This award supports research in algebraic geometry, a central area of mathematics. The investigator will study certain geometric properties of spaces that parameterize mathematical objects characterized by algebraic properties. One such space parameterizes one-complex-dimensional sub-objects in an algebraic space; the other space parameterizes fields on an algebraic space. These are important constructions to study in mathematics, in part due to their intrinsically beautiful structures and the applications of these structural results to geometry, topology, and representation theories, and in part due to the fact that these spaces are mathematical analogues of objects (fields, strings) in theoretical physics. The completion of this project will improve our understanding of mathematics and of theoretical physics in general. This research project addresses the long-term goal of broadening mathematical research by understanding new ideas from theoretical physics, and contributing to the development of theoretical physics by providing the mathematical foundations vital to its advancement.
This research project will develop a new theory to study all genus Gromov-Witten invariants of the quintic Calabi-Yau threefolds. This theory is a geometric realization of that envisioned in superstring theories. It will provide new avenues to study similar moduli spaces in algebraic geometry. The investigator will further develop a theory to understand the generalized Donaldson-Thomas invariants of Calabi-Yau threefolds through the angle of Kirwan desingularizations, and to make an inroad to understand the recently constructed perverse sheaves on the moduli of sheaves on Calabi-Yau threefolds.
