Schedule, abstracts and list of participants: [ PS : PDF ]
| Organizers: | V.P. Maslov (Moscow) | islc@dol.ru | G.L. Litvinov (Moscow) | glitvinov@mail.ru |
The workshop is concerned with Idempotent Mathematics (or Idempotent Analysis, or Idempotent Calculus) and its applications, especially in Mathematical Physics. Idempotent Mathematics is a new mathematical area. It is based on replacing the usual arithmetic operations by a new set of basic operations (such as maximum or minimum). There is a lot of such new arithmetics, which are associated with sufficiently rich algebraic structures called idempotent semirings (i.e. semirings with idempotent addition; this means that x+x=x). One of the most important examples is the well-known max-plus algebra.
In a sense, the traditional Mathematics over numerical fields can be treated as a quantum theory, while the Idempotent Mathematics can be treated as a `classical shadow (or counterpart)' of the traditional one. There exists the corresponding procedure of an idempotent dequantization. This dequantization is based on the so-called logarithmic transform used by E. Schroedinger (1926) and E. Hopf (1950). In this case the parameter of the dequantization coincides with the Planck constant taking pure imaginary values. A similar idea (a passage to the imaginary time) is used in the Euclidean quantum field theory (there is the well-known duality between energy and time).
There exists a correspondence between interesting, useful and important constructions and results in the traditional Mathematics and similar constructions and results in Idempotent Mathematics. This heuristic correspondence can be formulated in the spirit of the well-known N. Bohr's correspondence principle in Quantum Mechanics; in fact, the two principles are intimately connected. For example, the Hamilton--Jacobi equation is an idempotent version of the Schrödinger equation, the variational principles of Classical Mechanics can be treated as an idempotent version of the Feynman path integral approach to Quantum Mechanics. The representation of solutions to the Schroedinger equation in terms of the Feynman integral corresponds to the Lax--Oleinik representation of solutions to the Hamilton--Jacobi equation. The Legendre transform turns out to be an idempotent version of the Fourier transform etc. A systematic and consistent application of the idempotent correspondence principle leads to a variety of results, often quite unexpected.
The abstract theory is well advanced and includes, in particular, a new integration theory, linear algebra and spectral theory, idempotent functional and harmonic analysis etc. Its applications include important problems in Mathematical Physics and Differential Equations, various optimization problems such as multi-criteria decision making, optimization on graphs, discrete optimization with a large parameter (asymptotic problems), optimal design of computer systems and computer media, optimal organization of parallel data processing, dynamic programming, applications to numerical analysis, discrete event systems, computer science, discrete mathematics, mathematical logic, etc.
The main objective of the workshop is to enhance collaboration between different scientific groups in the world working on the methods in Idempotent Mathematics and their applications in different areas including Mathematical Physics, Differential Equations, Optimization, Analysis and Numerical Analysis, stochastic problems, computer applications.
| FEBRUARY 3, MONDAY | |
| 10.00 -- 13.00 | Registration (in the Institute) |
| 13.00 -- 15.00 | Lunch |
| 15.00 -- 15.10 | Official start |
| 15.10 -- 16.00 | G. Litvinov (joint talk with V. Maslov): "IDEMPOTENT MATHEMATICS AND MATHEMATICAL PHYSICS" |
| 16.15 -- 16.50 | E. Pap: "APPLICATIONS OF GENERAL PSEUDO-ANALYSYS ON NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS" |
