James Joseph Sylvester
James Joseph Sylvester FRS HFRSE LLD was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, combinatorics, he played a leadership role in American mathematics in the half of the 19th century as a professor at the Johns Hopkins University and as founder of the American Journal of Mathematics. At his death, he was professor at Oxford. James Joseph was born in London on the son of Abraham Joseph, a merchant. James adopted the surname Sylvester when his older brother did so upon emigration to the United States—a country which at that time required all immigrants to have a given name, a middle name, a surname. At the age of 14, Sylvester was a student of Augustus De Morgan at the University of London, his family withdrew him from the University after he was accused of stabbing a fellow student with a knife. Subsequently, he attended the Liverpool Royal Institution. Sylvester began his study of mathematics at St John's College, Cambridge in 1831, where his tutor was John Hymers.
Although his studies were interrupted for two years due to a prolonged illness, he ranked second in Cambridge's famous mathematical examination, the tripos, for which he sat in 1837. However, Sylvester was not issued a degree, because graduates at that time were required to state their acceptance of the Thirty-Nine Articles of the Church of England, Sylvester could not do so because he was Jewish. For the same reason, he was unable to obtain a Smith's prize. In 1838, Sylvester became professor of natural philosophy at University College London and in 1839 a Fellow of the Royal Society of London. In 1841, he was awarded an MA by Trinity College, Dublin. In the same year he moved to the United States to become a professor of mathematics at the University of Virginia, but left after less than four months following a violent encounter with two students he had disciplined, he moved to New York City and began friendships with the Harvard mathematician Benjamin Peirce and the Princeton physicist Joseph Henry.
However, he left in November 1843 after being denied appointment as Professor of Mathematics at Columbia College, again for his Judaism, returned to England. On his return to England, he was hired in 1844 by the Equity and Law Life Assurance Society for which he developed successful actuarial models and served as de facto CEO, a position that required a law degree; as a result, he studied for the Bar, meeting a fellow British mathematician studying law, Arthur Cayley, with whom he made significant contributions to invariant theory and matrix theory during a long collaboration. He did not obtain a position teaching university mathematics until 1855, when he was appointed professor of mathematics at the Royal Military Academy, from which he retired in 1869, because the compulsory retirement age was 55; the Woolwich academy refused to pay Sylvester his full pension, only relented after a prolonged public controversy, during which Sylvester took his case to the letters page of The Times. One of Sylvester's lifelong passions was for poetry.
Following his early retirement, Sylvester published a book entitled The Laws of Verse in which he attempted to codify a set of laws for prosody in poetry. In 1872, he received his B. A. and M. A. from Cambridge, having been denied the degrees due to his being a Jew. In 1876 Sylvester again crossed the Atlantic Ocean to become the inaugural professor of mathematics at the new Johns Hopkins University in Baltimore, Maryland, his salary was $5,000. After negotiation, agreement was reached on a salary, not paid in gold. In 1878 he founded the American Journal of Mathematics; the only other mathematical journal in the US at that time was the Analyst, which became the Annals of Mathematics. In 1883, he returned to England to take up the Savilian Professor of Geometry at Oxford University, he held this chair until his death, although in 1892 the University appointed a deputy professor to the same chair. He was on the governing body of Abingdon School. Sylvester died in London on 15 March 1897, he is buried in Balls Pond Road Jewish Cemetery on Kingsbury Road in London.
