Singularity theory
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In
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. These situations are called perestroika (Russian: перестройка), bifurcations or catastrophes. Classifying the types of changes and characterizing sets of parameters which give rise to these changes are some of the main mathematical goals. Singularities can occur in a wide range of mathematical objects, from matrices depending on parameters to wavefronts.[1]
How singularities may arise
In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes. Projection is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our eyes); in looking at classical statuary the folds of drapery are amongst the most obvious features. Singularities of this kind include caustics, very familiar as the light patterns at the bottom of a swimming pool.
Other ways in which singularities occur is by degeneration of manifold structure. The presence of symmetry can be good cause to consider orbifolds, which are manifolds that have acquired "corners" in a process of folding up, resembling the creasing of a table napkin.
Singularities in algebraic geometry
Algebraic curve singularities
Historically, singularities were first noticed in the study of algebraic curves. The double point at (0, 0) of the curve
and the cusp there of
are qualitatively different, as is seen just by sketching.
It was then a short step to define the general notion of a singular point of an algebraic variety; that is, to allow higher dimensions.
The general position of singularities in algebraic geometry
Such singularities in
The smooth theory and catastrophes
At about the same time as Hironaka's work, the
Arnold's view
While Thom was an eminent mathematician, the subsequent fashionable nature of elementary
Duality
An important reason why singularities cause problems in mathematics is that, with a failure of manifold structure, the invocation of
Other possible meanings
The theory mentioned above does not directly relate to the concept of
See also
Notes
- ^ Arnold, V. I. (2000). "Singularity Theory". www.newton.ac.uk. Isaac Newton Institute for Mathematical Sciences. Retrieved 31 May 2016.
- ^ Arnold 1992
References
- V.I. Arnold (1992). Catastrophe Theory. Springer-Verlag. ISBN 978-3540548119.
- E. Brieskorn; H. Knörrer (1986). Plane Algebraic Curves. Birkhauser-Verlag. ISBN 978-3764317690.
- R. Abraham and J. Marsden (1987). Foundations of Mechanics, Second Edition. Benjamin/Cummings Publishing Company.