Symplectic Surfaces, Lefschetz Fibrations, and Arboreal Skeleta

该奖项支持一个迅速发展的数学领域的研究，该领域被称为辛拓扑，它源于对物理学中哈密尔顿运动方程的研究。位置和动量坐标被记录在更高维的空间中，而连接位置和动量的方程在空间的几何结构中被跟踪。20世纪80年代和90年代发展了强大的技术来研究这些辛几何空间，但许多基本问题仍然没有解决。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。国际数学家协会将与不同的年轻数学家群体分享这些新技术，将新一代的学生带入尖端研究。这个项目的活动将为学生提供有效沟通的培训，促进全球合作，让公众参与到数学中来，并确保我们的科学进步能够有效地建立在我们的前辈和同行所做的工作的基础上。第一个是在长期存在的辛等距问题上取得进展，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，这样就可以在没有大亏格的情况下获得较大的度数，从而避免了在21世纪初阻碍进展的技术障碍。第二个目标是利用Lefschetz函数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原函数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开启猜想和与复代数几何相关的新的研究方向。第三个目标是发展新兴的树栖骨骼技术，利用一组简单的奇点类型，通过拓扑学方法研究温斯坦流形。PI已经在最近发表的一篇论文中开始了该计划的初始阶段，并将继续发展理论和应用，特别是在低维领域。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。

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.