Sylvester invented a great number of mathematical terms such as "matrix", "graph" and "discriminant". He coined the term "totient" for Euler's totient function φ, his collected scientific work fills four volumes. In 1880, the Royal Society of London awarded Sylvester the Copley Medal, its highest award for scientific achievement. In Discrete geometry he is remembered for a result on the orchard problem. Sylvester House, a portion of an undergraduate dormitory at Johns Hopkins University, is named in his honor. Several professorships there are named in his honor also. Sylvester, James Joseph, The Laws of Verse Or Principles of Versification Exemplified in Metrical Translations: together with an annotated reprint of the inaugural presidential address to the mathematical and physical section of the British Association at Exeter, London: Longmans, Green and Co, ISBN 978-1-177-91141-2 Sylvester, James Joseph, Henry Frederick, ed; the collected mathematical papers of James Joseph Sylvester, I, New York: AMS Chelsea Publishing, ISBN 978-0-8218-3654-5 Sylvester, James Jose
Hilbert's eighth problem
Hilbert's eighth problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns number theory, in particular the Riemann hypothesis, although it is concerned with the Goldbach Conjecture; the problem as stated asked for more work on the distribution of primes and generalizations of Riemann hypothesis to other rings where prime ideals take the place of primes. This problem has yet to be resolved. Hilbert calls for a solution to the Riemann hypothesis, which has long been regarded as the deepest open problem in mathematics. Given the solution, he calls for more thorough investigation into Riemann's zeta function and the prime number theorem, he calls for a solution to the Goldbach conjecture, as well as more general problems, such as finding infinitely many pairs of primes solving a fixed linear diophantine equation. He calls for mathematicians to generalize the ideas of the Riemann hypothesis to counting prime ideals in a number field. English translation of Hilbert's original address
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Hilbert's fourth problem
In mathematics, Hilbert's fourth problem in the 1900 Hilbert problems is a foundational question in geometry. In one statement derived from the original, it was to find geometries whose axioms are closest to those of Euclidean geometry if the ordering and incidence axioms are retained, the congruence axioms weakened, the equivalent of the parallel postulate omitted; the original statement of Hilbert has been judged too vague to admit a definitive answer. A solution was sought with the German mathematician Georg Hamel being the first who tried to solve the problem. A recognized solution for dimensions 2 and 3 was given by Armenian mathematician Rouben V. Ambartzumian. Hilbert discusses the existence of non-Euclidean geometry and non-Archimedean geometry, as well as the idea that a'straight line' is defined as the shortest path between two points, he mentions how congruence of triangles is necessary for Euclid's proof that a straight line in the plane is the shortest distance between two points.
He summarizes as follows: The theorem of the straight line as the shortest distance between two points and the equivalent theorem of Euclid about the sides of a triangle, play an important part not only in number theory but in the theory of surfaces and in the calculus of variations. For this reason, because I believe that the thorough investigation of the conditions for the validity of this theorem will throw a new light upon the idea of distance, as well as upon other elementary ideas, e. g. upon the idea of the plane, the possibility of its definition by means of the idea of the straight line, the construction and systematic treatment of the geometries here possible seem to me desirable. One popular interpretation of this problem is that it is asking for all metrics on convex portions of the plane where the geodesics are straight Euclidean lines; the solution of Hilbert's fourth problem in dimension 2 was obtained in 1976 by Rouben V. Ambartzumian in the framework of his theory of combinatorial integral geometry by application of measure continuation starting from "Buffonic" valuations in the space of lines in the plane.
An attempt was made by Ambartzumian to apply the same techniques starting from similar valuations that live in the space of planes in 3 dimensional Euclidean space. The paper puts forward the concept of wedge metrics and formulates some conditions for a wedge metric to generate a measure in the space of planes; the definition of a wedge metrics is based on certain tetrahedral inequalities of combinatorial nature. The latter inequalities replace the usual triangle inequality. A gnomonic map projection of the sphere displays all great circles as straight lines, resulting in any line segment on a gnomonic map showing the shortest route between the segment's two endpoints; this is achieved by casting surface points of the sphere onto a tangent plane, each landing where a ray from the center of the earth passes through the point on the surface and on to the plane. This projection allows one to give a spherical metric to the portion of the plane it maps onto. In geometry, the Klein disk model is a model of 2-dimensional hyperbolic geometry in which points are represented by the points in the interior of the unit disk and lines are represented by the chords, straight line segments with endpoints on the boundary circle.
Busemann, Herbert. "Problem IV. Desarguesian spaces". In Browder, Felix E. Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. XXVIII. American Mathematical Society. Pp. 131–141. ISBN 0-8218-1428-1. Zbl 0352.50010. Papadopoulos, Athanase. "Hilbert's fourth problem". Handbook of Hilbert geometry. IRMA Lectures in Mathematics and Theoretical Physics. 22. European Mathematical Society. Pp. 391–432. ISBN 978-3-03719-147-7
Algebraic varieties are the central objects of study in algebraic geometry. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.:58Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility; the concept of an algebraic variety is similar to that of an analytic manifold. An important difference is that an algebraic variety may have singular points, while a manifold cannot; the fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial in one variable with complex number coefficients is determined by the set of its roots in the complex plane.
Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory; this correspondence is a defining feature of algebraic geometry. An affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define quasi-projective varieties in a similar way; the most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, but Nagata gave an example of such a new variety in the 1950s. For an algebraically closed field K and a natural number n, let An be affine n-space over K; the polynomials f in the ring K can be viewed as K-valued functions on An by evaluating f at the points in An, i.e. by choosing values in K for each xi.
