Objective
The aim of this project is the comprehensive study of discriminant sets and their complements with the emphasis on their analytical and differential properties and on adjacent problems. Many problems in analysis, algebraic geometry, topology etc. may be reduced to the study of discriminant sets. A lot of problems in mechanics and optics may be naturally formulated in terms of Lagrangian and/or Legendre maps, which caustics and fronts are also discriminants. In papers by V.A.Vassiliev there was discovered a close relation between the topology of discriminants and the characteristic classes and invariants of geometric objects. The construction of Vassiliev invariants in knot theory was based on the study of the discriminant in the space of immersions of the circle into the space. Those invariants attracted a high interest and initiated a lot of articles.
In the study of the discriminant sets both real and complex cases are of great interest. In the real case the discriminant divides the sets of objects with constant topological properties. In the complex case the connectedness of the complement to the discriminant implies the independence of discrete invariants of generic objects, which may be calculated in terms of Newton polyhedra. The theory of Newton polyhedra connects the algebraic geometry, analysis and topology with the theory of convex bodies.
The main jects of the proposed research are the following:
Global topology of discriminants and invariants,
Topological invariants of singular points of varieties and maps,
Newton polyhedra,
Study of topological and differential structures connected with Lagrangian
and Legendre manifolds and maps,
Topology of complements of singular hypersurfaces. The work at these directions are carried out in Russia mainly by Arnold's school, in Western Europe, USA and Japan. Remarkable results on the jects mentioned above were obtained in the Netherlands, Germany, France, Great Britain, Denmark.
This project is aimed at the intensification of scientific collaboration between the groups from Russia and Georgia and the scientists from these countries and at the carrying out the joint researches.
Topic(s)
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3508 TA Utrecht
Netherlands