Geometry and topology of 4-manifolds

这个项目的首要目标是加深对四维流形结构的理解，四维流形是在我们生活的四维时空上局部建模的几何对象。这些物体上光滑、辛和复杂的结构在理论物理中有着广泛的特征：例如，在经典力学、粒子物理、弦理论和量子场论中。这个项目结合了现代数学工具来探索配备了这种结构的四维流形的异同。该研究计划将调用各种新的想法和技术，通过研究曲面上曲线之间的代数关系来解决一些重要问题，研究生和高级本科生可以很容易地接触到这一点。这一努力的更广泛影响是通过指导和外展。这是一个低维几何和拓扑学的研究项目，旨在为光滑和辛四维流形、接触三维流形及其填充等各种深刻问题提供新的见解。该项目的一个中心目标是探索四维空间中是否存在奇异的光滑和辛结构，并着眼于得出合理的分类方案。其中一些项目是设计奇异有理曲面和直纹曲面、伪辛射影平面、球商和Calabi-Yau曲面的新结构，深入研究曲面上曲面丛的地理学，以及研究各种稳定的四维分类方案。该研究计划将借鉴拓扑学、几何学、分析和代数领域的知识，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重要作用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.