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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 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
Cornell University
. CONTENTS* 1 Mathematical contributions * 1.1 Foliations * 1.2 The geometrization conjecture * 1.3 Orbifold
Orbifold
theorem * 2 Education and career * 2.1 Selected works * 3 See also * 4 References * 5 Further reading * 6 External links MATHEMATICAL CONTRIBUTIONSFOLIATIONSHis early work, in the early 1970s, was mainly in foliation theory, where he had a dramatic impact
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John N. Mather
JOHN NORMAN MATHER (June 9, 1942 – January 28, 2017) was a mathematician at Princeton University
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
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)
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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 is a circle, and the 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 . Thus, the n-sphere would be defined by: S n = { x R n + 1 : x = r }
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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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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 nowadays is 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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Wolfgang Haken
WOLFGANG HAKEN (born June 21, 1928 in Berlin
Berlin
, Germany
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
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
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Euler Characteristic
In mathematics , and more specifically in algebraic topology and polyhedral combinatorics , the 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 {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
Leonhard Euler
, for whom the concept is named, was responsible for much of this early work. In modern mathematics, the Euler characteristic
Euler characteristic
arises from homology and, more abstractly, homological algebra
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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. CONTENTS * 1 Definition * 2 Haefliger structure and foliations * 3 Classifying space * 4 References DEFINITIONA Haefliger structure on a space X is determined by a HAEFLIGER COCYCLE. A codimension-q Haefliger cocycle consists of a covering of X by open sets Uα, together with continuous maps Ψαβ from Uα ∩ Uβ to the sheaf of germs of local diffeomorphisms of Rq, satisfying the 1-cocycle condition ( u ) = ( u ) ( u ) {displaystyle displaystyle Psi _{gamma alpha }(u)=Psi _{gamma beta }(u)Psi _{beta alpha }(u)} for u U U U
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National Academy Of Sciences
The NATIONAL ACADEMY OF SCIENCES (NAS) is a United States
United States
nonprofit , non-governmental organization . NAS is part of the National Academies of Sciences, Engineering, and Medicine
Medicine
, along with the National Academy of Engineering
Engineering
(NAE) and the National Academy of Medicine (NAM). As a national academy , new members of the organization are elected annually by current members, based on their distinguished and continuing achievements in original research. Election to the National Academies is one of the highest honors in the scientific field. Members serve pro bono as "advisers to the nation" on science , engineering , and medicine . The group holds a congressional charter under Title 36 of the United States
United States
Code
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Washington, D.C.
WASHINGTON, D.C., formally the DISTRICT OF COLUMBIA and commonly referred to as "WASHINGTON", "THE DISTRICT", or simply "D.C.", is the capital of the United States. The signing of the Residence Act on July 16, 1790, approved the creation of a capital district located along the Potomac River on the country's East Coast . The U.S. Constitution provided for a federal district under the exclusive jurisdiction of the Congress and the District is therefore not a part of any state. The states of Maryland and Virginia
Virginia
each donated land to form the federal district, which included the pre-existing settlements of Georgetown and Alexandria . Named in honor of President George Washington
George Washington
, the City of Washington was founded in 1791 to serve as the new national capital
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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 . He wrote his thesis in 1970 at the University of Copenhagen under the joint supervision of Werner Fenchel and Bent Fuglede
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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. Its fields can be divided into a variety of theoretical and practical disciplines
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Mathematician
A MATHEMATICIAN is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems . Mathematics
Mathematics
is concerned with numbers , data , quantity , structure , space , models , and change . CONTENTS * 1 History * 2 Required education * 3 Activities * 3.1 Applied mathematics
Applied mathematics
* 3.2 Abstract mathematics * 3.3 Mathematics
Mathematics
teaching * 3.4 Consulting * 4 Occupations * 5 Quotations about mathematicians * 6 Prizes in mathematics * 7 Mathematical autobiographies * 8 See also * 9 Notes * 10 References * 11 Further reading * 12 External links HISTORY This section is on the history of mathematicians
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Normal Surface
In mathematics , a NORMAL SURFACE is a surface inside a triangulated 3-manifold that intersects each tetrahedron so that each component of intersection is a triangle or a quad (see figure). A triangle cuts off a vertex of the tetrahedron while a quad separates pairs of vertices. A normal surface may have many components of intersection, called NORMAL DISKS, with one tetrahedron, but no two normal disks can be quads that separate different pairs of vertices since that would lead to the surface self-intersecting. A normal surface intersects a tetrahedron in (possibly many) triangles (see above left) and quads (see above right) Dually, a normal surface can be considered to be a surface that intersects each handle of a given handle structure on the 3-manifold in a prescribed manner similar to the above. The concept of normal surface can be generalized to arbitrary polyhedra. There are also related notions of ALMOST NORMAL SURFACE and SPUN NORMAL SURFACE
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University Of California, Davis
The UNIVERSITY OF CALIFORNIA, DAVIS (also referred to as UCD, UC DAVIS, or DAVIS), is a public research university and land-grant university as well as one of the 10 campuses of the University of California
California
(UC) system. It is located in Davis, California , just west of Sacramento
Sacramento
, and has the third-largest enrollment in the UC System after UCLA
UCLA
and UC Berkeley . The university has been labeled one of the "Public Ivies ," a publicly funded university considered to provide a quality of education comparable to those of the Ivy League
Ivy League
. The Carnegie Foundation classifies UC Davis as a comprehensive doctoral research university with a medical program, and very high research activity
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Cornell University
CORNELL UNIVERSITY (/kɔːrˈnɛl/ kor-NEL ) is an American private Ivy League
Ivy League
and public land-grant doctoral university located in Ithaca, New York
Ithaca, New York
. Founded in 1865 by Ezra Cornell
Ezra Cornell
and Andrew Dickson White , the university was intended to teach and make contributions in all fields of knowledge—from the classics to the sciences , and from the theoretical to the applied. These ideals, unconventional for the time, are captured in Cornell's motto, a popular 1865 Ezra Cornell quotation: "I would found an institution where any person can find instruction in any study." The university is broadly organized into seven undergraduate colleges and seven graduate divisions at its main Ithaca campus, with each college and division defining its own admission standards and academic programs in near autonomy
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