In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More each point of an n-dimensional manifold has a neighbourhood, homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold. One-dimensional manifolds include circles, but not figure eights. Two-dimensional manifolds are called surfaces. Examples include the plane, the sphere, the torus, which can all be embedded in three dimensional real space, but the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space. Although a manifold locally resembles Euclidean space, meaning that every point has a neighbourhood homeomorphic to an open subset of Euclidean space, globally it may not: manifolds in general are not homeomorphic to Euclidean space. For example, the surface of the sphere is not homeomorphic to the Euclidean plane, because it has the global topological property of compactness that Euclidean space lacks, but in a region it can be charted by means of map projections of the region into the Euclidean plane.
When a region appears in two neighbouring charts, the two representations do not coincide and a transformation is needed to pass from one to the other, called a transition map. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described and understood in terms of the simpler local topological properties of Euclidean space. Manifolds arise as solution sets of systems of equations and as graphs of functions. Manifolds can be equipped with additional structure. One important class of manifolds is the class of differentiable manifolds. A Riemannian metric on a manifold allows angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. A surface is a two dimensional manifold, meaning that it locally resembles the Euclidean plane near each point. For example, the surface of a globe can be described by a collection of maps, which together form an atlas of the globe.
Although no individual map is sufficient to cover the entire surface of the globe, any place in the globe will be in at least one of the charts. Many places will appear in more than one chart. For example, a map of North America will include parts of South America and the Arctic circle; these regions of the globe will be described in full in separate charts, which in turn will contain parts of North America. There is a relation between adjacent charts, called a transition map that allows them to be patched together to cover the whole of the globe. Describing the coordinate charts on surfaces explicitly requires knowledge of functions of two variables, because these patching functions must map a region in the plane to another region of the plane. However, one-dimensional examples of manifolds can be described with functions of a single variable only. Manifolds have applications in computer-graphics and augmented-reality given the need to associate pictures to coordinates. In an augmented reality setting, a picture can be seen as something associated with a coordinate and by using sensors for detecting movements and rotation one can have knowledge of how the picture is oriented and placed in space.
After a line, the circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Consider, for instance, the top part of the unit circle, x2 + y2 = 1, where the y-coordinate is positive. Any point of this arc can be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous, invertible, mapping from the upper arc to the open interval: χ t o p = x; such functions along with the open regions they map are called charts. There are charts for the bottom and right parts of the circle: χ b o t t o m = x χ l e f t = y χ r i g h t = y. Together, these parts cover the four charts form an atlas for the circle; the top and right charts, χ t o
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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.
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James Dillon Stasheff is an American mathematician, a professor emeritus of mathematics at the University of North Carolina at Chapel Hill. He works in algebraic algebra as well as their applications to physics. Stasheff did his undergraduate studies in mathematics at the University of Michigan, graduating in 1956. Stasheff began his graduate studies at Princeton University. After his second year at Princeton, he moved to Oxford University on a Marshall Scholarship. Two years in 1961, with a pregnant wife, needing an Oxford degree to get reimbursed for his return trip to the US, yet still feeling attached to Princeton, he split his thesis into two parts and earned two doctorates, a D. Phil. From Oxford under the supervision of Ioan James and a Ph. D. the same year from Princeton under the supervision of John Coleman Moore. From 1961 to 1962, Stasheff was a C. L. E. Moore instructor at the Massachusetts Institute of Technology. In 1962 joined the faculty of University of Notre Dame as an assistant professor.
He visited Princeton University from 1968 to 1969 and stayed there the next year as a Sloan Fellow. In 1970 he moved to Temple University, where he held a position until 1978. In 1976, he joined the UNC faculty, he has visited the Institute for Advanced Study, Lehigh University, Rutgers University, the University of Pennsylvania. Stasheff was an editor of the Transactions of the American Mathematical Society, he has two children. Stasheff's research contributions include the study of associativity in loop spaces and the construction of the associahedron, ideas leading to the theory of operads. In the 1960s he wrote fundamental papers on higher homotopy homotopy algebras, he introduced Stasheff algebras and Stasheff polytopes. In the 1980s he turned to the application of characteristic classes and other topological and algebraic concepts in mathematical physics, first in the algebraic structure of anomalies in quantum field theory, where he worked with among others, Tom Kephart and Paolo Cotta-Ramusino.
