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On Computing the Homology Type of a Triangulation
, 1994
"... :We analyze an algorithm for computing the homology type of a triangulation. By triangulation we mean a finite simplicial complex; its homology type is given by its homology groups (with integer coefficients). The algorithm could be used in computeraided design to tell whether two finiteelement me ..."
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:We analyze an algorithm for computing the homology type of a triangulation. By triangulation we mean a finite simplicial complex; its homology type is given by its homology groups (with integer coefficients). The algorithm could be used in computeraided design to tell whether two finiteelement meshes or B'ezierspline surfaces are of the same "topological type," and whether they can be embedded in R³. Homology computation is a purely combinatorial problem of considerable intrinsic interest. While the worstcase bounds we obtain for this algorithm are poor, we argue that many triangulations (in general) and virtually all triangulations in design are very "sparse," in a sense we make precise. We formalize this sparseness measure, and perform a probabilistic analysis of the sparse case to show that the expected running time of the algorithm is roughly quadratic in the geometric complexity (number of simplices) and linear in the dimension.
What do topologists want from SeibergWitten theory
 Int. J. Mod. Phys. A
"... In 1983, Donaldson shocked the topology world by using instantons from physics to prove new theorems about fourdimensional manifolds, and he developed new topological invariants. In 1988, Witten showed how these invariants could be obtained by correlation functions for a twisted N = 2 SUSY gauge th ..."
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In 1983, Donaldson shocked the topology world by using instantons from physics to prove new theorems about fourdimensional manifolds, and he developed new topological invariants. In 1988, Witten showed how these invariants could be obtained by correlation functions for a twisted N = 2 SUSY gauge theory. In 1994, Seiberg and Witten discovered dualities for such theories, and in particular, developed a new way of looking at fourdimensional manifolds that turns out to be easier, and is conjectured to be equivalent to, Donaldson theory. This review describes the development of this mathematical subject, and shows how the physics played a pivotal role in the current understanding of this area of topology. Keywords: Seiberg–Witten; instantons; Donaldson theory; topology; fourdimensional manifolds.
A Signal Subspace Approach to Multiple Emitter Location and Spectral Estimation
 J. London Math. Soc
, 1980
"... In previously studied cases in varieties of algebras, the word and isomorphism problems have had the same solution. For abelian groups and loops, for example, both are soluble [15, Section 3.3; 9; 10]; for groups, semigroups and latticeordered groups, for example, both are insoluble [7, 19, 1, 18, ..."
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In previously studied cases in varieties of algebras, the word and isomorphism problems have had the same solution. For abelian groups and loops, for example, both are soluble [15, Section 3.3; 9; 10]; for groups, semigroups and latticeordered groups, for example, both are insoluble [7, 19, 1, 18, 12, 11]. In contrast, Miller [17, p. 77], shows that there are recursive classes of finitely presented groups with uniformly soluble word problem but with insoluble isomorphism problem. Our contribution is to provide a variety of algebras defined by a finite number of laws where this dichotomy also occurs. Specifically, we have the following. THEOREM A. (1) The class of finitely presented abelian latticeordered groups has uniformly soluble word problem. (2) The isomorphism problem for the class of finitely presented latticeordered groups is insoluble. In other words, there is a single algorithm which when given an arbitrary finitely presented abelian latticeordered group (from any recursive listing of them all) and a word in its generators determines whether or not the word is 0, yet there is no algorithm which can always distinguish nonisomorphic finitely presented abelian latticeordered groups. Part (1) of Theorem A was already known. Part (2) of the theorem is established using the ideas of Baker and Beynon to show that with each compact polyhedron we can associate—in an effective manner—a finitely generated one relator abelian latticeordered group so that two compact polyhedra are piecewise linear homeomorphic if and only if the associated finitely presented abelian latticeordered groups are isomorphic. Since Markov [16] has proved that the piecewise linear homeomorphism problem for compact polyhedra is insoluble, Theorem A follows. Actually, the duality and the full strength of Markov's superb theorem give a very strong dichotomy, as follows. THEOREM B. The isomorphism problem for m generator 1 relator abelian latticeordered groups is insoluble whenever m ^ 10.
UNSOLVABLE PROBLEMS ABOUT HIGHERDIMENSIONAL KNOTS AND RELATED GROUPS
, 908
"... containing the preceding one, related to codimension 2 smooth embeddings of manifolds. Kn is the class of groups of complements of nspheres in S n+2; S (resp. M, G) is the class of groups of complements of orientable, closed surfaces in S 4 (resp. a 1connected 4manifold, ..."
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containing the preceding one, related to codimension 2 smooth embeddings of manifolds. Kn is the class of groups of complements of nspheres in S n+2; S (resp. M, G) is the class of groups of complements of orientable, closed surfaces in S 4 (resp. a 1connected 4manifold,