Enumerative geometry of Hilbert schemes

本课题的主要目的是研究低维数列的模空间的几何性质，并揭示该学科与其他数学领域的可能联系。著名的Gromov-Witten/Donaldson-Thomas对应关系是三次元曲线Hilbert格式上的全称理想束特征类的积分与映射到三次元的模空间上的积分之间的一种推测对应关系。证明了该猜想适用于环三折和最低次非平凡特征类。提出的项目旨在将已知结果扩展到环面情况之外，并研究更高度特征类的情况。项目的另一部分致力于研究奇异平面曲线上点的希尔伯特格式的拓扑结构。PI和V. Shende提出了Hilbert格式的庞加莱多项式在曲线奇异点的连杆不变量下的一个猜想公式。证明这个公式是该项目的目标之一。轮轴的模空间是场论的数学模型。数学物理预测，如场论和弦论之间的对偶性或结不变量的路径积分公式，可以转化为迷人的数学猜想。这个项目的目标是找到其中一些猜想的证明。

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.