RTG: Frontiers in Applied Analysis

在应用领域中越来越多地使用复杂的数学模型，需要一支具有强大理论基础的数学队伍，并清楚地了解如何应用分析概念来应对科学和技术前沿的挑战。该RTG的研究和培训侧重于应用分析，包括偏微分方程组、变分、几何分析、随机分析、数值分析、最优运输，以及它们在材料科学、几何处理和机器学习中对相关模型的应用。这个RTG将创造一个丰富的活动生态系统，关注每个级别的受训者的需求。本科生将被介绍到应用分析研究，并将与研究生和教职员工一起进行创新的基于课程的本科研究和高强度的夏季本科研究项目。本科生研究会议将提供一个展示、网络和学习应用数学职业机会的场所。这个RTG将在一个刺激的环境中为研究生和博士后提供全面的指导，包括专题课程、每周工作小组、研讨会(包括一个新的研讨会系列，重点是跨学科应用分析的使用)、研讨会和暑期学校。将通过定期的专业发展活动和欧洲主要研究小组对国际合作伙伴的访问来加强培训。将特别注意从任职人数不足的群体招聘受训人员并确保他们取得成功。总体而言，RTG将有助于吸引学生学习应用数学，并将创造一支受过技术培训的美国劳动力队伍，拥有先进应用分析工具的专业知识，随时准备迎接应用学科中未来出现的挑战。从科学上讲，RTG将激发合作努力，以解决应用分析中引人注目的问题；特别是新型材料的变分问题、最小化问题中的几何结构、几何处理任务的新描述、混合和增强耗散的定量研究、允许在高维中进行准确计算的创新几何和梯度流，以及涉及热力学的问题的建模和模拟。这些合作将架起学科之间的桥梁，并导致创造解决应用挑战所需的新数学。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。

The increased use of sophisticated mathematical models in applied fields calls for a mathematical workforce with a strong theoretical foundation and a clear vision of how concepts of analysis can be applied to meet challenges at the frontiers of science and technology. The research and training of this RTG focuses on applied analysis, which encompasses partial differential equations, calculus of variations, geometric analysis, stochastic analysis, numerical analysis, optimal transportation, and their applications to relevant models in materials science, geometry processing, and machine learning. This RTG will create a rich ecosystem of activities that is attentive to the needs of trainees at each level. Undergraduate students will be introduced to research in applied analysis and will work alongside graduate students and faculty in innovative course-based undergraduate research and intense summer undergraduate research programs. An undergraduate research conference will provide a venue for presentations, networking, and learning about career opportunities in applied mathematics. This RTG will provide comprehensive mentorship for graduate students and postdocs in a stimulating environment with topics courses, weekly working groups, seminars (including a new seminar series focused on uses of applied analysis across disciplines), workshops, and summer schools. The training will be enhanced by regular professional development activities and visits to international partners from leading research groups in Europe. Particular attention will be given to recruiting and ensuring the success of trainees from underrepresented groups. Overall, the RTG will help attract students to applied mathematics, and will create a technically trained US workforce with expertise in advanced tools of applied analysis ready to engage with future challenges that arise in applied disciplines.Scientifically, the RTG will spark collaborative efforts to address compelling problems in applied analysis; in particular, variational problems for novel materials, geometric structures in minimization problems, new descriptions of geometry processing tasks, quantitative study of mixing and enhanced dissipation, innovative geometries and gradient flows allowing for accurate computation in high dimensions, and modeling and simulation of problems involving thermomechanics. These collaborations will bridge disciplines, and lead to the creation of new mathematics necessary to address applied challenges.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.