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William Thurston
William Paul Thurston (October 30, 1946 – August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal
Fields Medal
for his contributions to the study of 3-manifolds. From 2003 until his death he was a professor of mathematics and computer science at Cornell University.Contents1 Mathematical contributions1.1 Foliations 1.2 The geometrization conjecture 1.3 Orbifold
Orbifold
theorem2 Education and career2.1 Selected works3 See also 4 References 5 Further reading 6 External linksMathematical contributions[edit] Foliations[edit] His early work, in the early 1970s, was mainly in foliation theory, where he had a dramatic impact
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Washington, D.C.
Washington, D.C., formally the District of Columbia
District of Columbia
and commonly referred to as Washington or D.C., is the capital of the United States of America.[4] Founded after the American Revolution
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Rochester, New York
Rochester (/ˈrɒtʃɪstər, ˈrɒtʃɛstər/) is a city on the southern shore of Lake Ontario
Lake Ontario
in western New York. With a population of 208,880 residents, Rochester is the seat of Monroe County and the third most populous city in New York state, after New York City
New York City
and Buffalo. The metropolitan area has a population of just over 1 million people.[4] Rochester was one of America's first boomtowns, initially due to its flour mills along the Genesee River, and then as a manufacturing hub.[5] Several of the region's universities (notably the University of Rochester and Rochester Institute of Technology) have renowned research programs. Rochester is the site of many important inventions and innovations in consumer products. The Rochester area has been the birthplace to Kodak, Western Union, Bausch & Lomb, Gleason and Xerox, which conduct extensive research and manufacturing of industrial and consumer products
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Wolfgang Haken
Wolfgang Haken (born June 21, 1928 in Berlin, Germany) is a mathematician who specializes in topology, in particular 3-manifolds. In 1962 he left Germany
Germany
to become a visiting professor at the University of Illinois at Urbana-Champaign, he became a full professor by 1965, and he is now an emeritus professor. In 1976 together with colleague Kenneth Appel, also at the University of Illinois at Urbana-Champaign, Haken solved one of the most famous problems in mathematics, the four-color problem. They proved that any map can be filled in with four colors without any adjacent "countries" sharing the same color. Haken has introduced several important ideas, including Haken manifolds, Kneser–Haken finiteness, and an expansion of the work of Kneser into a theory of normal surfaces. Much of his work has an algorithmic aspect, and he is one of the influential figures in algorithmic topology
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Figure-eight Knot
The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in both sailing and rock climbing as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under strain, often requiring the rope to be cut, the figure-of-eight will also jam, but is usually more easily undone than the overhand knot.The figure-eight or figure-of-eight knot is also called (in books) the Flemish knot. The name figure-of-eight knot appears in Lever's Sheet Anchor; or, a Key to Rigging (London, 1808). The word "of" is nowadays usually omitted. The knot is the sailor's common single-strand stopper knot and is tied in the ends of tackle falls and running rigging, unless the latter is fitted with monkey's tails. It is used about ship wherever a temporary stopper knot is required
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Troels Jørgensen
Troels Jørgensen is a Danish mathematician at Columbia University working on hyperbolic geometry and complex analysis, who proved Jørgensen's inequality
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Compact-open Topology
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and functional analysis. It was introduced by Ralph Fox
Ralph Fox
in 1945.[1] If the codomain of the functions under consideration have a uniform structure or a metric structure then the compact-open topology is the "topology of uniform convergence on compact sets." That is to say, a sequence of functions converges in the compact-open topology precisely when it converges uniformly on every compact subset of the domain.[2][better source needed]Contents1 Definition 2 Properties 3 Fréchet differentiable functions 4 See also 5 ReferencesDefinition[edit] Let X and Y be two topological spaces, and let C(X, Y) denote the set of all continuous maps between X and Y
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Discrete Space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set, i.e., it defines all subsets as open sets
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Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος (homoios) = similar or same (French pareil) and μορφή (morphē) = shape, form, introduced to mathematics by Henri Poincaré
Henri Poincaré
in 1895.[1][2] Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not
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Cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology
Cohomology
can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications
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John N. Mather
John Norman Mather (June 9, 1942 – January 28, 2017) was a mathematician at Princeton University known for his work on singularity theory and Hamiltonian dynamics. He was descended from Atherton Mather (1663-1734), a cousin of Cotton Mather. His early work dealt with the stability of smooth mappings between smooth manifolds of dimensions n (for the source manifold N) and p (for the target manifold P). He determined the precise dimensions (n,p) for which smooth mappings are stable with respect to smooth equivalence by diffeomorphisms of the source and target (i.e., infinitely differentiable coordinate changes). He also proved the conjecture of the French topologist René Thom that under topological equivalence smooth mappings are generically stable: the subset of the space of smooth mappings between two smooth manifolds consisting of the topologically stable mappings is a dense subset in the smooth Whitney topology
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N-sphere
In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension. It is an n-dimensional manifold that can be embedded in Euclidean (n + 1)-space. The 0-sphere is a pair of points, the 1-sphere
1-sphere
is a circle, and the 2-sphere
2-sphere
is an ordinary sphere. Generally, when embedded in an (n + 1)-dimensional Euclidean space, an n-sphere is the surface or boundary of an (n + 1)-dimensional ball. That is, for any natural number n, an n-sphere of radius r may be defined in terms of an embedding in (n + 1)-dimensional Euclidean space
Euclidean space
as the set of points that are at distance r from a central point, where the radius r may be any positive real number
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Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the height of the defining ideal. For this reason, the height of an ideal is often called its codimension. The dual concept is relative dimension.Contents1 Definition 2 Additivity of codimension and dimension counting 3 Dual interpretation 4 In geometric topology 5 See also 6 ReferencesDefinition[edit] Codimension is a relative concept: it is only defined for one object inside another
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Euler Characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic
Euler characteristic
(or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by χ displaystyle chi (Greek lower-case letter chi). The Euler characteristic
Euler characteristic
was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. Leonhard Euler, for whom the concept is named, was responsible for much of this early work
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Haefliger Structure
In mathematics, a Haefliger structure on a topological space is a generalization of a foliation of a manifold, introduced by Haefliger (1970, 1971). Any foliation on a manifold induces a Haefliger structure, which uniquely determines the foliation.Contents1 Definition 2 Haefliger structure and foliations 3 Classifying space 4 ReferencesDefinition[edit] A Haefliger structure on a space X is determined by a Haefliger cocycle
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Computer Science
Computer science
Computer science
is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers. It is the scientific and practical approach to computation and its applications and the systematic study of the feasibility, structure, expression, and mechanization of the methodical procedures (or algorithms) that underlie the acquisition, representation, processing, storage, communication of, and access to, information. An alternate, more succinct definition of computer science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems.[1] Its fields can be divided into a variety of theoretical and practical disciplines
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