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C. T. C. Wall
Charles Terence Clegg "Terry" Wall (born 14 December 1936) is a British mathematician, educated at Marlborough College, Marlborough and Trinity College, Cambridge. He is an :wikt:emeritus, emeritus professor of the University of Liverpool, where he was first appointed professor in 1965. From 1978 to 1980 he was the president of the London Mathematical Society. Work His early work was in cobordism theory in algebraic topology; this includes his 1959 Cambridge PhD thesis entitled "Algebraic aspects of cobordism", written under the direction of Frank Adams and Christopher Zeeman. His research was then mainly in the area of manifolds, particularly geometric topology and related abstract algebra included in surgery theory, of which he was one of the founders. In 1964 he introduced the Brauer–Wall group of a field. His 1970 research monograph "Surgery on Compact Manifolds" is a major reference work in geometric topology. In 1971 he conjectured that every finitely generated group ...
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Bristol
Bristol () is a City status in the United Kingdom, city, Ceremonial counties of England, ceremonial county and unitary authority in England. Situated on the River Avon, Bristol, River Avon, it is bordered by the ceremonial counties of Gloucestershire to the north and Somerset to the south. Bristol is the most populous city in South West England. The wider Bristol Built-up Area is the List of urban areas in the United Kingdom, eleventh most populous urban area in the United Kingdom. Iron Age hillforts and Roman villas were built near the confluence of the rivers River Frome, Bristol, Frome and River Avon, Bristol, Avon. Around the beginning of the 11th century, the settlement was known as (Old English: 'the place at the bridge'). Bristol received a royal charter in 1155 and was historic counties of England, historically divided between Gloucestershire and Somerset until 1373 when it became a county corporate. From the 13th to the 18th century, Bristol was among the top three E ...
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Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. Homology ...
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Isolated Singularity
In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it. In other words, a complex number ''z0'' is an isolated singularity of a function ''f'' if there exists an open disk ''D'' centered at ''z0'' such that ''f'' is holomorphic on ''D'' \ , that is, on the set obtained from ''D'' by taking ''z0'' out. Formally, and within the general scope of general topology, an isolated singularity of a holomorphic function a function f: \Omega\to \mathbb is any isolated point of the boundary \partial \Omega of the domain \Omega. In other words, if U is an open subset of \mathbb , a\in U and f: U\setminus \\to \mathbb is a holomorphic function, then a is an isolated singularity of f. Every singularity of a meromorphic function on an open subset U\subset \mathbb is isolated, but isolation of singularities alone is not sufficient to guarantee a function is meromorphic. Many important tools of complex analysis such a ...
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Singularity Theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. In some places the flat string will cross itself in an approximate "X" shape. The points on the floor where it does this are one kind of singularity, the double point: one bit of the floor corresponds to more than one bit of string. Perhaps the string will also touch itself without crossing, like an underlined "U". This is another kind of singularity. Unlike the double point, it is not ''stable'', in the sense that a small push will lift the bottom of the "U" away from the "underline". Vladimir Arnold defines the main goal of singularity theory as describing how objects depend on parameters, particularly in cases where the properties undergo sudden change under a small variation of the parameters. Thes ...
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Finitely Presented Group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set ''R'' of relations among those generators. We then say ''G'' has presentation :\langle S \mid R\rangle. Informally, ''G'' has the above presentation if it is the "freest group" generated by ''S'' subject only to the relations ''R''. Formally, the group ''G'' is said to have the above presentation if it is isomorphic to the quotient of a free group on ''S'' by the normal subgroup generated by the relations ''R''. As a simple example, the cyclic group of order ''n'' has the presentation :\langle a \mid a^n = 1\rangle, where 1 is the group identity. This may be written equivalently as :\langle a \mid a^n\rangle, thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity. ...
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Martin Dunwoody
Martin John Dunwoody (born 3 November 1938) is an emeritus professor of Mathematics at the University of Southampton, England. He earned his PhD in 1964 from the Australian National University. He held positions at the University of Sussex before becoming a professor at the University of Southampton in 1992. He has been emeritus professor since 2003. Dunwoody works on geometric group theory and low-dimensional topology. He is a leading expert in splittings and accessibility of discrete groups, groups acting on graphs and trees, JSJ-decompositions, the topology of 3-manifolds and the structure of their fundamental groups. Since 1971 several mathematicians have been working on Wall's conjecture, posed by Wall in a 1971 paper, which said that all finitely generated groups are accessible. Roughly, this means that every finitely generated group can be constructed from finite and one-ended groups via a finite number of amalgamated free products and HNN extensions over finite sub ...
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Group (mathematics)
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of th ...
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Stallings Theorem About Ends Of Groups
In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group ''G'' has more than one end if and only if the group ''G'' admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup. In the modern language of Bass–Serre theory the theorem says that a finitely generated group ''G'' has more than one end if and only if ''G'' admits a nontrivial (that is, without a global fixed point) action on a simplicial tree with finite edge-stabilizers and without edge-inversions. The theorem was proved by John R. Stallings, first in the torsion-free case (1968) and then in the general case (1971). Ends of graphs Let Γ be a connected graph where the degree of every vertex is finite. One can view Γ as a topological space by giving it the natural structure of a one-dimensional cell complex. Then the ends of Γ are the ends of this topological space. A more explicit definition of ...
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Finitely Generated Group
In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of such elements. By definition, every finite group is finitely generated, since ''S'' can be taken to be ''G'' itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated. Examples * Every quotient of a finitely generated group ''G'' is finitely generated; the quotient group is generated by the images of the generators of ''G'' under the canonical projection. * A subgroup of a finitely generated group need not be finitely generated. * A group that is generated by a single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of the ...
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Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by ...
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Brauer–Wall Group
In mathematics, the Brauer–Wall group or super Brauer group or graded Brauer group for a field ''F'' is a group BW(''F'') classifying finite-dimensional graded central division algebras over the field. It was first defined by as a generalization of the Brauer group. The Brauer group of a field ''F'' is the set of the similarity classes of finite dimensional central simple algebras over ''F'' under the operation of tensor product, where two algebras are called similar if the commutants of their simple modules are isomorphic. Every similarity class contains a unique division algebra, so the elements of the Brauer group can also be identified with isomorphism classes of finite dimensional central division algebras. The analogous construction for Z/2Z-graded algebras defines the Brauer–Wall group BW(''F'').Lam (2005) pp.98–99 Properties * The Brauer group B(''F'') injects into BW(''F'') by mapping a CSA ''A'' to the graded algebra which is ''A'' in grade zero. * showed that th ...
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Surgery Theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold ''M'' of dimension n=p+q+1, could be described as removing an imbedded sphere of dimension ''p'' from ''M''. Originally developed for differentiable (or, smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds. Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions. More technically, the idea is to start with a well-understood manifold ''M'' and perform surgery on it to produce a manifold ''M''′ having some desired property, in such a way ...
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