FRG Collaborative Research: Homological Mirror Symmetry and its applications

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The main aim of this collaborative project is an in-depth study ofthe homological mirror symmetry conjecture. Auroux, Katzarkov,Kontsevich, Orlov and Seidel will lead a concerted effort to formulateand understand homological mirror symmetry in systematic manner, andextend 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 toinvestigate applications of mirror symmetry to classical problems inalgebraic geometry (for example studying the rationality of certainalgebraic varieties) and symplectic topology (in particular, Lagrangiansubmanifolds). From a wider perspective, the project aims to provide a mathematicalcounterpart to some recent advances in theoretical physics. Thecontribution that mathematics can make is to verify the soundness andconsistency of physical intuition, and to prepare the general groundon which further development can occur. This is particularly importantin those situations where developments in physics suggest the presenceof deep and complex structures, which are difficult to detect by directexperiment. At the same time, this effort will make it possible to answersome purely mathematical (geometric) questions, some of which have beenopen for a long time. The collaborative effort will be carried out through regular meetingsand workshops, and by fostering interaction between leading experts inthe field and younger mathematicians (graduate students and postdocs);dissemination of knowledge in this rapidly evolving area of mathematicswill be facilitated by regularly held winter and summer schools andconferences.