André Weil was an influential French mathematician of the 20th century, known for his foundational work in number theory and algebraic geometry. He was the de facto early leader of the mathematical Bourbaki group; the philosopher Simone Weil was his sister. André Weil was born in Paris to agnostic Alsatian Jewish parents who fled the annexation of Alsace-Lorraine by the German Empire after the Franco-Prussian War in 1870–71; the famous philosopher Simone Weil was Weil's only sibling. He studied in Paris, Rome and Göttingen and received his doctorate in 1928. While in Germany, Weil befriended Carl Ludwig Siegel. Starting in 1930, he spent two academic years at Aligarh Muslim University. Aside from mathematics, Weil held lifelong interests in classical Greek and Latin literature, in Hinduism and Sanskrit literature: he taught himself Sanskrit in 1920. After teaching for one year in Aix-Marseille University, he taught for six years in Strasbourg, he married Éveline in 1937. Weil was in Finland, his wife Éveline returned to France without him.
Weil was mistakenly arrested in Finland at the outbreak of the Winter War on suspicion of spying. Weil returned to France via Sweden and the United Kingdom, was detained at Le Havre in January 1940, he was charged with failure to report for duty, was imprisoned in Le Havre and Rouen. It was in the military prison in Bonne-Nouvelle, a district of Rouen, from February to May, that Weil completed the work that made his reputation, he was tried on 3 May 1940. Sentenced to five years, he requested to be attached to a military unit instead, was given the chance to join a regiment in Cherbourg. After the fall of France, he met up with his family in Marseille, he went to Clermont-Ferrand, where he managed to join his wife Éveline, living in German-occupied France. In January 1941, Weil and his family sailed from Marseille to New York, he spent the remainder of the war in the United States, where he was supported by the Rockefeller Foundation and the Guggenheim Foundation. For two years, he taught undergraduate mathematics at Lehigh University, where he was unappreciated and poorly paid, although he didn't have to worry about being drafted, unlike his American students.
But, he hated Lehigh much for their heavy teaching workload and he swore that he would never talk about "Lehigh" any more. He quit the job at Lehigh, he moved to Brazil and taught at the Universidade de São Paulo from 1945 to 1947, where he worked with Oscar Zariski, he returned to the United States and taught at the University of Chicago from 1947 to 1958, before moving to the Institute for Advanced Study, where he would spend the remainder of his career. He was a Plenary Speaker at the ICM in 1950 in Cambridge, Massachusetts, in 1954 in Amsterdam, in 1978 in Helsinki. In 1979, Weil shared the second Wolf Prize in Mathematics with Jean Leray. Weil made substantial contributions in a number of areas, the most important being his discovery of profound connections between algebraic geometry and number theory; this began in his doctoral work leading to the Mordell–Weil theorem. Mordell's theorem had an ad hoc proof. Both aspects of Weil's work have developed into substantial theories. Among his major accomplishments were the 1940s proof of the Riemann hypothesis for zeta-functions of curves over finite fields, his subsequent laying of proper foundations for algebraic geometry to support that result.
The so-called Weil conjectures were hugely influential from around 1950. Weil introduced the adele ring in the late 1930s, following Claude Chevalley's lead with the ideles, gave a proof of the Riemann–Roch theorem with them. His'matrix divisor' Riemann–Roch theorem from 1938 was a early anticipation of ideas such as moduli spaces of bundles; the Weil conjecture on Tamagawa numbers proved resistant for many years. The adelic approach became basic in automorphic representation theory, he picked up another credited Weil conjecture, around 1967, which under pressure from Serge Lang became known as the Taniyama–Shimura conjecture based on a formulated question of Taniyama at the 1955 Nikkō conference. His attitude towards conjectures was that one should not dignify a guess as a conjecture and in the Taniyama case, the evidence was only there after extensive computational work carried out from the late 1960s. Other significant results were on Pontryagin differential geometry, he introduced the concept of a uniform space in general topology, as a by-product of his collaboration with Nicolas Bourbaki.
His work on sheaf theory hardly appears in his published papers, but correspondence with Henri Cartan in the late 1940s, reprinted in his collected papers, proved most influential. He created the ∅, he discovered that the so-called Weil representation introduced in quantum mechanics by Irving Segal an
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
In algebraic topology, simplicial homology formalizes the idea of the number of holes of a given dimension in a simplicial complex. This generalizes the number of connected components. Simplicial homology arose as a way to study topological spaces whose building blocks are n-simplices, the n-dimensional analogs of triangles; this includes a line segment, a triangle and a tetrahedron. By definition, such a space is homeomorphic to a simplicial complex; such a homeomorphism is referred to as a triangulation of the given space. Many topological spaces of interest can be triangulated, including every smooth manifold. Simplicial homology is defined by a simple recipe for any abstract simplicial complex, it is a remarkable fact. As a result, it gives a computable way to distinguish one space from another. Singular homology is a related theory, better adapted to theory rather than computation. Singular homology is defined for all topological spaces and depends only on the topology, not any triangulation.