For each set S of polynomials in K, define the zero-locus Z to be the set of points in An on which the functions in S vanish, to say Z =. A subset V of An is called an affine algebraic set if V = Z for some S.:2 A nonempty affine algebraic set V is called irreducible if it cannot be written as the union of two proper algebraic subsets.:3 An irreducible affine algebraic set is called an affine variety.:3 Affine varieties can be given a natural topology by declaring the closed sets to be the affine algebraic sets. This topology is called the Zariski topology.:2Given a subset V of An, we define I to be the ideal of all polynomial functions vanishing on V: I =. For any affine algebraic set V, the coordinate ring or structure ring of V is the quotient of the polynomial ring by this ideal.:4 Let k be an algebraically closed field and let Pn be the projective n-space over k. Let f in k be a homogeneous polynomial of degree d, it is not well-defined to evaluate f on points in Pn in homogeneous coordinates.
However, because f is homogeneous, meaning that f = λd f , it does make sense to ask whether f vanishes at a point. For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in Pn on which the functions in S vanish: Z =. A subset V of Pn is called a projective algebraic set if V = Z for some S.:9 An irreducible projective algebraic set is called a projective variety.:10Projective varieties are equipped with the Zariski topology by declaring all algebraic sets to be closed. Given a subset V of Pn, let I be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.:10A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every affine variety is quasi-projective. Notice that the complement of an algebraic set in an affine variety is a quasi-projective variety. In classical algebraic geometry, a
Arthur Cayley was a British mathematician. He helped; as a child, Cayley enjoyed solving complex maths problems for amusement. He entered Trinity College, where he excelled in Greek, French and Italian, as well as mathematics, he worked as a lawyer for 14 years. He postulated the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial, verified it for matrices of order 2 and 3, he was the first to define the concept of a group in the modern way—as a set with a binary operation satisfying certain laws. When mathematicians spoke of "groups", they had meant permutation groups. Cayley tables and Cayley graphs as well. Arthur Cayley was born in Richmond, England, on 16 August 1821, his father, Henry Cayley, was a distant cousin of Sir George Cayley, the aeronautics engineer innovator, descended from an ancient Yorkshire family. He settled in Russia, as a merchant, his mother was daughter of William Doughty. According to some writers she was Russian, his brother was the linguist Charles Bagot Cayley.
Arthur spent his first eight years in Saint Petersburg. In 1829 his parents were settled permanently near London. Arthur was sent to a private school. At age 14 he was sent to King's College School; the school's master observed indications of mathematical genius and advised the father to educate his son not for his own business, as he had intended, but to enter the University of Cambridge. At the unusually early age of 17 Cayley began residence at Cambridge; the cause of the Analytical Society had now triumphed, the Cambridge Mathematical Journal had been instituted by Gregory and Robert Leslie Ellis. To this journal, at the age of twenty, Cayley contributed three papers, on subjects, suggested by reading the Mécanique analytique of Lagrange and some of the works of Laplace. Cayley's tutor at Cambridge was George Peacock and his private coach was William Hopkins, he finished his undergraduate course by winning the place of Senior Wrangler, the first Smith's prize. His next step was to take the M.
A. degree, win a Fellowship by competitive examination. He continued to reside at Cambridge University for four years; because of the limited tenure of his fellowship it was necessary to choose a profession. He made a specialty of conveyancing, it was while he was a pupil at the bar examination that he went to Dublin to hear Hamilton's lectures on quaternions. His friend J. J. Sylvester, his senior by five years at Cambridge, was an actuary, resident in London. During this period of his life, extending over fourteen years, Cayley produced between two and three hundred papers. At Cambridge University the ancient professorship of pure mathematics is denominated by the Lucasian, is the chair, occupied by Isaac Newton. Around 1860, certain funds bequeathed by Lady Sadleir to the University, having become useless for their original purpose, were employed to establish another professorship of pure mathematics, called the Sadleirian; the duties of the new professor were defined to be "to explain and teach the principles of pure mathematics and to apply himself to the advancement of that science."
To this chair Cayley was elected. He gave up a lucrative practice for a modest salary, he at once settled down in Cambridge. More fortunate than Hamilton in his choice, his home life was one of great happiness, his friend and fellow investigator, once remarked that Cayley had been much more fortunate than himself. At first the teaching duty of the Sadleirian professorship was limited to a course of lectures extending over one of the terms of the academic year. For many years the attendance was small, came entirely from those who had finished their career of preparation for competitive examinations; the subject lectured on was that of the memoir on which the professor was for the time engaged. The other duty of the chair — the advancement of mathematical science — was discharged in a handsome manner by the long series of memoirs that he published, ranging over every department of pure mathematics, but it was discharged in a much less obtrusive way. In 1872 he was made an honorary fellow of Trinity College, three years an ordinary fellow, which meant stipend as well as honour.
