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.
I discovered and developed new methodology for linearizing complex mathematical models and quantative systems. I applied this methodology to study and solve fundamental problems in geometry, classical and quantum field theory, and high energy particle physics. Four projects were completed. In the first one I constructed new Hodge theoretic invariants of symplectic and complex manifolds. To build and compute these invariants I constructed the deformations of spaces with potentials and their tame compactifications and described their symplectic mirrors. In the second project I introduced the new concept of a shifted symplectic structure and developed a toolbox for analysing the intricate geometry of the parameter spaces for boundary conditions. In the third project I applied this toolbox to quantize the space of objects in Fukaya categories and to capture these spaces as critical points of global potentials. In the fourth project I built a geometric realization of the Standard Model of particle physcs based on an F-theory compactification with a Wilson line symetry breaking. 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 common zeroes of polynomial equations 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 category theory, combinatorial group theory, the theory of integrable systems, string theory, quantum gravity and cosmology. The project outlines concrete interdisciplinary applications to string dualities and the quantization of three dimensional field theories. While completing this research program I organized 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 was achieved by training three postdoctoral researchers, six Ph.D. and two M.Sc students in mathematics and physics, and by a curriculum development of courses on Mirror Symmetry, non-commutative geometry, and algebraic constructions of Fukaya categories. The work was disseminated through talks at multidisciplinary conferences, courses at graduate schools, research seminars, and publications in peer reviewed scientific journals.
