Calibrations and Manifolds with Special Holonomy

摘要：本项目的研究课题是由低维流形的几何和拓扑问题以及数学物理问题引起的。首先，P.I.计划研究G_2和Spin(7)流形中定标子流形的变形。在与S.Akbuut的合作中，P.I.研究了复结合和复Cayle子流形，并证明了这些子流形的模空间是光滑的，对这种变形没有任何障碍。她还介绍了镜像Calabi-Yau和G_2流形的数学定义。本项目的目的是跟随这些足迹，得到这些模空间的紧化，并研究Calabi-Yau流形和G_2流形中的镜像对偶。这将导致新的计数不变量的构造，并提供对镜像对称现象的更好的理解。在第二部分中，P.I.计划继续她在非紧G_2流形上的工作。在与D.Joyce的合作中，P.I.研究了渐近柱面共结子流形的变形及其拓扑量子场理论。此外，在最近与C·罗伯斯的合作中，她研究了关于结合和Cayley嵌入到G_2和Spin(7)流形中的Cartan-Kahler理论。她计划利用这些结果来构造G_2和自旋(7)流形的新例子。类似地，Calabi-Yau流形的特殊拉格朗日子流形有望给出类似的拓扑量子场理论。了解退化Calabi-Yau流形中特殊的拉格朗日模空间将为Floer同调程序提供一个严格的框架。因此，P.I.计划研究Ascp的模空间。研究具有特殊完整的流形的几何和拓扑的长期目标是在物理学中对M-理论有更广泛的数学理解。尽管M理论具有很高的概念性，但事实证明，它引起了美国公众的极大兴趣，产生了描述这一潜在的“万物理论”本质的流行文学和电视特别节目。P.I.认为，这使得M理论成为一门极好的学科，可以吸引可能决定学习数学和科学的学生的兴趣，作为这一令人兴奋的领域的一名女性研究人员，她感到有特别的责任鼓励年轻女性开始并继续她们的数学研究。她目前正在指导两名博士生，并担任本科生数学学生协会的教师顾问。她还在密歇根大学组织几何研讨会和学术讲座。此外，她还是康奈尔大学几何研讨会的联合组织者。她将继续鼓励本科生和研究生，并在这一研究领域与他们合作，预计这将对数学和理论物理产生长期影响。

Abstract:The research topics of this project are motivated by questions in geometry and topology of low dimensional manifolds and mathematical physics. Firstly, the P.I. plans to study the deformations of calibrated submanifolds in G_2 and Spin(7) manifolds. In joint work with S. Akbulut, the P.I. studied complex associative and complex Cayley submanifoldsand 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. This project aims to follow these footsteps and to obtain a compactification of these moduli spaces and to study the mirror dualities in 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. In the second part, the P.I. plans to continue her work in noncompact G_2 manifolds. In joint work with D. Joyce, the P.I. studied deformations of asymptotically cylindrical coassociative submanifolds and their Topological Quantum Field Theories. Also, in recent joint work with C. Robles, she studied the Cartan-Kahler Theory for associative and Cayley embeddings into G_2 and Spin(7) manifolds. She plans to use these results to construct new examples of G_2 and Spin(7) manifolds. Similarly, special Lagrangian submanifolds of Calabi-Yau manifolds are expected to give analogous Topological Quantum Field Theories. Understanding the special Lagrangian moduli spaces inside degenerating Calabi-Yau manifolds will provide a rigorous framework for the Floer homology program. Therefore, the P.I. plans to study the moduli spaces of asymp. cylindrical special Lagrangian submanifolds with boundary.The long range goal of studying the geometry and topology of manifolds with special holonomy is to bring a broader mathematical understanding of M-theory in physics. Despite its highly conceptual nature, M-theory has proven to be of great interest to the U.S. public, yielding popular literature and television specials that describe the essence of this potential ``theory of everything''. The P.I. believes that this makes M-theory an excellent subject with which to catch the interest of 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 is currently supervising two Ph.D students and is serving as the faculty advisor of the Society of Undergraduate Mathematics Students. She also organizes the geometry seminars and the colloquium talks at UR. Additionally, she is the co-organizer of the joint Cornell-UR geometry seminars. 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 andtheoretical physics.