Nonetheless, because it is possible to compute the simplicial homology of a simplicial complex automatically and efficiently, simplicial homology has become important for application to real-life situations, such as image analysis, medical imaging, data analysis in general. A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as, with the rule that two orderings define the same orientation if and only if they differ by an permutation, thus every simplex has two orientations, switching the order of two vertices changes an orientation to the opposite orientation. For example, choosing an orientation of a 1-simplex amounts to choosing one of the two possible directions, choosing an orientation of a 2-simplex amounts to choosing what "counterclockwise" should mean. Let S be a simplicial complex. A simplicial k-chain is a finite formal sum ∑ i = 1 N c i σ i, where each ci is an integer and σi is an oriented k-simplex.
In this definition, we declare that each oriented simplex is equal to the negative of the simplex with the opposite orientation. For example, = −; the group of k-chains on S is written Ck. This is a free abelian group which has a basis in one-to-one correspondence with the set of k-simplices in S. To define a basis explicitly, one has to choose an orientation of each simplex. One standard way to do this is to choose an ordering of all the vertices and give each simplex the orientation corresponding to the induced ordering of its vertices. Let σ = be an oriented k-simplex, viewed as a basis element of Ck; the boundary operator ∂ k: C k → C k − 1 is the homomorphism defined by: ∂ k = ∑ i = 0 k i, where the oriented simplex is the ith face of σ, obtained by deleting its ith vertex. In Ck, elements of the subgroup Z k = ker ∂ k are referred to as cycles, the subgroup B k = im ∂ k + 1 is said to consist of boundaries. A direct computation shows that ∂2 = 0. In geometric terms, this says. Equivalently, the abelian groups form a chain complex.
Another equivalent statement is. The kth homology group Hk of S is defined to be the quotient abelian group H k = Z k / B k, it follows that the homology group Hk is nonzero when there are k-cycles on S which are not boundaries. In a sense, this means. For example, consider the complex S obtained by gluing two triangles along one edge, shown in the image; the edges of each triangle can be oriented so as to form a cycle
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within society at large; the press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton, its first book was a new 1912 edition of John Witherspoon's Lectures on Moral Philosophy. Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two existing local publishers, that of the Princeton Alumni Weekly and the Princeton Press; the new press printed both local newspapers, university documents, The Daily Princetonian, added book publishing to its activities. Beginning as a small, for-profit printer, Princeton University Press was reincorporated as a nonprofit in 1910.
Since 1911, the press has been headquartered in a purpose-built gothic-style building designed by Ernest Flagg. The design of press’s building, named the Scribner Building in 1965, was inspired by the Plantin-Moretus Museum, a printing museum in Antwerp, Belgium. Princeton University Press established a European office, in Woodstock, north of Oxford, in 1999, opened an additional office, in Beijing, in early 2017. Six books from Princeton University Press have won Pulitzer Prizes: Russia Leaves the War by George F. Kennan Banks and Politics in America from the Revolution to the Civil War by Bray Hammond Between War and Peace by Herbert Feis Washington: Village and Capital by Constance McLaughlin Green The Greenback Era by Irwin Unger Machiavelli in Hell by Sebastian de Grazia Books from Princeton University Press have been awarded the Bancroft Prize, the Nautilus Book Award, the National Book Award. Multi-volume historical documents projects undertaken by the Press include: The Collected Papers of Albert Einstein The Writings of Henry D. Thoreau The Papers of Woodrow Wilson The Papers of Thomas Jefferson Kierkegaard's WritingsThe Papers of Woodrow Wilson has been called "one of the great editorial achievements in all history."
Princeton University Press's Bollingen Series had its beginnings in the Bollingen Foundation, a 1943 project of Paul Mellon's Old Dominion Foundation. From 1945, the foundation had independent status and providing fellowships and grants in several areas of study, including archaeology and psychology; the Bollingen Series was given to the university in 1969. Annals of Mathematics Studies Princeton Series in Astrophysics Princeton Series in Complexity Princeton Series in Evolutionary Biology Princeton Series in International Economics Princeton Modern Greek Studies The Whites of Their Eyes: The Tea Party's Revolution and the Battle over American History, by Jill Lepore The Meaning of Relativity by Albert Einstein Atomic Energy for Military Purposes by Henry DeWolf Smyth How to Solve It by George Polya The Open Society and Its Enemies by Karl Popper The Hero With a Thousand Faces by Joseph Campbell The Wilhelm/Baynes translation of the I Ching, Bollingen Series XIX. First copyright 1950, 27th printing 1997.
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