He referred to the research field as cohomological physics. In 2012 he became a fellow of the American Mathematical Society. Milnor, John W.. Characteristic Classes, Annals of Mathematics Studies, 76, Princeton, NJ: Princeton University Press, ISBN 0-691-08122-0. Stasheff, James D. "Homotopy associativity of H-spaces. I, II", Transactions of the American Mathematical Society, 108: 275–292, 293–312, doi:10.2307/1993609, MR 0158400. Stasheff, James D. H-spaces from a Homotopy Point of View, Berlin: Springer. Homepage
Sylvain Edward Cappell, is a Belgian American mathematician and former student of William Browder at Princeton University, is a topologist who has spent most of his career at the Courant Institute of Mathematical Sciences at NYU, where he is now the Silver Professor of Mathematics. He was born in Brussels and immigrated with his parents to New York City in 1950 and grew up in this city. In 1963, as a senior at the Bronx High School of Science, he won first place in the Westinghouse Science Talent Search for his work on "The Theory of Semi-cyclical Groups with Special Reference to Non-Aristotelian Logic." He is best known for his "codimension one splitting theorem", a standard tool in high-dimensional geometric topology, a number of important results proven with his collaborator Julius Shaneson. Their work includes aspects of low-dimensional topology, they gave the first nontrivial examples of topological conjugacy of linear transformations, which led to a flowering of research on the topological study of spaces with singularities.
More they combined their understanding of singularities, first to lattice point counting in polytopes to Euler-Maclaurin type summation formulae, most to counting lattice points in the circle. This last problem is a classical one, initiated by Gauss, the paper is still being vetted by experts. In 2012 he became a fellow of the American Mathematical Society. Cappell was elected and served as a vice president of the AMS for the term of February 2010 through January 2013. Official website at NYU Sylvain Cappell at the Mathematics Genealogy Project
In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting and bending, but not tearing or gluing. An n-dimensional topological space is a space with certain properties of connectedness and compactness; the space discrete. It can be closed. Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space and transformation; such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems; the term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics. Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries.
Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realised the importance of the edges of a polyhedron; this led to his polyhedron formula, V − E + F = 2. Some authorities regard this analysis as the first theorem. Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term "Topologie" in Vorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print; the English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". The term "topologist" in the sense of a specialist in topology was used in 1905 in the magazine Spectator.
Their work was corrected and extended by Henri Poincaré. In 1895, he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a special case of a general topological space, with any given topological space giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space. A topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski. Modern topology depends on the ideas of set theory, developed by Georg Cantor in the part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series.
For further developments, see point-set topology and algebraic topology. Topology can be formally defined as "the study of qualitative properties of certain objects that are invariant under a certain kind of transformation those properties that are invariant under a certain kind of invertible transformation." Topology is used to refer to a structure imposed upon a set X, a structure that characterizes the set X as a topological space by taking proper care of properties such as convergence and continuity, upon transformation. Topological spaces show up in every branch of mathematics; this has made topology one of the great unifying ideas of mathematics. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects and both separate the plane into two parts, the part inside and the part outside.
In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg that would cross each of its seven bridges once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks; this Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory. The hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is convincing to most people though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of t
Andrew Alexander Ranicki was a British mathematician who worked on algebraic topology. He was a professor of mathematics at the University of Edinburgh. Ranicki was the only child of the well-known literary critic Marcel Reich-Ranicki and the artist Teofila Reich-Ranicki, he lived in Warsaw, in Frankfurt am Main and Hamburg and attended school in England from the age of sixteen. Ranicki graduated from the University of Cambridge in Mathematics in 1969. At Cambridge he was a student of John Frank Adams, he earned his doctoral degree in 1973 with a thesis on algebraic L-theory. Ranicki received numerous honors for his scientific achievements during his studies. From 1972 to 1977 he was a Fellow of Cambridge. From 1977 to 1982, he was Assistant Professor at Princeton University. In 1982 he began at the University of Edinburgh as a lecturer. In 1992, he became a Fellow of the Royal Society of Edinburgh. Since 1995, Ranicki has been the Chair of Algebraic Surgery at the University of Edinburgh. Several times he stayed as a visiting scientist at the Max Planck Institute for Mathematics in Bonn, most in 2011.
Ranicki was married to American paleontologist Ida Thompson in 1979. Ranicki suffered from leukemia. Exact sequences in the algebraic theory of surgery, Princeton University Press, 1981. Lower K and L Theory, London Mathematical Society Lecture Notes, Vol. 178, Cambridge University Press. 1992. Algebraic L-Theory and Topological Manifolds', Cambridge Tracts in Mathematics Vol. 102, Cambridge University Press, 1992. Algebraic and Geometric Surgery, Oxford University Press, 2002. High dimensional knot theory, Springer, 1998. With Bruce Hughes: Ends of Complexes, Cambridge Tracts in Mathematics Vol. 123, Cambridge University Press, 1996. With Norman Levitt and Frank Quinn: "Algebraic and geometric topology", Springer 1985, Lecture Notes in Mathematics Vol. 1126. Editor with David Lewis and Eva Bayer-Fluckiger: "Quadratic forms and their applications", Contemporary Mathematics Vol. 272, American Mathematical Society, 2000. Publisher: Noncommutative Localization in Algebra and Topology, London Mathematical Society Vol. 330, Cambridge University Press, 2006.
