 He introduced the notion of quantum Weyl group, studied algebras of functions on compact quantum groups, and meromorphic braided monoidal categories.His recent work (mainly in collaboration with Maxim Kontsevich) is devoted to various algebraic and differential-geometric aspects of homological mirror symmetry, including a proof of Deligne conjecture about operations on the cohomological Hochschild complex, and a direct construction of Calabi-Yau varieties based on SYZ conjecture and non-archimedean geometry.In a recent preprint a new hypothetical wall-crossing formula for changes of BPS states was proposed.