This award supports research that aims to further build the bridge between abstract mathematics and theoretical physics (in particular string theory). This relationship follows naturally from the concept of supersymmetry, which postulates that at very small length scales there is an exact parity between bosonic and fermionic particles in nature. The resulting algebraic structure leads to new and deep relations between string theory and black hole entropy on one side, and abstract algebraic geometry and topology on the other. This unique interaction leads in turn to important advances and novel ideals in both disciplines. It also promotes a new way of thinking combining mathematical rigor with the physical insight and flexibility of string theory. The goal of this project is to make significant advances in the refined Donaldson-Thomas theory of orbifolds as well as develop a new approach to cohomological invariants of moduli spaces of sheaves on Calabi-Yau threefolds. The first direction aims to prove new combinatorial formulas for orbifold refined stable pairs invariants using localization and wall-crossing. This is an important component of at least two current major open problems, concerning the cohomology of tamely ramified character varieties, as well as an algebraic geometric construction of knot invariants. The second part of the project aims to develop a new approach to the cohomology of moduli spaces of two dimensional sheaves on Calabi-Yau threefolds using string duality. The main idea is to build a concrete relation between the cohomology of such moduli spaces sheaves and chiral algebras on Laumon spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
|Effective start/end date||8/1/18 → 7/31/21|
- National Science Foundation (National Science Foundation (NSF))