The main aim of this collaborative project is an in-depth study of the homological mirror symmetry conjecture. Auroux, Katzarkov, Kontsevich, Orlov and Seidel will lead a concerted effort to formulate and understand homological mirror symmetry in systematic manner, and extend it to varieties of general type and to noncommutative varieties. This will require some foundational work in homological algebra, noncommutative geometry, and symplectic geometry. Another goal is to investigate applications of mirror symmetry to classical problems in algebraic geometry (for example studying the rationality of certain algebraic varieties) and symplectic topology (in particular, Lagrangian submanifolds).
From a wider perspective, the project aims to provide a mathematical counterpart to some recent advances in theoretical physics. The contribution that mathematics can make is to verify the soundness and consistency of physical intuition, and to prepare the general ground on which further development can occur. This is particularly important in those situations where developments in physics suggest the presence of deep and complex structures, which are difficult to detect by direct experiment. At the same time, this effort will make it possible to answer some purely mathematical (geometric) questions, some of which have been open for a long time. The collaborative effort will be carried out through regular meetings and workshops, and by fostering interaction between leading experts in the field and younger mathematicians (graduate students and postdocs); dissemination of knowledge in this rapidly evolving area of mathematics will be facilitated by regularly held winter and summer schools and conferences.
