RTG: Analysis and Applications

威斯康星大学分析与应用研究和培训小组(RTG)项目包括各种倡议，旨在增加学习数学并在数学领域从事教学和研究工作的美国公民和永久居民的数量。该项目将显著增加威斯康星大学麦迪逊分校的活动强度和范围，并为博士后、研究生和本科生创造一种广泛教育和强化全面培训的文化。该提案的研究范围既广泛又连贯，由共同的主题和想法交织在一起。调和分析的研究主题包括关于傅里叶乘子的经典问题、傅里叶变换的限制以及次黎曼几何与分析的相互作用。在数学流体动力学方面，研究将集中在活动标量的正则性和爆破问题，研究建立在经典Boussinesq系统基础上的实际大气动力学模型，以及位于偏微分方程组、动力系统和概率论界面上的混合问题。在数学生物学中，我们将致力于建立反应网络，以模拟生物化学、系统生物学和生态学中的广泛过程，并研究趋化作用在加强生物反应中的应用。本项目的研究将集中在分析和应用上。应用问题将来自不同的领域和科学。将有一些应用程序涉及大气科学中用于天气和气候研究的模型。在从内燃模型到食品加工的各种应用中，流体中的混合效率将会出现问题。最后，将会有生物化学中感兴趣的反应网络的问题，以及生态学和海洋学中感兴趣的生物对化学物质的感知问题。该项目旨在为年轻参与者提供适应各种职业道路和追求最有希望的方向所需的经验和深度。它提供广泛的教学和研究活动，包括为即将到来的研究生提供暑期培训计划，为研究生和博士后举办年度研究研讨会，为本科生提供年度研究体验计划，研究研讨会和外展机会。本科生研究体验项目旨在让学生尽早了解数学研究的兴奋，并为研究生和博士后提供领导力培训机会。这些活动在本学年将有大量的创新教学部分，随后将在夏季进行密集的研究项目。研究生研究研讨会将涵盖各种活跃的主题，让研究人员和学生接触先进的分析技术以及可以使用这些技术的具体应用问题。研究讲习班将有助于拓宽受训人员的视野，并提出新的研究方向。该项目赞助的外联活动将包括为高中生举办的定期数学研讨会，参加这些研讨会的将是RTG学员。RTG助学金支持的活动将大大加强威斯康星大学数学系的研究和培训计划，这些计划有助于培养一支具有数学技能的劳动力队伍。这一贡献代表了该项目在全国和全州范围内的影响。

The University of Wisconsin Research and Training Groups (RTG) project in Analysis and Applications consists of a variety of initiatives with the aim of increasing the number of US citizens and permanent residents who study mathematics and then pursue teaching and research careers in mathematics. The project will significantly increase both the intensity and range of activities at UW Madison, and create a culture of broad education and intensive well-rounded training, for trainees at the postdoctoral, graduate and undergraduate level. The research scope of the proposal is both broad and coherent, interwoven by common themes and ideas. The harmonic analysis research topics include classical questions on Fourier multipliers, restriction of Fourier transforms and the interplay of sub-Riemannian geometry and analysis. In mathematical fluid dynamics, the research will focus on regularity and blow up problems for active scalars, on investigating realistic models of atmospheric dynamics building upon the classical Boussinesq system, and on mixing problems that lie on the interface of partial differential equations, dynamical systems and probability. In mathematical biology, we will work on reaction networks that model a broad range of processes in biochemistry, systems biology and ecology, and on applications of chemotaxis to enhancement of biological reactions. The research of this project will be focused on analysis and applications. The applied problems will come from a variety of areas and sciences. There will be applications that involve models used in atmospheric science for weather and climate studies. There will be problems on efficiency of mixing in fluids, of interest in applications ranging from internal combustion modeling to food processing. Finally, there will be problems on reaction networks of interest in biochemistry, and problems on sensing of chemicals by living organisms of interest in ecology and oceanography. The project is designed to give young participants experience and depth necessary to adapt to a variety of career paths and to pursue the most promising directions available. It offers a wide range of teaching and research activities, including summer training programs for incoming graduate students, annual research seminars for graduate students and postdocs, annual research experience programs for undergraduates, research workshops and outreach opportunities. The research experience programs for undergraduates seek to early acquaint students with the excitement of mathematics research and provide leadership training opportunities for graduate students and postdocs. Those activities will have a large innovative teaching component during the academic year, and will be followed by intensive research programs in the summer. The graduate research seminars will cover a diverse range of active topics, exposing researchers and students to a mix of advanced analysis techniques along with concrete applied problems where these techniques can be used. The research workshops will serve to broaden the perspective of the trainees and suggest new directions of research. Outreach activities sponsored by the project will include regular mathematics seminars for high-school students, with the participation of RTG trainees. The activities supported by the RTG grant will significantly strengthen the UW mathematics department's research and training programs which contribute to the effort of creating a mathematically skilled workforce. This contribution represents the national and statewide effect of this project.