RTG Analysis

该项目的目标是增加能够将先进的数学思想用于其他领域的独立研究或应用的美国公民和永久居民的数量。该项目的主要重点将是教育想要获得数学博士学位的研究生，以及促进隶属于加州大学洛杉矶分校分析小组的博士后研究人员的职业生涯。让分析小组成员参与的各种措施旨在吸引所有层次的学生学习数学，并可能吸引他们学习分析领域。这样做的目的是确保有一批想要学习高级数学概念的合适候选人，并帮助他们在这一探索中取得成功。对于小学生、初中生和高中生，这些举措包括与洛杉矶数学圈的合作，该圈在周日定期举行会议，提供说明性讲座、解决问题的会议和学习小组。对于本科生，将有暑期复习课程，以期为GRE数学科目考试做准备。即将入学的研究生将受益于“训练营”，这是一个为期六周的密集复习课程，讨论数学的核心主题，旨在促进本科生和研究生培训之间的过渡。这些额外的活动将被“垂直整合”，以确保各级学生和研究人员之间的互动，并着眼于两个目标：提高研究实习生的沟通和陈述技能，以及提高学生，特别是在代表不足的群体中，对数学作为一个有趣和令人兴奋的研究领域的认识。自莱布尼茨和牛顿时代以来，分析一直是数学的核心学科之一。它在科学和工程上有许多壮观的应用。到目前为止，在基于定量推理的所有调查领域中，分析工具无处不在。最近数学领域中许多比较突出的进展在很大程度上依赖复杂的分析方法。例如，Perelman对3-流形的瑟斯顿几何猜想的解，Lawler-Schramm-Werner关于随机Loewner方程的工作及其在随机过程中的应用，或者Tao-Vu对随机矩阵理论中普适性定律的研究。今天，分析是数学研究的一个充满活力的领域，有着广泛的应用。加州大学洛杉矶分校数学系的分析小组包括许多杰出的研究人员，在比较排名中享有很高的地位。它的成员有广泛的研究兴趣，包括算子代数、偏微分方程、复分析、调和分析、数学物理、概率和解析数论。除了其研究人员的实力外，这个分析小组在培训数学家方面也有着悠久而成功的历史。凭借集团内部的不同兴趣和良好的过去记录，它拥有独特的专业知识，可以向学生传授该领域的广泛视野，并培养下一代分析专家。

The goal of this project is to increase the number of US citizens and permanent residents who are able to employ advanced mathematical ideas for independent research or applications in other fields. The major emphasis of the project will be the education of graduate students who want to obtain a PhD in mathematics, and the promotion of the careers of postdoctoral researchers who are affiliated with the analysis group at the University of California, Los Angeles. A variety of measures involving members of the analysis group is intended to attract students on all levels to mathematics and possibly to the area of analysis. The goal is to secure a "pipeline" of suitable candidates who want to learn advanced mathematical concepts and help them to be successful with this quest. For elementary, middle, and high school students these initiatives include the cooperation with the Los Angeles Math Circle that has regular meetings on Sundays and offers expository lectures, problem-solving sessions, and study groups. For undergraduates there will be summer review sessions with a view towards preparation for the GRE subject test in mathematics. Incoming graduate students will benefit from the "Boot Camp", an intensive six-week long review session on central topics in mathematics that is intended to facilitate the transition between undergraduate and graduate training. These additional activities will be "vertically integrated" to ensure interactions between students and researchers on all levels with a twofold goal in mind: improving the communication and presentation skills of the research trainees and raising the awareness of students, in particular among underrepresented groups, for mathematics as an interesting and exciting field of study.Since the times of Leibniz and Newton, analysis has been one of the core subjects of mathematics. It has led to many spectacular applications in science and engineering. By now analytic tools are ubiquitous in all areas of investigation based on quantitative reasoning. Many of the more prominent recent advances within mathematics itself rely heavily on sophisticated analytic methods. Examples are Perelman's solution of Thurston's geometrization conjecture for 3-manifolds, the work by Lawler-Schramm-Werner on the stochastic Loewner equation and its applications for random processes, or the investigations by Tao-Vu on universality laws in random matrix theory. Today analysis is a vibrant area of mathematical investigation with a wide range of applications. The analysis group in the Department of Mathematics at UCLA includes many distinguished researchers and has a high standing in comparative rankings. Its members have a broad range of research interests including operator algebras, partial differential equations, complex analysis, harmonic analysis, mathematical physics, probability, and analytic number theory. In addition to the strength of its researchers, the analysis group has a long and successful history of training mathematicians. With the diversity of interests within the group and its strong past record, it has the unique expertise to impart a broad vision of the field to its students and to train the next generation of experts in analysis.