New Advances on Flat Surfaces

平面的新进展

• 批准号: 2301030
• 负责人: Dawei Chen
• 金额: $ 25万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2301030&HistoricalAwards=false
• 关键词: New; Advances; Flat; Surfaces

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.

该项目着重于研究平面表面并探索其在各个领域的应用。平面表面是具有识别为平行边缘对的多边形，其中平面的顶点被粘合到形成圆锥形的奇点。扁平表面为研究几何，动力学和代数结构之间的复杂联系提供了一个宝贵的框架。对平面表面的理解已经导致有关空间量和台球轨迹的著名发现。首席研究人员致力于通过继续他的研究工作并在扁平表面研究中取得新的成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。除了他的研究外，首席研究人员还将参与指导学生，组织研讨会和参加外展活动。重点将是促进该领域内的多样性，并准备下一代科学家以寻找未来的挑战和机会。Flat表面对应于Riemann表面上的差分形式，在Riemann表面上，平坦表面的圆锥形奇异性与差分的零相对应。这些等效但不同的描述使平坦的表面位于许多领域的界面上，作为研究热点。该项目将涵盖扁平表面研究的各种焦点，包括动力不变，交叉理论，残基理论，生育几何形状，紧凑型，光彩理论，拓扑理论，拓扑，周期类别，更高的差异和仿射结构。为了应对与这些领域相关的复杂挑战，主要研究者将采用代数几何，分析几何，动力学和枚举几何形状的技术组合。通过利用这些不同的方法，主要研究者将全面分析不同类型的平面结构，以了解其几何特性。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并揭示新的研究途径的宝贵工具。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估来评估值得支持的。