germany to the northeast and Italy to the east, Andorra and Spain to the south. The overseas territories include French Guiana in South America and several islands in the Atlantic and Indian oceans; the country's 18 integral regions span a combined area of 643,801 square kilometres and a total population of 67.3 million. France, a sovereign state, is a unitary semi-presidential republic with its capital in Paris, the country's largest city and main cultural and commercial centre. Other major urban areas include Lyon, Toulouse, Bordeaux and Nice. During the Iron Age, what is now metropolitan France was inhabited by a Celtic people. Rome annexed the area in 51 BC, holding it until the arrival of Germanic Franks in 476, who formed the Kingdom of Francia.
The Treaty of Verdun of 843 partitioned Francia into Middle Francia and West Francia. West Francia which became the Kingdom of France in 987 emerged as a major European power in the Late Middle Ages following its victory in the Hundred Years' War. During the Renaissance, French culture flourished and a global colonial empire was established, which by the 20th century would become the second largest in the world; the 16th century was dominated by religious civil wars between Protestants. France became Europe's dominant cultural and military power in the 17th century under Louis XIV. In the late 18th century, the French Revolution overthrew the absolute monarchy, established one of modern history's earliest republics, saw the drafting of the Declaration of the Rights of Man and of the Citizen, which expresses the nation's ideals to this day. In the 19th century, Napoleon established the First French Empire, his subsequent Napoleonic Wars shaped the course of continental Europe. Following the collapse of the Empire, France endured a tumultuous succession of governments culminating with the establishment of the French Third Republic in 1870.
France was a major participant in World War I, from which it emerged victorious, was one of the Allies in World War II, but came under occupation by the Axis powers in 1940. Following liberation in 1944, a Fourth Republic was established and dissolved in the course of the Algerian War; the Fifth Republic, led by Charles de Gaulle, remains today. Algeria and nearly all the other colonies became independent in the 1960s and retained close economic and military connections with France. France has long been a global centre of art and philosophy, it hosts the world's fourth-largest number of UNESCO World Heritage Sites and is the leading tourist destination, receiving around 83 million foreign visitors annually. France is a developed country with the world's sixth-largest economy by nominal GDP, tenth-largest by purchasing power parity. In terms of aggregate household wealth, it ranks fourth in the world. France performs well in international rankings of education, health care, life expectancy, human development.
France is considered a great power in global affairs, being one of the five permanent members of the United Nations Security Council with the power to veto and an official nuclear-weapon state. It is a leading member state of the European Union and the Eurozone, a member of the Group of 7, North Atlantic Treaty Organization, Organisation for Economic Co-operation and Development, the World Trade Organization, La Francophonie. Applied to the whole Frankish Empire, the name "France" comes from the Latin "Francia", or "country of the Franks". Modern France is still named today "Francia" in Italian and Spanish, "Frankreich" in German and "Frankrijk" in Dutch, all of which have more or less the same historical meaning. There are various theories as to the origin of the name Frank. Following the precedents of Edward Gibbon and Jacob Grimm, the name of the Franks has been linked with the word frank in English, it has been suggested that the meaning of "free" was adopted because, after the conquest of Gaul, only Franks were free of taxation.
Another theory is that it is derived from the Proto-Germanic word frankon, which translates as javelin or lance as the throwing axe of the Franks was known as a francisca. However, it has been determined that these weapons were named because of their use by the Franks, not the other way around; the oldest traces of human life in what is now France date from 1.8 million years ago. Over the ensuing millennia, Humans were confronted by a harsh and variable climate, marked by several glacial eras. Early hominids led a nomadic hunter-gatherer life. France has a large number of decorated caves from the upper Palaeolithic era, including one of the most famous and best preserved, Lascaux. At the end of the last glacial period, the climate became milder. After strong demographic and agricultural development between the 4th and 3rd millennia, metallurgy appeared at the end of the 3rd millennium working gold and bronze, iron. France has numerous megalithic sites from the Neolithic period, including the exceptiona
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques from commutative algebra, for solving geometrical problems about these sets of zeros; the fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, parabolas, hyperbolas, cubic curves like elliptic curves, quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis and number theory. A study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, it becomes more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution. In the 20th century, algebraic geometry split into several subareas; the mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more to the points with coordinates in an algebraically closed field. Real algebraic geometry is the study of the real points of an algebraic variety. Diophantine geometry and, more arithmetic geometry is the study of the points of an algebraic variety with coordinates in fields that are not algebraically closed and occur in algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, p-adic fields.
A large part of singularity theory is devoted to the singularities of algebraic varieties. Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers, it consists of algorithm design and software development for the study of properties of explicitly given algebraic varieties. Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way, similar to its use in the study of differential and analytic manifolds; this is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring.
This means that a point of such a scheme may be either a subvariety. This approach enables a unification of the language and the tools of classical algebraic geometry concerned with complex points, of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach. In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R3 could be defined as the set of all points with x 2 + y 2 + z 2 − 1 = 0. A "slanted" circle in R3 can be defined as the set of all points which satisfy the two polynomial equations x 2 + y 2 + z 2 − 1 = 0, x + y + z = 0. First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed.
We consider the affine space of dimension n over denoted An. When one fixes a coordinate system, one may identify An with kn; the purpose of not working with kn is to emphasize that one "forgets" the vector space structure that kn carries. A function f: An → A1 is said to be polynomial if it can be written as a polynomial, that is, if there is a polynomial p in k such that f = p for every point M with coordinates in An; the property of a function to be polynomial does not depend on the choice of a coordinate system in An. When a coordinate system is chosen, the regular functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k. Therefore, the set of the