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Stratified Morse Theory
In mathematics, stratified Morse theory is an analogue to Morse theory for general stratified spaces, originally developed by Mark Goresky and Robert MacPherson. The main point of the theory is to consider functions f : M \to \mathbb R and consider how the stratified space f^(-\infty,c] changes as the real number c \in \mathbb R changes. Morse theory of stratified spaces has uses everywhere from pure mathematics topics such as braid groups and Lawrence–Krammer representation, representations to robot motion planning and potential theory. A popular application in pure mathematics is Morse theory on manifolds with boundary, and manifolds with corners. See also * Digital Morse theory * Discrete Morse theory * Level-set method Level-set methods (LSM) are a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. The advantage of the level-set model is that one can perform numerical computations involving curves and surfaces o ... ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morse Theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology. Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics ( critical points of the energy functional on the space of paths). These techniques were used in Raoul Bott's proof of his periodicity theorem. The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory. Basic concepts To illustrate, consider a mountainous landscape surface M (more generally, a manifold). If f is the functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stratified Space
In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat). A basic example is a subset of a smooth manifold that admits a Whitney stratification. But there is also an abstract stratified space such as a Thom–Mather stratified space. On a stratified space, a constructible sheaf can be defined as a sheaf that is locally constant on each stratum. Among the several ideals, Grothendieck's ''Esquisse d’un programme'' considers (or proposes) a stratified space with what he calls the tame topology. A stratified space in the sense of Mather Mather gives the following definition of a stratified space. A ''prestratification'' on a topological space ''X'' is a partition of ''X'' into subsets (called strata) such that (a) each stratum is locally closed, (b) it is locally finite and (c) (axiom of frontier) if two strata ''A'', '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mark Goresky
Robert Mark Goresky is a Canadian mathematician who invented intersection homology with his advisor and life partner Robert MacPherson. Career Goresky received his Ph.D. from Brown University in 1976. His thesis, titled ''Geometric Cohomology and Homology of Stratified Objects'', was written under the direction of MacPherson. Many of the results in his thesis were published in 1981 by the American Mathematical Society. He has taught at the University of British Columbia in Vancouver, and Northeastern University. Awards Goresky received a Sloan Research Fellowship in 1981. He received the Coxeter–James Prize in 1984. In 2002, Goresky and MacPherson were jointly awarded the Leroy P. Steele Prize for Seminal Contribution to Research by the American Mathematical Society. In 2012 Goresky became a fellow of the American Mathematical Society. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert MacPherson (mathematician)
Robert Duncan MacPherson (born May 25, 1944) is an American mathematician at the Institute for Advanced Study and Princeton University. He is best known for the invention of intersection homology with Mark Goresky, whose thesis he directed at Brown University, and who became his life partner. MacPherson previously taught at Brown University, the University of Paris, and the Massachusetts Institute of Technology. In 1983 he gave a plenary address at the International Congress of Mathematicians in Warsaw. Education and career Educated at Swarthmore College and Harvard University, MacPherson received his PhD from Harvard in 1970. His thesis, written under the direction of Raoul Bott, was entitled ''Singularities of Maps and Characteristic Classes''. Among his many PhD students are Kari Vilonen and Mark Goresky. Honors and awards In 1992, MacPherson was awarded the NAS Award in Mathematics from the National Academy of Sciences. In 2002 he and Goresky were awarded th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lawrence–Krammer Representation
In mathematics the Lawrence–Krammer representation is a representation of the braid groups. It fits into a family of representations called the Lawrence representations. The first Lawrence representation is the Burau representation and the second is the Lawrence–Krammer representation. The Lawrence–Krammer representation is named after Ruth Lawrence and Daan Krammer. Definition Consider the braid group B_n to be the mapping class group of a disc with ''n'' marked points, P_n. The Lawrence–Krammer representation is defined as the action of B_n on the homology of a certain covering map, covering space of the Configuration space (physics), configuration space C_2 P_n. Specifically, the first integral Homology (mathematics), homology group of C_2 P_n is isomorphic to \mathbb Z^, and the subgroup of H_1 (C_2 P_n,\mathbb) invariant under the action of B_n is primitive, free abelian, and of rank 2. Generators for this invariant subgroup are denoted by q, t. The cover ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digital Morse Theory
In mathematics, digital Morse theory is a digital adaptation of continuum Morse theory for scalar volume data. This is not about the Samuel Morse's Morse code of long and short clicks or tones used in manual electric telegraphy. The term was first promulgated by DB Karron based on the work of JL Cox and DB Karron. The main utility of a digital Morse theory is that it serves to provide a theoretical basis for isosurfaces (a kind of embedded manifold submanifold) and perpendicular streamlines in a digital context. The intended main application of DMT is in the rapid semiautomatic segmentation objects such as organs and anatomic structures from stacks of medical images such as produced by three-dimensional computed tomography by CT or MRI technology. DMT Tree A DMT tree is a digital version of a Reeb graph or contour tree graph, showing the relationship and connectivity of one isovalued defined object to another. Typically, these are nested objects, one inside another, giving a p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Morse Theory
Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces, homology computation, denoising, mesh compression, and topological data analysis. Notation regarding CW complexes Let X be a CW complex and denote by \mathcal its set of cells. Define the ''incidence function'' \kappa\colon\mathcal \times \mathcal \to \mathbb in the following way: given two cells \sigma and \tau in \mathcal, let \kappa(\sigma,~\tau) be the degree of the attaching map from the boundary of \sigma to \tau. The boundary operator is the endomorphism \partial of the free abelian group generated by \mathcal defined by :\partial(\sigma) = \sum_\kappa(\sigma,\tau)\tau. It is a defining property of boundary operators that \partial\circ\partial \equiv 0. In more axiomatic definitions one can find the requirement that \forall \sigma,\tau^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Level-set Method
Level-set methods (LSM) are a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. The advantage of the level-set model is that one can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects (this is called the ''Eulerian approach''). Also, the level-set method makes it very easy to follow shapes that change topology, for example, when a shape splits in two, develops holes, or the reverse of these operations. All these make the level-set method a great tool for modeling time-varying objects, like inflation of an airbag, or a drop of oil floating in water. The figure on the right illustrates several important ideas about the level-set method. In the upper-left corner we see a shape; that is, a bounded region with a well-behaved boundary. Below it, the red surface is the graph of a level set function \varphi determining this shape, and the flat blue regio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Manifolds
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
