Calibrations and Manifolds with Special Holonomy

具有特殊 Holonomy 的校准和歧管

• 批准号: 0805858
• 负责人: Sema Salur
• 金额: $ 11.66万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-15 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805858&HistoricalAwards=false
• 关键词: Calibrations; Manifolds; Special; Holonomy

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.

翻译后摘要：这个项目的研究课题的动机是低维流形和数学物理的几何和拓扑问题。首先，P.I.计划研究G_2流形和Spin（7）流形中标定子流形的形变。在与S。阿克布卢特那个私家侦探研究了复结合子流形和复Cayley子流形，并证明了这些子流形的模空间是光滑的，没有任何障碍。她还介绍了镜像卡-丘流形和G_2流形的数学定义。本项目的目标是沿着这些思路，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建设，并提供了一个更好的理解镜像对称现象。在第二部分中，P.I.计划继续她的工作在非紧G_2流形。与D.乔伊斯那个私家侦探研究了渐近圆柱余结合子流形的形变及其拓扑量子场论。此外，在最近与C。Robles，她研究了结合和凯莱嵌入到G_2和Spin（7）流形的Cartan-Kahler理论。她计划利用这些结果来构造G_2和Spin（7）流形的新例子。类似地，卡-丘流形的特殊拉格朗日子流形被期望给出类似的拓扑量子场论。了解退化卡-丘流形内部的特殊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，PI。计划研究asymp的模空间。研究具有特殊完整性的流形的几何和拓扑的长期目标是为物理学中的M-理论带来更广泛的数学理解。尽管M理论具有高度的概念性，但它已被证明对美国公众产生了极大的兴趣，产生了描述这种潜在的“万物理论”本质的流行文学和电视特辑。 私家侦探认为，这使得M理论一个很好的主题，以赶上学生的兴趣谁可能决定学习数学和科学，并作为一个女性研究人员在这个令人兴奋的领域，她觉得特别有责任鼓励年轻女性开始，并继续他们的数学研究。 她目前正在指导两名博士生，并担任本科数学学生协会的指导老师。她还在UR组织几何研讨会和学术讨论会。此外，她是联合康奈尔大学几何研讨会的共同组织者。她将继续鼓励本科生和研究生，并与他们在这一研究领域的合作，预计将有一个长期的影响数学和理论物理。