New Advances on Flat Surfaces

该项目专注于研究平面并探索其在不同领域的应用。平面是具有确定的一对平行边的多边形，其中平面的顶点粘合在一起形成圆锥形奇点。平面为研究几何、动力学和代数结构之间的复杂联系提供了一个有价值的框架。对平面的理解已经导致了关于空间体积和台球轨迹的引人注目的发现。首席研究人员致力于通过继续他的研究努力并在平面研究方面取得新的结果来促进这一领域的知识。该项目还将为学生和博士后学者创造许多机会。除了他的研究，首席调查员还将参与指导学生、组织研讨会和参加外联活动。一个关键的重点将是促进该领域的多样性，并为下一代科学家为未来的挑战和机会做好准备。平面对应于黎曼曲面上的微分形式，其中平面的锥形奇点对应于微分的零点。这些等价而又截然不同的描述使得平面成为许多领域的研究热点。这个项目将涵盖平面研究的各个焦点，包括动力学不变量、交集理论、留数理论、双曲面几何、紧化、Brill-Noether理论、拓扑学、循环类、高阶微分和仿射结构。为了解决与这些领域相关的复杂挑战，首席研究员将使用代数几何、解析几何、动力学和计数几何的技术组合。通过利用这些不同的方法，首席研究员将全面分析不同类型的平面结构，以深入了解它们的几何特性。这些平坦的表面结构将成为加深我们对基本数学概念的理解和发现新的研究途径的宝贵工具。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 project focuses on studying flat surfaces and exploring their applications across various fields. Flat surfaces are polygons with identified pairs of parallel edges, where the vertices of a flat surface are glued to form conical singularities. Flat surfaces provide a valuable framework for investigating the intricate connections between geometry, dynamics, and algebraic structures. The understanding of flat surfaces has already led to notable discoveries about space volumes and billiard trajectories. The principal investigator is dedicated to advancing knowledge in this area by continuing his research efforts and achieving new results in the study of flat surfaces. This project will also create many opportunities for students and postdoctoral scholars. Alongside his research, the principal investigator will engage in mentoring students, organizing workshops, and participating in outreach activities. A key focus will be on fostering diversity within the field and preparing the next generation of scientists for future challenges and opportunities.Flat surfaces correspond to differential forms on Riemann surfaces, where the conical singularities of a flat surface correspond to the zeros of a differential. These equivalent yet distinct descriptions make flat surfaces lie at the interface of many fields as a research hotspot. This project will cover various focal points in the study of flat surfaces, including dynamical invariants, intersection theory, residue theory, birational geometry, compactification, Brill-Noether theory, topology, cycle classes, higher differentials, and affine structures. To address the complex challenges associated with these areas, the principal investigator will employ a combination of techniques from algebraic geometry, analytic geometry, dynamics, and enumerative geometry. By utilizing these diverse methodologies, the principal investigator will comprehensively analyze different types of flat surface structures to gain insights into their geometric properties. These flat surface structures will serve as valuable tools for deepening our understanding of fundamental mathematical concepts and uncovering new avenues of research.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.