RTG: Analysis and Differential Equations

这是芝加哥大学数学系“分析与微分方程”RTG项目的提案。这五位共同作者（Csornyei, Fefferman, Kenig， Schlag和Souganidis）在学年期间提出了针对本科生、研究生和博士后的几项活动。分析和微分方程是数学的核心、深奥和丰富的领域。从应用的角度来看，微分方程是所有科学和工程中最基本的数学模型。它们描述了从量子力学到天气预报和气候的现象，一般来说，它们构成了许多科学技术的基本理论工具。现代计算机性能的指数级提高使日益复杂的微分方程组的实际应用成为可能。虽然最近的成果解决了许多突出的问题，但仍有许多问题有待解决。理解这些模型的数学性质是必不可少的。学术界和工业界对这些领域受过良好训练的年轻数学家的需求不断增长。pi认为，在（应用）分析和微分方程方面受过训练的美国博士严重短缺。该项目的中心目标是发展一个全面的现代教育计划，以增加从事应用分析和微分方程研究的美国研究人员的数量，并提高从本科数学专业到博士后研究人员的未来数学科学家（在学术界或工业界工作）的培训质量。pi期望该计划的几个组成部分（课程改革，新课程，综合研讨会，外展活动，本科生研究经验和为期四周的夏季教程/课程）将成为全国类似举措的典范。该计划不仅将为有抱负的数学科学家提供分析和微分方程的坚实基础，而且还将通过几项计划的活动为这些不断发展的数学领域之间可能的现代联系做好准备。无论在学术界还是产业界，要想成为一名成功的应用数学研究者，这种多样化的训练是必不可少的。在RTG资金的帮助下，芝加哥大学的pi和其他研究人员将促进形成、形成和发展一种环境，在这种环境中，成功地利用本科生、研究生、博士后和教职员工的综合才能的活动将更加突出。该项目旨在将芝加哥大学在分析方面的伟大传统提升到新的高度。通过开展全方位的活动，pi相信该计划有潜力成为当地以及全国的分析和微分方程培训和研究范例。

This is a proposal for an RTG project in ``Analysis and Differential Equations" at the Department of Mathematics of the University of Chicago. The five co-PIs (Csornyei, Fefferman, Kenig, Schlag and Souganidis) propose several activities during the academic year geared towards undergraduate and graduate students and postdocs. Analysis and differential equations are central, deep and rich fields of mathematics. From an applied perspective, differential equations are the most fundamental mathematical models in all of science and engineering. They describe phenomena from quantum mechanics to weather prediction and climate,and generally speaking constitute the basic theoretical tools for much of science and technology. The exponential improvement in the capabilities of modern computers allows for the practical use of increasingly complex systems of differential equations. Although recent results have solved numerous outstanding problems, many still remain open. The understanding of the mathematical properties of these models is essential. The need in both academia and industry for young, well-trained mathematicians in these areas is ever growing.The PIs believe that there is a severe shortage of US PhDs trained in (applied) analysis and differential equations. The proposed project's central goals are to develop a well-rounded and modern educational program to increase the number of American researcher working in the applied analysis and differential equations, and to improve the quality of training of future mathematical scientists (working in either academia or industry) from undergraduate mathematics majors to postdoctoral researchers. The PIs expect that the several components of this program (curriculum reform, new courses, integrated seminars, outreach activities, undergraduate research experience and four week long summer tutorials/courses) will serve as a model for similar initiatives across the country. The proposed project will not only provide aspiring mathematical scientists with a firm grounding in analysis and differential equations, but also prepare them for possible modern connections among these ever developing areas of mathematics through several planned activities. This diversified training is definitely needed in order to become a successful researcher in applied mathematics in either academia or industry. With the aid of the RTG funding, the PIs and other researchers at the University of Chicago will stimulate the formation and the formation and the development of an environment in which activities that successfully exploit the combined talents of undergraduate and graduate students, postdoctoral fellows and faculty members are more prominent features. The proposed project is aiming towards bringing the great tradition of the University of Chicago in analysis to renewed heights. Running the whole spectrum of activities the PIs believe that the program has the potential of becoming both a local as well as a national paradigm of training and research in analysis and differential equations.