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，需要具有强大理论基础和对如何应用分析概念以应对科学和技术前沿挑战的清晰愿景的数学劳动力。该RTG的研究和培训侧重于应用分析，包括偏微分方程、变分法、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等相关模型中的应用。这个RTG将创造一个丰富的活动生态系统，关注每一级受训人员的需求。本科学生将被引入应用分析研究，并将与研究生和教师一起在创新的基于课程的本科研究和激烈的暑期本科研究项目中工作。本科生研究会议将提供一个展示、交流和学习应用数学职业机会的场所。RTG将通过主题课程、每周工作组、研讨会（包括跨学科应用分析的新系列研讨会）、讲习班和暑期学校，为研究生和博士后提供全面的指导。将通过定期的专业发展活动和访问欧洲主要研究小组的国际伙伴来加强培训。将特别注意从人数不足的群体中征聘和确保受训人员的成功。总体而言，RTG将有助于吸引学生学习应用数学，并将培养一支技术训练有素的美国劳动力队伍，他们精通先进的应用分析工具，随时准备应对应用学科未来出现的挑战。在科学上，RTG将激发合作努力，以解决应用分析中的紧迫问题；特别是，新材料的变分问题，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，允许在高维精确计算的创新几何和梯度流动，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，并创造出应对应用挑战所必需的新数学。该奖项反映了美国国家科学基金会的法定使命，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。

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.