Orbifolds, Higher Spin Curves, and Algebraic Structures

DMS-0204824 Takashi Kimura这个提议的目的是研究几何构造，这是由曲线的模空间与更高的自旋结构和曲线的模空间与映射到orbifolds之间的类比所启发的。前者与更高的KdV可积族有关，而后者与orbifolds的量子上同调有关。我们将分析由此产生的代数结构所建议的可积系统和这些空间的拓扑结构。KdV方程是一个非线性方程的行为的水波在一个直的渠道，是第一个层次的方程，其解决方案是相互兼容的。这样的层次结构和它们的推广与具有内部对称性的附加场的曲面密切相关。我们将调查这一关系及其后果。

DMS-0204824Takashi KimuraThe goal of this proposal is to investigate geometric constructions inspired by the analogy between the moduli space of curves together with higher spin structures and the moduli space of curves together with maps into orbifolds.The former is related to higher KdV integrable hierarchies while the latter is related to the quantum cohomology of orbifolds. We will analyze the resulting algebraic structures suggested by integrable systems and the topology of these spaces.The KdV equation is a nonlinear equation modeling the behavior of water waves in a straight channel and is the first of a hierarchy of equations whose solutions are mutually compatible. Such hierarchies and their generalizations are intimately related to surfaces endowed with additional fields possessinginternal symmetries. We will investigate this relationship and its ramifications.