Symplectic Surfaces, Lefschetz Fibrations, and Arboreal Skeleta

该奖项支持一个快速发展的数学领域的研究，即辛拓扑，它源于对物理学中汉密尔顿运动方程的研究。在高维空间中记录位置和动量坐标，并在空间上的几何结构中跟踪连接位置和动量的方程。在20世纪80年代和90年代，人们开发了强大的技术来研究这些辛几何空间，但许多基本问题仍未得到解决。例如，我们还不了解在这些空间中可以找到的所有表面。这个项目将探索这些基本问题，并开发新的技术来理解辛空间并提取它们的性质。PI将与不同群体的年轻数学家分享这些新技术，将新一代学生带入前沿研究。这个项目包括的活动将为学生提供有效沟通的培训，促进全球合作，吸引公众参与数学，并确保我们的科学进步能够有效地建立在前人和同龄人的工作基础上。这个项目有三个主要的科学目标。首先是在长期存在的辛同位素问题上取得进展，研究复射影平面上的辛曲面。提出的新方法是简化为奇异曲面，这使得人们可以在没有大属的情况下获得大度，从而避免了在21世纪初停滞不前的技术障碍。第二个目标是利用Lefschetz纤维从辛拓扑的角度研究代数几何和奇点理论中出现的问题。Lefschetz纤束是构造和计算辛流形不变量的有效工具，PI计划将这一有效工具应用于与复杂代数几何相关的开放猜想和新的研究方向。第三个目标是发展新兴的树骨架技术，利用简单的奇点类型集合通过拓扑方法研究Weinstein流形。在最近发表的一篇论文中，PI已经开始了该计划的初始阶段，并将继续发展理论和应用，特别是在低维度。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。

This award supports research in a rapidly growing field of mathematics known as symplectic topology that grew out of studying Hamilton's equations of motion in physics. Position and momentum coordinates are recorded in a higher dimensional space and the equations connecting position and momentum are kept track of in a geometric structure on the space. Powerful techniques were developed in the 1980s and 1990s to study these symplectic geometric spaces, but many fundamental questions remain unsolved. For example, we do not yet understand all of the surfaces that can be found inside these spaces. This project will explore such fundamental questions and develop new techniques for understanding symplectic spaces and for extracting their properties. The PI will share these new techniques with diverse groups of young mathematicians to bring the entering generation of students into cutting edge research. Activities included in this project will provide training in effective communication for students, to promote global collaborations, to engage the public in mathematics, and to ensure our scientific progress can effectively build off of work done by our predecessors and peers.There are three main scientific goals of this project. The first is to make progress on the longstanding symplectic isotopy problem, studying symplectic surfaces in the complex projective plane. The new approach proposed is to reduce to singular surfaces, which allow one to obtain large degrees without large genus and thus avoid the technical roadblock that stalled progress in the early 2000s. The second goal is to utilize Lefschetz fibrations to study questions arising in algebraic geometry and singularity theory from the symplectic topological perspective. Lefschetz fibrations have been an effective tool to construct and calculate invariants of symplectic manifolds, and the PI plans to apply this effective tool to open conjectures and new lines of research related to complex algebraic geometry. The third goal is to develop the emerging technique of arboreal skeleta to study Weinstein manifolds through topological methods utilizing a simple collection of singularity types. The PI has begun initial stages of this program in a recently published paper and will continue development of the theory and applications particularly in low dimensions.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.