RTG Analysis

该项目的目标是增加能够运用先进数学思想进行独立研究或在其他领域应用的美国公民和永久居民的数量。该项目的主要重点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。 涉及分析小组成员的各种措施旨在吸引各个级别的学生学习数学，并可能吸引分析领域的学生。目标是确保有合适的候选人“管道”，他们想要学习高级数学概念并帮助他们成功完成这一任务。 对于小学生、初中生和高中生来说，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，并提供说明性讲座、问题解决课程和学习小组。本科生将参加夏季复习课程，为 GRE 数学科目考试做准备。 即将入学的研究生将受益于“新兵训练营”，这是一个为期六周的强化数学中心主题复习课程，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“垂直整合”，以确保学生和研究人员之间在各个层面上的互动，并牢记双重目标：提高研究实习生的沟通和表达技巧，并提高学生（特别是代表性不足的群体）对数学作为一个有趣且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。它在科学和工程领域带来了许多引人注目的应用。到目前为止，分析工具在基于定量推理的所有研究领域中已经无处不在。数学本身最近取得的许多更突出的进展在很大程度上依赖于复杂的分析方法。例如 Perelman 对 3 流形的 Thurston 几何化猜想的解、Lawler-Schramm-Werner 对随机 Loewner 方程及其在随机过程中的应用的研究，或者 Tai-Vu 对随机矩阵理论中的普遍性定律的研究。如今，分析是数学研究的一个充满活力的领域，具有广泛的应用。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，在比较排名中具有很高的地位。其成员具有广泛的研究兴趣，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。除了研究人员实力雄厚外，分析小组在培养数学家方面也有着悠久而成功的历史。凭借集团内部兴趣的多样性和过去的良好记录，它拥有独特的专业知识，可以向学生传授该领域的广阔视野，并培训下一代分析专家。

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.