RTG: Analysis on Manifolds

流形分析是一个广泛的数学领域，历史上出现在19世纪和20世纪的物理问题的研究。 它涉及到诸如肥皂泡的形状、鼓的泛音等问题，甚至涉及到数论中的基本问题，如素数的分布。 爱因斯坦的广义相对论成功地描述了引力，最近发现的希格斯玻色子理论也依赖于流形上微分方程的研究。 该研究培训小组项目旨在增加对研究分析感兴趣的美国学生数量，更具体地说，流形分析。该补助金将为流形分析领域的本科生，研究生和博士后研究员提供丰富的教育环境，由该领域的初级和高级教师指导。 我们将为高年级本科生和研究生学习的学生建立一个夏季研讨会，以提高他们对现代分析语言的流利程度。 我们将加强本科生的分析课程，并扩大本科生的研究经验范围。 我们还将在芝加哥地区的高中学生和教师中开展外展活动。 更大的目标是增加在数学科学领域从事职业的准备充分的美国公民的数量，特别是增加在这一领域代表性不足的少数民族的数量。

Analysis on manifolds is a broad area of mathematics that arose historically in the nineteenth and twentieth centuries from the study of physical problems. It relates to questions such as the shape of a soap bubble, the overtones of a drum, and even to basic questions in number theory, such as the distribution of primes. The gravitational force is described with great success by Einstein's theory of General Relativity, and the theory of the recently discovered Higgs boson also depends on the study of differential equations on manifolds. This Research Training Group project aims to increase the number of US students interested in pursuing the study of analysis, and more specifically, analysis on manifolds.This grant will support a rich educational environment, for undergraduates, graduate students, and postdoctoral fellows in the field of analysis on manifolds, mentored by junior and senior faculty in the area. We will establish a summer workshop for advanced undergraduates and students at the threshold of graduate study, to increase their fluency in the language of modern analysis. We will enhance the courses in analysis available to our undergraduate students, and expand the range of research experiences for undergraduates. We will also conduct outreach activities among high school students and teachers in the Chicago region. The larger goal is to increase the number of well-prepared US citizens who pursue careers in the mathematical sciences, with a particular effort to increase the numbers of under-represented minorities in this field.