RTG: Partial Differential Equations on Manifolds

RTG：流形上的偏微分方程

• 批准号: 2135998
• 负责人: Jason Metcalfe
• 金额: $ 241.66万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2135998&HistoricalAwards=false
• 关键词: RTG; Partial; Differential; Equations; Manifolds

Partial differential equations are ubiquitous in models of physical phenomena. Discovering the statistics of the sets where quantum particles are least likely to exist, describing the behavior of light waves near black holes, explaining the existence of bands on the planet Jupiter by analyzing the equations of fluid dynamics on a rotating sphere, and establishing the existence of solutions to systems modeling ferromagnetism or plasmas require an analysis of partial differential equations in the presence of background geometry. All of these are fundamental topics that members of the team of principal investigators have explored, and for which they are ideally situated to pass on their expertise. The purpose of this project is to create a sustainable program that enhances and modernizes the curriculum, enriches the training, and improves recruitment. This will create larger and better prepared cohorts of mathematicians to advance the understanding of the interplay between geometry and solutions to partial differential equations.The associated faculty will thoughtfully mentor postdoctoral, graduate student, undergraduate student, and high school research trainees. Graduate courses that may be taken asynchronously will be developed to broaden the curriculum. A regular workshop will be formed to get new graduate students to engage with research early. Developmental training groups will be designed to improve trainees’ technical writing and to guide them on aspects of academic life such as seeking funding. An online topics course collaborative will be established to promote sharing across academic institutions of specialized courses on cutting-edge material. An online undergraduate research seminar will be created for the dissemination of results from a newly formalized undergraduate research program at the host institution and for the recruitment of new graduate students who are conducting interesting research at other colleges and universities. The host institution’s partial differential equations mini-school program, which allows for trainees to interact with a principal speaker and their students, postdocs, and co-authors in an intimate environment that permits in-depth presentations, will be revived. Outreach activities such as a program to allow for interested local high school students to engage with mathematical research and the Girls Talk Math summer program will be supported. Many of the concrete needs that are addressed are not unique to the host institution, and these initiatives offer great benefit to a broad community of trainees.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.

偏微分方程在物理现象的模型中无处不在。发现量子粒子最不可能存在的集合的统计，描述黑洞附近光波的行为，通过分析旋转球体上的流体动力学方程来解释木星上带的存在，以及建立模拟铁磁性或等离子体的系统的解的存在性，都需要在背景几何的存在下分析偏微分方程。所有这些都是主要研究人员团队成员探索的基本主题，他们非常适合传递他们的专业知识。该项目的目的是创建一个可持续的计划，增强和现代化的课程，丰富培训，并提高招聘。这将创造更大和更好的数学家群体，以促进几何和偏微分方程的解决方案之间的相互作用的理解。相关的教师将周到地指导博士后，研究生，本科生和高中研究实习生。将开发可以异步学习的研究生课程，以拓宽课程。将定期举办研讨会，让新的研究生尽早参与研究。发展培训小组将旨在提高学员的技术写作能力，并指导他们在学术生活的各个方面，如寻求资金。将建立一个在线专题课程协作，以促进学术机构之间分享关于尖端材料的专门课程。将创建一个在线本科研究研讨会，用于传播主办机构新正式的本科研究计划的结果，并招募正在其他学院和大学进行有趣研究的新研究生。主办机构的偏微分方程迷你学校计划，允许学员与主要发言人和他们的学生，博士后和合作者在一个亲密的环境，允许深入的演示，将被恢复。将支持外联活动，如让感兴趣的当地高中生参与数学研究的方案和女孩谈论数学暑期方案。许多具体的需求并不是主办机构所独有的，这些举措为广大的受训者社区带来了巨大的利益。该奖项反映了NSF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。