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Tropical Structures in Geometry and Physics |
| November 30, 2009 to December 04, 2009 |
| Organized By: Mark Gross ( University of California San Diego), Kentaro Hori (University of Toronto), Viatcheslav Kharlamov (Université de Strasbourg (Louis Pasteur), Richard Kenyon* (Brown University) |
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| Parent Programs: |
| Tropical Geometry |
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| Return to Workshop Description |
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Dequantization and tropical structures in classical mechanics and classical geometry
Thursday December 3, 2009
04:15PM - 05:15PM
Speakers:
Grigory Litvinov
Abstract:
Tropical mathematics can be treated as a result of a
dequantization of the traditional mathematics as the Planck
constant tends to zero taking imaginary values. This kind of dequantization is known as the Maslov dequantization and it leads to a
mathematics over tropical algebras like the max-plus algebra. For example,tropical algebraic geometry can be treated as a result of the
Maslov dequantization applied to traditional algebraic geometry ( O. Viro, G. Mikhalkin and others). The Maslov dequantization generates other
interesting and useful dequantization procedures.
In the spirit of N.Bohr's correspondence principle there
is a (heuristic) correspondence between important, useful, and interesting constructions and results over fields and similar results over
tropical algebras. A systematic application of this correspondence principle leads to a variety of theoretical and applied results.
Last time the Maslov dequantization and related dequantization procedures are applied to different concrete mathematical objects and
structures.
Examples:
1. The Hamilton-Jacobi equation (which is the main equation of classical mechanics) can be treated as a result of the Maslov dequantization
of the Schroedinger equation (G.L. Litvinov and V.P. Maslov). The Hamilton-Jacobi uation is linear over tropical algebras (V.P. Maslov).
2. The Legendre transform can be treated as a result of the Maslov dequantization of the Fourier-Laplace transform (V.P. Maslov).
3.If F is a polynomial, then a dequantization procedure leads to the Newton polytope of F. Using the so-called dequantization transform it is
possible to generalize this result to a wide class of functions and convex sets (G.L. Litvinov and G.B. Shpiz).
4.An application of a dequantization procedure to metrics leads to the Hausdorff-Besicovich dimension including the fractal dimension. An
application of a dequantization procedure to measures and differential forms leads to a notion of dimension at a point. This dimension can be
real-valued,e.g. negative (G.L. Litvinov and G.B. Shpiz).
Other ezamples will also be examined.
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