This research will develop a relative Hodge theory of families of complex algebraic varieties. More precisely, there is associated to any variety a category of objects called stratified variations of Hodge structure. Given an algebraic map, the data of the cohomology along the fibers together with their mixed Hodge structures should give rise to such an object. The principal investigator will examine this situation, as well as consider the category of polarizable mixed Hodge structures. This is research in the field of algebraic geometry, which is one of the oldest parts of modern mathematics. 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 also from analysis and topology. It is finding application in those areas of mathematics as well as in theoretical computer science and in robotics.
