Harvard University is a private Ivy League research university in Cambridge, with about 6,700 undergraduate students and about 15,250 postgraduate students. Established in 1636 and named for its first benefactor, clergyman John Harvard, Harvard is the United States' oldest institution of higher learning, its history and wealth have made it one of the world's most prestigious universities; the Harvard Corporation is its first chartered corporation. Although never formally affiliated with any denomination, the early College trained Congregational and Unitarian clergy, its curriculum and student body were secularized during the 18th century, by the 19th century, Harvard had emerged as the central cultural establishment among Boston elites. Following the American Civil War, President Charles W. Eliot's long tenure transformed the college and affiliated professional schools into a modern research university. A. Lawrence Lowell, who followed Eliot, further reformed the undergraduate curriculum and undertook aggressive expansion of Harvard's land holdings and physical plant.
James Bryant Conant led the university through the Great Depression and World War II and began to reform the curriculum and liberalize admissions after the war. The undergraduate college became coeducational after its 1977 merger with Radcliffe College; the university is organized into eleven separate academic units—ten faculties and the Radcliffe Institute for Advanced Study—with campuses throughout the Boston metropolitan area: its 209-acre main campus is centered on Harvard Yard in Cambridge 3 miles northwest of Boston. Harvard's endowment is worth $39.2 billion, making it the largest of any academic institution. Harvard is a large residential research university; the nominal cost of attendance is high, but the university's large endowment allows it to offer generous financial aid packages. The Harvard Library is the world's largest academic and private library system, comprising 79 individual libraries holding over 18 million items; the University is cited as one of the world's top tertiary institutions by various organizations.
Harvard's alumni include eight U. S. presidents, more than thirty foreign heads of state, 62 living billionaires, 359 Rhodes Scholars, 242 Marshall Scholars. As of October 2018, 158 Nobel laureates, 18 Fields Medalists, 14 Turing Award winners have been affiliated as students, faculty, or researchers. In addition, Harvard students and alumni have won 10 Academy Awards, 48 Pulitzer Prizes and 108 Olympic medals, have founded a large number of companies worldwide. Harvard was established in 1636 by vote of the Great and General Court of the Massachusetts Bay Colony. In 1638, it acquired British North America's first known printing press. In 1639, it was named Harvard College after deceased clergyman John Harvard, an alumnus of the University of Cambridge, who had left the school £779 and his scholar's library of some 400 volumes; the charter creating the Harvard Corporation was granted in 1650. A 1643 publication gave the school's purpose as "to advance learning and perpetuate it to posterity, dreading to leave an illiterate ministry to the churches when our present ministers shall lie in the dust".
It offered a classic curriculum on the English university model—many leaders in the colony had attended the University of Cambridge—but conformed to the tenets of Puritanism. It was never affiliated with any particular denomination, but many of its earliest graduates went on to become clergymen in Congregational and Unitarian churches; the leading Boston divine Increase Mather served as president from 1685 to 1701. In 1708, John Leverett became the first president, not a clergyman, marking a turning of the college from Puritanism and toward intellectual independence. Throughout the 18th century, Enlightenment ideas of the power of reason and free will became widespread among Congregational ministers, putting those ministers and their congregations in tension with more traditionalist, Calvinist parties; when the Hollis Professor of Divinity David Tappan died in 1803 and the president of Harvard Joseph Willard died a year in 1804, a struggle broke out over their replacements. Henry Ware was elected to the chair in 1805, the liberal Samuel Webber was appointed to the presidency of Harvard two years which signaled the changing of the tide from the dominance of traditional ideas at Harvard to the dominance of liberal, Arminian ideas.
In 1846, the natural history lectures of Louis Agassiz were acclaimed both in New York and on the campus at Harvard College. Agassiz's approach was distinctly idealist and posited Americans' "participation in the Divine Nature" and the possibility of understanding "intellectual existences". Agassiz's perspective on science combined observation with intuition and the assumption that a person can grasp the "divine plan" in all phenomena; when it came to explaining life-forms, Agassiz resorted to matters of shape based on a presumed archetype for his evidence. This dual view of knowledge was in concert with the teachings of Common Sense Realism derived from Scottish philosophers Thomas Reid and Dugald Stewart, whose works were part of the Harvard curriculum at the time; the popularity of Agassiz's efforts to "soar with Plato" also derived from other writings to which Harvard students
David Allen Bayer is an American mathematician known for his contributions in algebra and symbolic computation and for his consulting work in the movie industry. He is a professor of mathematics at Columbia University. Bayer was educated at Swarthmore College as an undergraduate, where he attended a course on combinatorial algorithms given by Herbert Wilf. During that semester, Bayer related several original ideas to Wilf on the subject; these contributions were incorporated into the second edition of Wilf and Albert Nijenhuis' influential book Combinatorial Algorithms, with a detailed acknowledgement by its authors. Bayer subsequently earned his PhD at Harvard University in 1982 under the direction of Heisuke Hironaka with a dissertation entitled The Division Algorithm and the Hilbert Scheme, he joined Columbia University thereafter. Bayer is the son of the inventor of the Bayer filter. Bayer has worked in various areas of algebra and symbolic computation, including Hilbert functions, Betti numbers, linear programming.
