Geometry and Topology of Manifolds with Special Holonomy

具有特殊完整和标定几何的流形与辛几何、复几何、代数几何、Cartan-Kahler理论和Seiberg-Witten理论有着重要的关系。在本项目中，P.I.计划继续对Calabi-Yau、G_2和Spin(7)流形的校准子流形进行研究。在与S. Akbulut的合作中，P.I.研究了复关联子流形和复Cayley子流形，并使用SW理论证明了这些子流形的模空间是光滑的，没有任何阻碍这种变形。她还介绍了镜像Calabi-Yau流形和G_2流形的数学定义。P.I.计划继续研究这些模空间的紧化问题，并研究其他Calabi-Yau和G_2流形中的镜像对偶性。这将导致构建新的计数不变量，并提供对镜像对称现象的更好理解。本课题的研究主题是由低维流形的几何和拓扑问题以及数学物理所激发的。私家侦探认为，这使得校准几何对那些可能决定学习数学和科学的学生来说是一门极好的学科，作为这个令人兴奋的领域的女性研究者，她感到有特殊的责任鼓励年轻女性开始并继续她们的数学研究。她将继续鼓励本科生和研究生在这一研究领域与他们合作，这一研究有望对数学和物理产生持久的影响。

Manifolds with special holonomy and calibrated geometry have important relations with symplectic geometry, complex geometry, algebraic geometry, Cartan-Kahler Theory and Seiberg-Witten Theory. In this project, the P.I. plans to continue her work on calibrated submanifolds of Calabi-Yau, G_2 and Spin(7) manifolds. In joint work with S. Akbulut, the P.I. studied complex associative and complex Cayley submanifolds and using the SW theory she showed that the moduli space of these submanifolds is smooth without any obstructions to such deformations. She also introduced the mathematical definitions of mirror Calabi-Yau and G_2 manifolds. The P.I. plans to continue her work on compactification problems of these moduli spaces and to study the mirror dualities in other Calabi-Yau and G_2 manifolds. These will lead to the construction of new counting invariants and provide a better understanding of the mirror symmetry phenomenon. The research topics of this project are motivated by questions in geometry and topology of low dimensional manifolds and mathematical physics. The P.I. believes that this makes calibrated geometry an excellent subject for students who might decide to study math and science, and as a female researcher in this exciting area she feels a particular responsibility to encourage young women to begin and to continue their studies of mathematics. She will continue to encourage both undergraduate and graduate students and to collaborate with them in this research field which is expected to have a long-lasting impact on both mathematics and physics.