Enumerative geometry of Hilbert schemes

这个项目的主要目标是研究低维簇上层的模空间的几何，并揭示该主题与其他数学领域的可能联系。著名的Gromov-Witten/Donaldson-Thomas对应是三重曲线的Hilbert概型上的泛理想层的特征类的积分与三重映射的模空间上的积分之间的一种拓扑对应。证明了该猜想的复曲面三重和最低程度的非平凡特征类。该项目提出的目的是扩展已知的结果超出环面的情况下，研究的情况下，更高程度的特征类。该项目的另一部分致力于研究奇异平面曲线上的点的希尔伯特方案的拓扑。PI和V. Shende根据曲线奇点的链环的链环不变量，提出了Hilbert格式的Poincare多项式的一个拓扑公式。证明这个公式是该项目的目标之一。层的模空间是场论的数学模型。数学物理预言，如场论与弦论之间的对偶或纽结不变量的路径积分公式，可以转化为迷人的数学公式。这个项目的目标是找到其中一些假设的证明。

The main objective of this project is to study the geometry of moduli spaces of sheaves on low dimensional varieties and reveal possible connections of the subject with other fields of mathematics. The celebrated Gromov-Witten/Donaldson-Thomas correspondence is a conjectural correspondence between the integrals of characteristic classes of the universal ideal-sheaf over the Hilbert scheme of curves in the threefold and integrals over the moduli space of maps into the threefold. The conjecture is proven for toric threefolds and the lowest degree nontrivial characteristic classes. The project proposed is intended to extend the known results beyond the toric case and study the case of higher degree characteristic classes. Another part of the project is devoted to studying the topology of the Hilbert scheme of points on singular planar curves. A conjectural formula was proposed by the PI and V. Shende for the Poincare polynomials of the Hilbert scheme in terms of link invariants of the links of the singularities of the curve. To prove the formula is one of the goals of the project.Moduli spaces of sheaves are mathematical models for the field theories.Mathematical physics predictions, such as duality between the field theories and string theory or path integral formulas for knot invariants, can be translated into fascinating mathematical conjectures. The goal of this project is to find a proof of some of these conjectures.