Gromov-Witten and Donaldson-Thomas theories in dimensions two and three

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String theory, a branch of physics, predicts that the building blocks of spacetime (which is the fabric of the universe) are certain geometric objects called Calabi-Yau manifolds. The PI will focus on the investigation of the geometric properties of these spaces, mainly by computing important invariants associated to them, called Gromov-Witten and Donaldson-Thomas invariants. Both Gromov-Witten (GW) and Donaldson-Thomas (DT) theories are inspired by theoretical physics and involve several mathematical subjects, including geometry, topology, algebra, combinatorics, and representation theory. The principal investigator will continue his research on the curious conjectural correspondences between GW and DT theories and will try to discover their links to these branches of mathematics and physics. The investigator plans to use the framework provided by these disciplines to create and teach courses for and mentor high school, undergraduate, and graduate students, as well as postdoctoral researchers.More specifically, the principal investigator will investigate the DT invariants of 2-dimensional sheaves in Calabi-Yau threefolds inspired by "M5-brane elliptic genera" and "BPS invariants of D4-D2-D0 systems" studied by string theorists. The PI plans to prove 1) the modularity of these DT invariants and 2) find their relation to the "BPS invariants of D6-D2-D0 systems" predicted from dualities in string theory. The latter BPS invariants are manifested in DT invariants of 1-dimensional sheaves as well as in GW invariants. The PI also plans to develop an algorithm for computing the rank 2 DT invariants of toric threefolds and then study their properties. This is known as the "Topological Vertex Algorithm" and has been a very powerful tool for computing GW and rank 1 DT invariants of the toric threefolds. The advanced tools that will be employed for these projects are localization, deformation, degeneration, categorification, wall-crossing, and mirror symmetry techniques.