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В четверг, 21 декабря 2006 года, в 15:40 в конференц-зале НМУ, Б. Власьевский 11, состоится доклад: Topics in max-plus or tropical spectral theory. Лектор – Stephane Gaubert (INRIA, France).

In the max-plus world, the addition is replaced by the maximum, and the multiplication is replaced by the usual addition. This world, in which many a familiar result has an analogue, has been brought to light by several schools under various names (idempotent analysis, tropical algebra, max-algebra, extremal algebra, …).

In this talk, I will present some recent developments concerning a basic max-plus theme: spectral theory.

One of the main motivations of max-plus spectral theory is the classical spectral theory, since the max-plus eigenproblem can be obtained as the limit of a «deformation» of the classical eigenproblem. The max-plus analogues of the eigenvalues can be found by solving a family of optimal assignment problems (i.e., special instances of the Monge-Kantorovitch mass transportation problem). I will show how max-plus techniques can be applied to solve some of the singular cases of a classical theorem in perturbation theory due to Vishik, Lusternik and Lidskii.

A second motivation comes from optimal control, or variations calculus: Maslov pointed out that the evolution semigroups of the Hamilton-Jacobi PDE associated to deterministic optimal control problems (Lax-Oleinik semigroups) are max-plus linear. I will discuss max-plus spectral problems over a non-compact spate space. The max-plus analogues of harmonic functions, which are the fixed points of Lax-Oleinik semigroups, can be determined by imitating the Martin representation arising in classical potential theory. The analogue of the Martin compactification turns out to coincide with the compactification of metric spaces by horofunctions introduced by Gromov. The harmonic functions which are extremal in the sense of max-plus convex analysis can be identified to the limits of semi-geodesics (Busemann points). An application to the representation of solutions of Hamilton-Jacobi PDE over a non-compact domain will be given.

A last motivation comes from non-linear Perron-Frobenius/Krein-Rutman theory. I will briefly discuss the extension of max-plus results to some non-linear order preserving maps arising in the study of stochastic control and game problems.

The materials of this talk are taken from joint works with several coauthors, specially M.Akian, R.Bapat, and C.Walsh.

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