Manifolds with Special Holonomy and Applications

自然界四种基本力--电磁力、引力、强核力和弱核力--的统一是物理学最大的未解之谜之一。在过去的几十年里，M理论，一个“万物理论”，已经成为这些力量统一的候选者。 这个项目是关于流形与特殊的holonomy，空间的无穷小对称性使他们在M理论的“紧化”中发挥了至关重要的作用-也就是说，他们模拟微小的“卷曲”维度潜伏在时空的每一点。 在这个项目中，主要研究者将特别关注6维卡-丘流形（扮演超弦理论卷曲维度的类似角色）和7维和8维空间，它们的对称性分别填充了称为G2和Spin（7）的特殊完整群。 尽管对Calabi-Yau流形进行了广泛的研究，但G2和Spin（7）流形的几何性质并没有得到很好的理解，并且存在校准的问题（即，体积最小化）子流形仍然是敞开的。 这个项目的一个目标是开发足够强大的技术来处理这些困难的存在问题。另一个目标是研究校准子流形的变形空间，因为理解这些空间最终将对M理论紧化有用。 PI还认为具有特殊完整性的流形是研究生研究的一个很好的课题，并打算继续指导博士生。 她计划鼓励女性和其他代表性不足的群体的成员参加研究生学习，并继续研究微分几何的职业生涯，通过活动，包括咨询，组织研讨会，特别会议，会议和“数学中的女性”讲习班。在这个项目中，PI计划继续她在Ricci平坦流形，其校准几何和模空间的紧致化方面的工作。在最近与F. Arikan和H. Cho证明了每个G2流形都是几乎切触流形。研究切触与G2结构之间的关系有助于找到7-流形上G2度量的存在条件（类似于Calabi-Yau度量的存在条件）。在另一个联合工作与赵和A. J.托德，她调查的性质G2流形从辛的观点。利用接触和辛结构，PI计划构造G2和Spin（7）流形的拉格朗日和勒让德型子流形。此外，在与C. Robles，她应用Cartan-Kahler理论将结合和Cayley嵌入到G2和Spin（7）流形中，她计划使用这些技术来构建G2和Spin（7）流形的新例子，并研究它们的接触和辛结构。在与D.乔伊斯，PI研究变形的渐近圆柱coassociative子流形和他们的拓扑量子场论，并与托德她也证明了类似的结果渐近圆柱特殊拉格朗日子流形。PI计划将这些技术应用于Calabi-Yau流形内的特殊拉格朗日模空间，以获得Floer同调程序的框架。理解这些子流形的模空间将提供对镜像对称现象的更好理解。

The unification of the four fundamental forces of nature--electromagnetism, gravity, the strong and weak nuclear forces--is one of the greatest unsolved mysteries of physics. Over the last few decades, M-theory, a "theory of everything",  has emerged as a candidate for such a unification of these forces.  This project is about manifolds with special holonomy, spaces whose infinitesimal symmetries allow them to play a crucial role in M-theory 'compactifications'---that is, they model the tiny 'curled up' dimensions lurking at every point of spacetime.   In this project, the principal investigator will focus in particular on 6-dimensional Calabi-Yau manifolds (which play the analogous role of the curled-up dimensions of superstring theory) and spaces of dimension 7 and 8 whose symmetries fill out the special holonomy groups known as G2 and Spin(7), respectively.  Despite extensive research on Calabi-Yau manifolds, the geometric properties of G2 and Spin(7) manifolds are not well understood, and the problem of the existence of calibrated (i.e., volume minimizing) submanifolds is still wide open.  One goal of this project is to develop techniques that are robust enough to handle these difficult existence questions. Another goal is to study the deformation spaces of calibrated submanifolds, as understanding these spaces will ultimately be useful for M-theory compactifications.  The PI also believes that manifolds with special holonomy is an excellent topic for graduate research, and intends to continue to supervise PhD students.  She plans to encourage women and members of other under-represented groups to take up graduate study and continue to research careers in differential geometry, through activities that include advising, organizing seminars, special sessions, conference and "Women in Math" workshops.In this project, the PI plans to continue her work on Ricci-flat manifolds, their calibrated geometries and the compactifications of moduli spaces. In recent joint work with F. Arikan and H. Cho, she showed that every G2 manifold is an almost contact manifold. Studying the relations between contact and G2 structures can be useful to find the existence conditions of a G2 metric  on 7-manifolds (similar to the existence conditions of the Calabi-Yau metric). In another joint work with Cho and A.J. Todd, she investigated the properties of G2 manifolds from a symplectic point of view. Using contact and symplectic structures, the PI plans to construct Lagrangian and Legendrian type submanifolds  of G2 and Spin(7) manifolds. Also, in joint work with C. Robles, she applied the Cartan-Kahler theory to associative and Cayley embeddings into G2 and Spin(7) manifolds, and she plans to use these techniques to construct new examples of G2 and Spin(7) manifolds and study their contact and symplectic structures. In other joint work with D. Joyce, the PI studied deformations of asymptotically cylindrical coassociative submanifolds and their topological quantum field theories, and with Todd she also proved similar results for asymptotically cylindrical special Lagrangian submanifolds. The PI plans to apply these techniques on special Lagrangian moduli spaces inside Calabi-Yau manifolds to obtain a framework for the Floer homology program. Understanding the moduli spaces of these submanifolds will provide a better understanding of the mirror symmetry phenomenon.