New Hodge theoretic invariants in geometry and physics

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This is a research in the field of algebraic geometry. The project addresses four problems providing novel interfaces between complex geometry and string theory and quantum physics. The first problem aims to construct new Hodge theoretic invariants of symplectic and complex manifolds. The building and computation of these invariants requires the development of new foundational formalism of Deligne cohomology, Griffiths groups, and normal functions in non-commutative geometry. The second problem seeks a new method for analysing the fine geometric structure of the moduli of objects in differential graded categories of Clalabi-Yau type. The third problem applies the method to a problem in symplectic topology and introduces a new concrete geometric description of the Fukaya category. The fourth problem addresses the construction of the mirror map for del Pezzo surfaces, and describes a strategy for proving the homological mirror symmetry conjecture in this context. The understanding of these questions is essential for unifying various linearization procedures in algebraic geometry, symplectic topology, theoretical and mathematical physics. 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 geometry and topology, the work proposed will be immediately relevant to deep questions in quantum gravity and cosmology. The project outlines concrete interdisciplinary applications to string dualities and the quantization of three dimensional field theories.The 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 Mirror Symmetry, non-commutative geometry, and algebraic constructions of Fukaya categories. Specific research opportunities on the interface of geometry and string theory for graduate students and postdocs are also discussed.