The proposal is concern with the mathematical study of the duality between String Theory and Gauge theory. One of the mathematical manifrestation of the duality is the conjecural Gromov-Witten/Donaldson-Thomas duality due to Maulik, Nekrasov, Okounkov and Pandharipande. The conjecture relates the generating function for the integrals over the module space of stable curves in the thefolds to the generating function for the integrals over Hilbert scheme of curves in the threfold. At the moment we are able to prove the conjecture for the toric threefolds. To prove the conjecture for the general threefold the relative version of the duality need to be addressed. This is the focus of our research. We also hope to explore the connections between Gromov-Witten/Donaldson-Thomas theory and the quantum cohomology of the Hilbert scheme of points on the surface. At the moment these relations are fully understood only for the surfaces which are resolutions of the simple singularities.
It is a classical problem in geometry to compute the number of curves satisfying some natural geometric conditions. For example, one can ask how many lines in three-dimensional space intersect four given lines. In this case the answer is either infinity, two or one (depending on the position of thefour lines). The first advances this type of problems are due to 19 century italian algebraic geometers. Recent advances are motivated by the new insights coming from physics. In particular String theory and Gauge theory give two ways of looking at the curves in three-dimensional space. The first theory treats the curves as the traces of the flying closed strings, for the second theory the curve is defined by simultanious vanishing of two equations. Interplay between this two approaches is in the center of our research.
