Geometric applications of dualities

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This is a research in the field of algebraic geometry - a classical subject studying the solutions to systems of polynomial equations. The project addresses four problems providing novel interfaces between complex geometry and string theory and quantum physics. The first one outlines a new way to extract Hodge theoretic invariants directly from the sheaf theory of commutative or noncommutative spaces. The formal structure of these linear entities will be studied and through the physical notion of quantum mirror symmetry used to produce new invariants of symplectic manifolds. In the second project a new method is proposed for proving the K-equivalence conjecture in birational geometry. The third project concerns the problem of deformation quantization of geometric dualities and symmetries in the complex analytic context. The fourth project analyzes the supersymmetry constraints for D-branes with non-trivial fluxes wrapping algebraic cycles in Calabi-Yau threefolds. The large N quantization of the branes is probed in the context of algebraically completely integrable systems.The understanding of these questions is essential for unifying variouslinearization procedures in algebraic geometry, symplectic topology,theoretical and mathematical physics. The project sets the stage forunderstanding the basic structure of algebraic varieties in a waysuitable for pragmatic use in a broad spectrum of applications. Theproject outlines concrete interdisciplinary applications to matrixquantum mechanics, string dualities and topological black holes.This project also aims to organize a concentrated effort on enhancingthe geometric arsenal of techniques used in high energy physics andcondensed matter theory. This will be achieved by training a group ofyoung researchers, and graduate and undergraduate students inmathematics and physics, and by a curriculum development of courses onHodge theory, non-commutative geometry and mirror symmetry, on thegraduate and undergraduate level.