Editor with Steven Ferry and Jonathan Rosenberg: "The Novikov conjectures, index theorems and rigidity", London Mathematical Society Lecture Notes, Vol. 226, 227, Cambridge University Press, 1995. Editor: The Hauptvermutung Book, Kluwer, 1996. Editor with Sylvain Cappell and Jonathan Rosenberg: Surveys on surgery theory. Papers dedicated to C. T. C. Wall
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. Milnor is a distinguished professor at Stony Brook University and one of the four mathematicians to have won the Fields Medal, the Wolf Prize, the Abel Prize. Milnor was born on February 1931 in Orange, New Jersey, his father was J. Willard Milnor and his mother was Emily Cox Milnor; as an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and proved the Fary–Milnor theorem. He continued on to graduate school at Princeton under the direction of Ralph Fox and submitted his dissertation, entitled "Isotopy of Links", which concerned link groups and their associated link structure, in 1954. Upon completing his doctorate he went on to work at Princeton, he was a professor at the Institute for Advanced Study from 1970 to 1990. His students have included Tadatoshi Akiba, Jon Folkman, John Mather, Laurent C. Siebenmann, Michael Spivak, his wife, Dusa McDuff, is a professor of mathematics at Barnard College.
One of his published works is his proof in 1956 of the existence of 7-dimensional spheres with nonstandard differential structure. With Michel Kervaire, he showed that the 7-sphere has 15 differentiable structures. An n-sphere with nonstandard differential structure is called an exotic sphere, a term coined by Milnor. Egbert Brieskorn found simple algebraic equations for 28 complex hypersurfaces in complex 5-space such that their intersection with a small sphere of dimension 9 around a singular point is diffeomorphic to these exotic spheres. Subsequently Milnor worked on the topology of isolated singular points of complex hypersurfaces in general, developing the theory of the Milnor fibration whose fiber has the homotopy type of a bouquet of μ spheres where μ is known as the Milnor number. Milnor's 1968 book on his theory inspired the growth of a huge and rich research area which continues to mature to this day. In 1961 Milnor disproved the Hauptvermutung by illustrating two simplicial complexes which are homeomorphic but combinatorially distinct.
In 1984 Milnor introduced a definition of attractor. The objects generalize standard attractors, include so-called unstable attractors and are now known as Milnor attractors. Milnor's current interest is dynamics holomorphic dynamics, his work in dynamics is summarized by Peter Makienko in his review of Topological Methods in Modern Mathematics: It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from the beginning, looking at the simplest nontrivial families of maps; the first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston. The case of a unimodal map, that is, one with a single critical point, turns out to be rich; this work may be compared with Poincaré's work on circle diffeomorphisms, which 100 years before had inaugurated the qualitative theory of dynamical systems.
Milnor's work has opened several new directions in this field, has given us many basic concepts, challenging problems and nice theorems. He was an editor of the Annals of Mathematics for a number of years after 1962, he has written a number of books. In 1962 Milnor was awarded the Fields Medal for his work in differential topology, he went on to win the National Medal of Science, the Lester R. Ford Award in 1970 and again in 1984, the Leroy P Steele Prize for "Seminal Contribution to Research", the Wolf Prize in Mathematics, the Leroy P Steele Prize for Mathematical Exposition, the Leroy P Steele Prize for Lifetime Achievement "... for a paper of fundamental and lasting importance, On manifolds homeomorphic to the 7-sphere, Annals of Mathematics 64, 399–405". In 1991 a symposium was held at Stony Brook University in celebration of his 60th birthday. Milnor was awarded the 2011 Abel Prize, for his "pioneering discoveries in topology and algebra." Reacting to the award, Milnor told the New Scientist "It feels good," adding that "ne is always surprised by a call at 6 o'clock in the morning."
In 2013 he became a fellow of the American Mathematical Society, for "contributions to differential topology, geometric topology, algebraic topology and dynamical systems". Milnor, John W.. Morse theory. Annals of Mathematics Studies, No. 51. Notes by M. Spivak and R. Wells. Princeton, NJ: Princeton University Press. ISBN 0-691-08008-9. ——. Lectures on the h-cobordism theorem. Notes by L. Siebenmann and J. Sondow. Princeton, NJ: Princeton University Press. ISBN 0-691-07996-X. OCLC 58324. ——. Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton, NJ: Princeton University Press. ISBN 0-691-08065-8. ——. Introduction to algebraic K-theory. Annals of Mathematics Studies, No. 72. Princeton, NJ: Princeton University Press. ISBN 978-0-691-08101-4. Husemoller, Dale. Symmetric bilinear forms. New York, NY: Springer-Verlag. ISBN 978-0-387-06009-5. Milnor, John W.. Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton, NJ: Princeton University Press. ISBN 0-691-08122-0. Milnor, John W..
Topology from the differentiable viewpoint. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-04833-9. —— (