He has written a number of cited papers in these areas with other notable mathematicians, including Bernd Sturmfels, Jeffrey Lagarias, Persi Diaconis, Irena Peeva, David Eisenbud. Bayer was a mathematics consultant for the film A Beautiful Mind, the biopic of John Nash, had a cameo as one of the "Pen Ceremony" professors. Bayer's homepage at Columbia University Dave Bayer on IMDb Dave Bayer at the Mathematics Genealogy Project Dave and Beautiful Math at Swarthmore College Bulletin
Lars Valerian Ahlfors was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his text on complex analysis. Ahlfors was born in Finland, his mother, Sievä Helander, died at his birth. His father, Axel Ahlfors, was a professor of engineering at the Helsinki University of Technology; the Ahlfors family was Swedish-speaking, so he first attended a private school where all classes were taught in Swedish. Ahlfors studied at University of Helsinki from 1924, graduating in 1928 having studied under Ernst Lindelöf and Rolf Nevanlinna, he assisted Nevanlinna in 1929 with his work on Denjoy's conjecture on the number of asymptotic values of an entire function. In 1929 Ahlfors published the first proof of this conjecture, now known as the Denjoy–Carleman–Ahlfors theorem, it states that the number of asymptotic values approached by an entire function of order ρ along curves in the complex plane going toward infinity is less than or equal to 2ρ. He completed his doctorate from the University of Helsinki in 1930.
Ahlfors worked as an associate professor at the University of Helsinki from 1933 to 1936. In 1936 he was one of the first two people to be awarded the Fields Medal. In 1935 Ahlfors visited Harvard University, he returned to Finland in 1938 to take up a professorship at the University of Helsinki. The outbreak of war led to problems, he was offered a post at the Swiss Federal Institute of Technology at Zurich in 1944 and managed to travel there in March 1945. He did not enjoy his time in Switzerland, so in 1946 he jumped at a chance to leave, returning to work at Harvard where he remained until he retired in 1977. Ahlfors was a visiting scholar at the Institute for Advanced Study in 1962 and again in 1966, he was awarded the Wihuri Prize in 1968 and the Wolf Prize in Mathematics in 1981. His book Complex Analysis is the classic text on the subject and is certainly referenced in any more recent text which makes heavy use of complex analysis. Ahlfors wrote several other significant books, including Conformal invariants.
He made decisive contributions to meromorphic curves, value distribution theory, Riemann surfaces, conformal geometry, quasiconformal mappings and other areas during his career. In 1933, he married Erna Lehnert, an Austrian who with her parents had first settled in Sweden and in Finland; the couple had three daughters. Ahlfors finiteness theorem Ahlfors measure conjecture Complex Analysis: an Introduction to the Theory of Analytic Functions of One Complex Variable Contributions to the Theory of Riemann Surfaces: Annals of Mathematics Studies Ahlfors, Lars. "FUNDAMENTAL POLYHEDRONS AND LIMIT POINT SETS OF KLEINIAN GROUPS". Proceedings of the National Academy of Sciences. 55: 251–254. Doi:10.1073/pnas.55.2.251. PMC 224131. PMID 16591331. Lars Ahlfors at the Mathematics Genealogy Project Ahlfors entry on Harvard University Mathematics department web site. Papers of Lars Valerian Ahlfors: an inventory Lars Valerian Ahlfors The MacTutor History of Mathematics page about Ahlfors The Mathematics of Lars Valerian Ahlfors, Notices of the American Mathematical Society.
45, no. 2. Lars Valerian Ahlfors, Notices of the American Mathematical Society. 45, no. 2. Frederick Gehring. "Lars Valerian Ahlfors: a biographical memoir". Biographical Memoirs. 87. National Academy of Sciences Biographical Memoir Author profile in the database zbMATH
Bernard Teissier is a French mathematician and a member of the Nicolas Bourbaki group. His research interests include algebraic geometry. Teissier attained his doctorate from Paris Diderot University in 1973, under supervision of Heisuke Hironaka, he was a member of Nicolas Bourbaki. In 2012 he became a fellow of the American Mathematical Society. Website
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
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