About this time his friends subscribed for a presentation portrait. Maxwell wrote an address to the committee of subscribers; the verses refer to the subjects investigated in several of Cay
Oscar Zariski was a Russian-born American mathematician and one of the most influential algebraic geometers of the 20th century. Zariski was born Oscher Zaritsky in 1918 studied at the University of Kiev, he left Kiev in 1920 to study at the University of Rome where he became a disciple of the Italian school of algebraic geometry, studying with Guido Castelnuovo, Federigo Enriques and Francesco Severi. Zariski wrote a doctoral dissertation in 1924 on a topic in Galois theory, proposed to him by Castelnuovo. At the time of his dissertation publication, he changed his name to Oscar Zariski. Zariski emigrated to the United States in 1927 supported by Solomon Lefschetz, he had a position at Johns Hopkins University where he became professor in 1937. During this period, he wrote Algebraic Surfaces as a summation of the work of the Italian school; the book was published in 1935 and reissued 36 years with detailed notes by Zariski's students that illustrated how the field of algebraic geometry had changed.
It is still an important reference. It seems to have been this work that set the seal of Zariski's discontent with the approach of the Italians to birational geometry, he addressed the question of rigour by recourse to commutative algebra. The Zariski topology, as it was known, is adequate for biregular geometry, where varieties are mapped by polynomial functions; that theory is too limited for algebraic surfaces, for curves with singular points. A rational map is to a regular map as a rational function is to a polynomial: it may be indeterminate at some points. In geometric terms, one has to work with functions defined on some open, dense set of a given variety; the description of the behaviour on the complement may require infinitely near points to be introduced to account for limiting behaviour along different directions. This introduces a need, in the surface case, to use valuation theory to describe the phenomena such as blowing up. After spending a year 1946–1947 at the University of Illinois at Urbana–Champaign, Zariski became professor at Harvard University in 1947 where he remained until his retirement in 1969.
In 1945, he fruitfully discussed foundational matters for algebraic geometry with André Weil. Weil's interest was in putting an abstract variety theory in place, to support the use of the Jacobian variety in his proof of the Riemann hypothesis for curves over finite fields, a direction rather oblique to Zariski's interests; the two sets of foundations weren't reconciled at that point. At Harvard, Zariski's students included Shreeram Abhyankar, Heisuke Hironaka, David Mumford, Michael Artin and Steven Kleiman—thus spanning the main areas of advance in singularity theory, moduli theory and cohomology in the next generation. Zariski himself worked on equisingularity theory; some of his major results, Zariski's main theorem and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme of Alexander Grothendieck that unified algebraic geometry. Zariski proposed the first example of a Zariski surface in 1958. Zariski was a Jewish atheist. Zariski was awarded the Steele Prize in 1981, in the same year the Wolf Prize in Mathematics with Lars Ahlfors.
He wrote Commutative Algebra in two volumes, with Pierre Samuel. His papers have been published in four volumes. Zariski, Abhyankar, Shreeram S.. Algebraic surfaces, Classics in mathematics, New York: Springer-Verlag, ISBN 978-3-540-58658-6, MR 0469915 Zariski, Introduction to the problem of minimal models in the theory of algebraic surfaces, Publications of the Mathematical Society of Japan, 4, The Mathematical Society of Japan, Tokyo, MR 0097403 Zariski, Cohn, James, ed. An introduction to the theory of algebraic surfaces, Lecture notes in mathematics, 83, New York: Springer-Verlag, doi:10.1007/BFb0082246, ISBN 978-3-540-04602-8, MR 0263819 Zariski, Oscar. Vol. II, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876 Zariski, Kmety, François; the moduli problem for plane branches, University Lecture Series, 39, Providence, R. I.: American Mathematical Society, ISBN 978-0-8218-2983-7, MR 0414561: Le problème des modules pour les branches planes Zariski, Collected papers. Vol. I: Foundations of algebraic geometry and resolution of singularities, Massachusetts-London: MIT Press, ISBN 978-0-262-08049-1, MR 0505100 Zariski, Collected papers.
Vol. II: Holomorphic functions and linear systems, Mathematicians of Our Time, Massachusetts-London: MIT Press, ISBN 978-0-262-01038-2, MR 0505100 Zariski, Artin, Michael. Collected papers. Volume III. Topology of curves and surfaces, special topics in the theory of algebraic varieties, Mathematicians of Our Time, Massachusetts-London: MIT Press, ISBN 978-0-262-24021-5, MR 0505104 Zariski, Lipman, Joseph. Collected papers. Vol. IV. Equisingularity on algebraic varieties, Mathematicians of Our Time, 16, MIT Press, ISBN 978-0-262-08049-1, MR 0545653 Zariski ring Zariski tangent space Zariski surface Zariski topology Zariski–Riemann surface Zariski space Zariski's lemma Zariski's main theorem