Recursive formulas for Gromov-Witten invariants

奖：DMS-0071393首席研究员：Eleny-Nicoleta Ionel第一个项目的目标是在有标记的Riemann曲面上复杂结构的模空间的同调中获得新的关系。利用她早期工作中开发的技术，PI发现了几个有趣的关系，其中一个可以证明关于自体环生成器的费伯猜想。第二个项目试图找到相对和绝对Gromov-Witten不变量之间的关系。这种关系似乎在镜像猜想计算中很有用，在计量学的其他几个公开问题中也是如此。最后一个项目提出了两种将Gromov型不变量扩展到“非一般”情况的方法。这将提供关于辛流形的更精细的信息。列举代数几何中的大多数问题都有一百多年的历史了。这些问题很容易问，但使用经典方法解决这些问题的进展仍然很慢。最近，在高能物理的二维拓扑量子场理论中也出现了类似的问题。在这些理论的启发下，新的方法在过去的几年里在该领域取得了惊人的进步。本文探索了处理这些老问题的两种新方法，这将进一步澄清二维拓扑量子场理论的结构。

Award: DMS-0071393Principal Investigator: Eleny-Nicoleta IonelThe goal of the first project is to obtain new relations in thecohomology of the moduli space of complex structures on a markedRiemann surface. Using techniques developed in her earlier work,the PI found several interesting relations, one of which couldprove the Faber conjecture about the generators of thetautological ring. The second project seeks to find relationsbetween the relative and absolute Gromov-Witten invariants. Suchrelations appear to be useful in mirror conjecture computations,as well as in several other open problems in enumerativegeometry. The final project suggests two ways of extending theGromov-type invariants to `nongeneric' situations. This wouldprovide more refined information about the symplectic manifold. Most of the problems in enumerative algebraic geometry are morethan a hundred years old. The questions are easy to ask, but theprogress in solving them using classical methods has been quiteslow. Recently, the same kind of questions arised in twodimensional topological quantuum field theories from high energyphysics. Inspired by these theories, new methods lead in the pastcouple of years to amazing progress in the field. The proposalexplores two new ways of approaching these old problems thatwould further clarify the structure of the two dimensionaltopological quantuum field theories.