The proposal contains two sections. Both are concentrated on the theory of variations of Hodge structures and singular germs of analytic maps. The first section is a continuation of the work of the principal investigator and J. Steenbrink. They computed the spectral pairs associated with a curve singularity with coefficients in a variation of Hodge structure with an abelian monodromy group. The PI plans to generalize this result for arbitrary dimension and for more general variations (e.g. variations with solvable monodromy group). As a first step, the PI plans to investigate the existence of the limit mixed Hodge structure for such a variations. In the second section, the PI studies a deformation of an isolated complete intersection singularity whose discriminant is a divisor with normal crossings. He constructs a limit mixed Hodge structure on the vanishing cohomology and proves the local analogs of several results of Cattani, Kaplan and Schmid, which correspond to the local analog of the "algebraic characterization" of the Nilpotent Orbit Theorem. The PI proposes to find also the geometric local analog of the classical (global) Nilpotent Orbit Theorem. This research is in the field of algebraic geometry and singularity theory. Algebraic geometry is one of the oldest parts of the modern mathematics, but one which has had a revolutionary flowering in the last thirty years. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays, the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics. Singularity theory studies the singular (exotic) points of the figures studied by the algebraic geometry. Even elementary results of the theory have surprising applications in physics, in the theory of dynamic systems, in chaos and catastrophe theory, and even in psychology.
