RTG: Analysis of Partial Differential Equations

这个项目的目标是在几个方向上加强得克萨斯大学奥斯汀分校偏微分方程分析领域的研究训练活动。偏微分方程的分析是数学的一个领域，对自然科学的发展至关重要。事实上，偏微分方程被用于许多现象的建模，如物质的原子特性、天体动力学、流体和气体力学（例如用于天气预报）以及电磁现象的研究（现代通信的基础）。这种多样性的应用体现在一个领域的复杂性和丰富的开放问题和丰硕的研究方向。因此，培养分析领域的研究生和本科生对美国的科学和经济竞争力至关重要，吸引学生到该领域的任何努力也是如此。与此同时，引导学生对该学科的看法不仅是技术性的和过于专业化的，而且要成功地强调该领域的多功能性和深度，这需要一个小组的协调工作。这五个共同的pi （Israel, Maggi, Patrizi, Pavlovic, Vasseur）与UT Austin的PDE专家（包括Beckner， Caffarelli, Chen， Gamba和Ren）一起，坚定地致力于研究培训，其联合研究活动涵盖了PDE中相互关联的广泛领域，并获得了国际认可。该项目将包括5个博士后奖学金和20个年度研究生奖学金，从而增加在德克萨斯大学奥斯汀分校接受PDE分析培训的美国公民的数量；以及一系列全面的行动，旨在提高我们的活动在本科生中的知名度，无论是在德州大学奥斯汀分校的内部还是外部，特别强调来自德克萨斯州中部大学的学生，这些大学为大量少数民族提供服务。该小组的教育活动（研讨会、课程、非正式讲座、阅读课程等）将被仔细地重新审视和加强，使受训者接触到新的想法，并增加指导、教学、专业发展和各级互动的机会。每年的暑期项目以及专业设计和维护的网站将提高我们项目在全国的知名度，并有助于传播其最成功的实践。最终的预期结果是在我们部门和其他部门的研究培训文化中创造积极持久的变化。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 goal of this project is enhancing in several directions the research-training activities in the area of the Analysis of partial differential equations (PDE) at the University of Texas (UT) at Austin. The Analysis of PDE is an area of Mathematics that is central to the development of Natural Sciences. Indeed, PDE are used in the modeling of many phenomena, like the atomic properties of matter, the dynamics of celestial bodies, the mechanics of fluids and gases (used for example in weather forecast), and the study of electromagnetic phenomena (fundamental to modern communications). This variety of applications is reflected in the complexity of a field rich in open problems and fruitful research directions. The training of graduate and undergraduate students in Analysis is thus crucial to the scientific and economic competitiveness of the US, and so is any effort to attract students to the area. At the same time, leading students to achieve a view of the subject that is not merely technical and overly specialized, but also succeeds in emphasizing the versatility and depth of the field, requires the coordinated work of a group.The five co-PIs (Israel, Maggi, Patrizi, Pavlovic, Vasseur), together with the larger group of PDE experts at UT Austin (including Beckner, Caffarelli, Chen, Gamba and Ren), are strongly committed to research training, with a combined research activity that covers a vast area of interconnected topics in PDE, and has received international recognition. This project will include 5 postdoctoral fellowships and 20 annual graduate fellowships, thus increasing the number of US citizens trained in the Analysis of PDE at UT Austin; and a comprehensive series of actions aimed at increasing the visibility of our activities among undergraduate students, both internally and externally to UT Austin, and with particular emphasis on students from universities in Central Texas serving large minority populations. The educational activities of the group (seminars, courses, informal lectures, reading courses, etc.) will be carefully revisited and enhanced, to expose trainees to new ideas, and to multiply the opportunities for mentoring, teaching, professional development, and interactions at all levels. An annual Summer Program and a professionally designed and maintained website will increase the national visibility of our program and help dispersing its most successful practices. The ultimate expected outcome is creating positive lasting changes in the research training culture at our department and beyond.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.