Geometry and topology of 4-manifolds

这个项目的首要目标是对四维流形的结构有更深的理解，四维流形是局部模拟我们生活的四维时空的几何物体。这些物体上的光滑、辛和复杂结构在理论物理中有广泛的特征：例如经典力学、粒子物理、弦理论和量子场论。本项目结合现代数学工具来探索配备这种结构的四维流形的异同。该研究项目将引入各种新的思想和技术，通过研究曲面上曲线之间的代数关系来解决一些重要的问题，这对研究生和高级本科生来说是相当容易的。这一努力的更广泛的影响是通过指导和推广。本课题是一项低维几何与拓扑学的研究项目，旨在为光滑、辛四流形、接触三流形及其填充的各种深奥问题提供新的见解。该项目的中心目标是探索四维空间中奇异的光滑和辛结构的存在，并着眼于推导合理的分类方案。一些项目包括奇异有理面和直纹面、假辛投影面、球商和Calabi-Yau面的新构造、表面上表面束的地理深入研究以及四维中各种稳定分类方案的研究。该研究项目将从拓扑学、几何学、分析学和代数等领域汲取知识，其中规范理论、新的映射类群技术和PI及其合作者最近发现的辛手术将发挥至关重要的作用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The overarching goal of this project is to develop a deeper understanding of the structure of four dimensional manifolds, which are geometric objects locally modeled on the four-dimensional space-time we live in. Smooth, symplectic and complex structures on these objects are broadly featured in theoretical physics: for example, in classical mechanics, particle physics, string theory, and quantum field theory. This project combines modern mathematical tools to explore similarities and differences of four dimensional manifolds equipped with such structures. The research program will invoke various new ideas and techniques that create leverage to address a number of important problems through a study of algebraic relations between curves on surfaces, which is fairly accessible to graduate and advanced undergraduate students. The broader impacts of this endeavor are through mentoring and outreach. This is a research project in low dimensional geometry and topology, aiming to provide new insight to a variety of profound questions on smooth and symplectic four-manifolds, contact three-manifolds and their fillings. A central goal of the project is to explore the existence of exotic smooth and symplectic structures in dimension four, with an eye towards deriving sensible classification schemes. Some of the projects are contriving novel constructions of exotic rational and ruled surfaces, fake symplectic projective planes, ball quotients and Calabi-Yau surfaces, an in-depth study of the geography of surface bundles over surfaces, and investigations of various stable classification schemes in dimension four. The research program will draw from the fields of topology, geometry, analysis and algebra, where gauge theory, new mapping class group techniques and symplectic surgeries recently discovered by the PI and his collaborators will play vital roles.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.