| 17.00 --17.40 | J.P. Quadrat (joint talk with P. Lotio and E. Mancinelli): "GIBBS SEMIRINGS AND TRAFFIC ASSIGNMENT" |
| FEBRUARY 4, TUESDAY | |
| 9.30 -- 10.20 | M. Gondran: "SCHROEDINGER DEMONSTRATION IN MINPLUS COMPLEX ANALYSIS" |
| 10.30 -- 11.20 | O. Viro: "DEQUANTIZATION OF REAL ALGEBRAIC GEOMETRY" |
| 11.30 -- 12.20 | G. Mikhalkin: "TROPICAL GROMOV-WITTEN INVARIANTS" |
| 12.20 -- 14.30 | Lunch |
| 14.30 -- 15.20 | F. Baccelli: "ASYMPTOTIC ANALYSIS OF STOCHASTIC NETWORKS AND TOPICAL MAPS" |
| 15.30 -- 16.20 | A. Di Nola: "ALGEBRAS OF LUKASIEWICZ'S LOGIC AND THEIR SEMIRING REDUCTS" |
| 16.30 -- 17.10 | K. Zimmermann: "MAX-SEPARABLE OPTIMIZATION PROBLEMS" |
| FEBRUARY 5, WEDNESDAY | |
| 9.30 -- 10.20 | P. Butkovic: "MAX-ALGEBRA: THE LINEAR ALGEBRA OF COMBINATORICS?" |
| 10.30 -- 11.20 | G. J. Olsder (joint talk with J. van der Woude): "ON (MIN,MAX,+)-INEQUALITIES" |
| 11.30 -- 12.10 | M. Akian (joint talk with R. Bapat and S. Gaubert): "PERTURBATION OF EIGENVALUES AND MIN-PLUS ALGEBRA" |
| 12.10 -- 14.30 | Lunch |
| 14.30 -- 15.20 | C. Walsh (joint talk with M. Akian and S. Gaubert): "MAX-PLUS MARTIN BOUNDARIES" |
| 15.30 -- 16.20 | A. Baklouti (joint talk with S. Dhieb and D. Manchon): "DEQUANTIZATION OF COADJOINT ORBITS AND CHARACTERISTIC MANIFOLDS" |
| 16.30 -- 17.10 | I. Roublev: "ON TWO NOTIONS OF GENERALIZED SOLUTION TO THE HAMILTON-JACOBI EQUATION" |
| FEBRUARY 6, THURSDAY | |
| 9.30 -- 10.20 | V. Kolokoltsov (joint work with S. Gaubert and M. Akian): "IDEMPOTENT ANALYSIS AND GALOIS CONNECTIONS" |
| 10.30 -- 11.20 | S. Gaubert: "EIGENVECTORS OF MONOTONE HOMOGENEOUS MAPS" |
| 11.30 -- 12.10 | G. Litvinov (joint work with G. Shpiz): "NUCLEAR SEMIMODULES AND KERNEL THEOREMS IN IDEMPOTENT ANALYSIS. AN ALGEBRAIC APPROACH" |
| 12.10 -- 14.30 | Lunch |
| 14.30 -- 15.20 | G. Mascari: "HIGHER DIMENSIONAL IDEMPOTENT MATHEMATICS: IDEMPOTENT MATHEMATICS AND HIGHER DIMENSIONAL ALGEBRA" |
| 15.30 -- 16.00 | M. Passare: "AMOEBAS AND THEIR SPINES" |
| 16.05 -- 16.35 | A. Tsikh: "AMOEBAS, THEIR CONTOURS, AND MULTIVARIATE ASYMPTOTICS" |
| 16.40 -- 17.10 | O. Gulinski: "QUANTUM LARGE DEVIATIONS AND IDEMPOTENT MEASURES" |
| FEBRUARY 7, FRIDAY | |
| 9.30 -- 10.20 | G. Cohen (joint talk with S. Gaubert, J.-P. Quadrat, and I. Singer): "MAX-PLUS CONVEX FUNCTIONS" |
| 10.30 -- 11.20 | I. Singer (joint talk with M. Akian) "DOWNWARD SETS AND CONJUGATIONS IN CONTINUOUS CONDITIONALLY COMPLETE LATTICE ORDERED GROUPS" |
| 11.30 -- 12.10 | A. Sobolevskii: "POSITIVE SEMIRINGS, IDEMPOTENT MATHEMATICS AND INTERVAL ANALYSIS" |
| 12.10 -- 14.30 | Lunch |
| 14.30 -- 5.20 | K. Khanin (joint talk with D. Khmelev and A. Sobolevskii): "LAGRANGIAN VARIATIONAL PROBLEMS IN UNBOUNDED DOMAINS AND BURGERS TURBULENCE" |
| 15.30 -- 16.05 | D. McCaffrey: "LAGRANGIAN MANIFOLDS, VISCOSITY SOLUTIONS AND MASLOV INDEX" |
| 16.15 -- 16.50 | A. Churkin: "OBJECT ORIENTED SOFTWARE FOR BASIC PROBLEMS OF IDEMPOTENT LINEAR ALGEBRA" (a computer demonstration) |
| FEBRUARY 8, SATURDAY | |
| 10.00 -- 10.50 | A. Soboleevskii: "IDEMPOTENT ANALYSIS, THE WEAK KAM THEORY, AND MONGE-KANTOROVICH MASS TRANSPORTATION ON SMOOTH MANIFOLDS" |
| 11.00 -- 11.30 | P. Loreti: "DYNAMIC PROGRAMMING: AN IDEMPOTENT APPROACH I." |
| 11.30 -- 12.00 | M. Pedicini: "DYNAMIC PROGRAMMING: AN IDEMPOTENT APPROACH II." |
| 12.00 -- 14.00 | Lunch |
| 14.00 -- 14.35 | E. Wagneur: "DECOMPOSITION OF IDEMPOTENT SEMIMODULES" |
| 14.40 -- 15.15 | V. Cafagna: "FOLDING FUNCTIONS ON AN INTERVAL: ALGEBRAIC AND ANALYTICAL ASPECTS" |
| 15.20 -- 15.55 | Z. Hucki: "GAME THEORETICAL APPROACH TO PRICING OF OPTIONS DEPENDING ON SEVERAL COMMON STOCKS" |
| FEBRUARY 10, MONDAY | |
| 9.30 -- 10.20 | E. Pap: "A GENERALIZATION OF THE UTILITY THEORY USING A HYBRID IDEMPOTENT-PROBABILISTIC MEASURE" |
| 10.30 -- 11.10 | G. Shpiz (joint work with G. Litvinov): "REPRESENTATIONS OF GROUPS IN IDEMPOTENT SPACES AND ENGEL TYPE THEOREMS" |
| 11.20 -- 11.30 | Official finish |