Enhanced moduli, Hodge theory, and quantization

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This is a research in the field of algebraic geometry. The project fuses techniques from geometry and quantum field theory to unravel the hidden complexity of algebraic singularities, to extract new enumerative invariants of varieties, and to construct novel dualities in representation theory. Five problems will be studied. The first one aims to construct new algebraic invariants of moduli spaces and of symplectic singularities. The building and computation of these invariants requires a theoretical development of the formalism of shifted symplectic structures in non-commutative geometry. In the second project a new method is proposed for constructing motivic enumerative invariants by shifted quantization of the moduli of objects in differential graded categories of Calabi-Yau type. The third project analyzes the way in which non-commutative Hodge theory controls the deformations of Landau-Ginzburg models and proposes a very general unobstructedness theorem. The fourth project gives a strategy for reformulating and proving the classical limit Langlands duality as a purely topological duality for perverse sheaves. The last project will establish the functoriality of the ramified and the twisted non-abelian Hodge correspondences.The resolution of these questions will consolidate and demystify several existing quantization schemes in geometry, symplectic topology, and field theory. The project sets the stage for understanding the basic structure of algebraic varieties in a way suitable for pragmatic use in a broad spectrum of applications. Aside from the natural applications to algebraic topology, the work proposed will be immediately relevant to deep questions in category theory, geometric representation theory, the theory of integrable systems, string theory, gauge theory, quantum gravity and cosmology. The project outlines concrete cross-discipline applications to the physics of mirror symmetry, renormalization group flow, and the quantization of gauge theories. This project also aims to organize a concentrated effort on enhancing and building a new geometric arsenal of techniques applicable to the theory of algebraic cycles, symplectic topology, and high energy physics. This will be achieved by training a group of young researchers, and graduate students in mathematics and physics, and by a curriculum development of a course on derived symplectic geometry, and a course on non-commutative Hodge theory. Specific research opportunities on the interface of geometry and string theory for graduate students and postdocs are discussed. The proposed work will be disseminated through talks at multidisciplinary conferences, research seminars and publications in peer reviewed scientific journals.