CAREER: Symplectic 4-Manifolds and Singular Symplectic Surfaces

这个项目研究四维空间几何学的新领域。这些空间在拓扑学和几何学的数学领域以及重要物理系统的模型中起着重要的作用。由于人类无法想象四维空间，因此有必要开发数学和图解工具来研究这些空间。 从历史上看，四维空间显示出不寻常的现象，与较低或较高维度的空间不同。在发现这种现象的一个持久的主题是辛几何的作用。虽然辛几何最初是为了编码哈密顿物理学的方程而制定的，但它已经发展成为一个重要的抽象数学主题，对于研究四维空间特别强大。该项目旨在使用尖端工具在四维辛几何中取得新的进展。由于这是一个快速增长的领域，培养准备充分和多样化的下一代尤为重要。为此，PI将开设一门新的本科生课程，为研究生开设演讲研讨会，开发一个研究生-本科生配对阅读计划，并组织研讨会、暑期学校和会议。该项目将研究辛4-流形和奇异辛曲面。第一个主要目标是创建图解方法来研究辛4-流形的同构类的微妙之处。第二个目标是产生新的辛合痕分类光滑和奇异辛曲面。第三个目标是对现有的辛不变量进行新的计算，并为辛4-流形及其子流形定义新的不变量。该项目将采用四种数学方法来研究辛4-流形和曲面：编织曲面的分支覆盖，三等分，（去）奇异化和Weinstein双曲体。第四个目标是建立一个新的数学家管道训练有素的研究流形和熟练的科学交流和研究。该项目包括几何学、拓扑学和四维空间的研究生和本科生培训，以及清晰的沟通和指导技能。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project studies new frontiers in the geometry of 4-dimensional spaces. These spaces play important roles in the mathematical areas of topology and geometry, as well as in models of important physical systems. Because humans cannot visualize four spatial dimensions, it is necessary to develop mathematical and diagrammatic tools to study these spaces. Historically, 4-dimensional spaces have displayed unusual phenomena, distinct from spaces of either lower or higher dimensions. A persistent theme in the discovery of such phenomena is the role of symplectic geometry. Although symplectic geometry was initially formulated to encode the equations of Hamiltonian physics, it has grown into an important abstract mathematical topic that is particularly powerful for studying 4-dimensional spaces. This project aims to use cutting-edge tools to make novel advances in 4-dimensional symplectic geometry. Because this is a rapidly growing field, training a well-prepared and diverse next generation is particularly important. To that end, the PI will create a new undergraduate course, run speaking workshops for graduate students, develop a paired graduate-undergraduate reading program, and organize workshops, summer schools, and conferences.The project will study symplectic 4-manifolds and singular symplectic surfaces. The first primary goal is to create diagrammatic methods to study the subtleties of diffeomorphism classes of symplectic 4-manifolds. The second goal is to produce new symplectic isotopy classifications for smooth and singular symplectic surfaces. The third goal is to perform new calculations of existing symplectic invariants and to define new invariants for symplectic 4-manifolds and their submanifolds. The project will employ four mathematical methods to study symplectic 4-manifolds and surfaces: branched coverings of braided surfaces, trisections, (de)singularization, and Weinstein handlebodies. The fourth goal is to create a pipeline of new mathematicians well trained in the study of manifolds and skilled in scientific communication and research. The project includes training of students at the graduate and undergraduate levels in geometry, topology, and 4-dimensional spaces, as well as in clear communication and mentoring skills.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.