Vertical Integration of Research and Education in Mathematics at Columbia University

[9810750 . pinkham]哥伦比亚大学VIGRE奖的主要目标是为数学家提供更符合当代科学需求和工作机会的数学培训。这需要对传统数学分支进行更有凝聚力的教学，并在早期阶段接触其他科学学科和非学术环境。因此，该提案的一个核心特征是对博士第一年的“核心课程”进行重组，这将为所有新的数学研究生提供一个广泛的，但紧密交织的现代数学基本主题和技术的介绍。材料范围从经典主题到最先进的发展，并跨越数学的传统细分到代数，分析和拓扑。它包括一系列为期一年的课程，其中有大量重叠，在这些课程中，相同的想法以不同的形式反复出现。除讲座外，所有课程都包括由提案支持的高级研究生和博士后主持的圆桌讨论。该项目侧重于哥伦比亚大学具有特别优势的数学科学的三个领域：1)核心数学，重点是几何；2)理论物理的数学；3)概率论，重点是金融数学。在每个领域，我们提供(a)重组课程和研究活动，反映该学科的前沿发展。(b)博士后和ritt助理教授联合职位，旨在为年轻数学家开始教师和研究生涯提供更好的支持。这些职位的教学负荷将是该系普通Ritt助理教授职位的一半。(c)其他部门或金融或工业公司等联合机构的实习职位。(d)在各级加强对数学教学问题的重视，其中大部分是通过与师范学院数学系合作进行的。(e)提高对研究生的资助，以缩短获得学位的时间。(f)改进对数学本科生的研究活动和资助。这将通过哥伦比亚大学的I.I. Rabi奖学金项目来完成，该项目已经在促进科学和数学方面的本科生研究方面建立了良好的记录，特别是通过一个暑期实习项目，允许Rabi学者加入活跃的研究小组，参与数学和科学调查和发现的实际过程。(g)研讨会和其他活动，旨在增加不同层次的学生和处于职业生涯不同阶段的教师之间的互动。纽约城市环境中的数学将是我们努力的一个重要方面。它将包括向纽约市的数学高中教师和高中学生伸出援手，重点是弱势学生和在弱势学生比例很高的学校工作的教师。在这方面，我们将与哥伦比亚大学两个完善的、全国公认的项目合作：a)双重探索中心（Double Discovery Center， DDC）， 1965年在哥伦比亚大学成立，是一个青年服务机构，为纽约市1000多名高危高中生提供服务。获资助的本科生和研究生将与DDC合作，为这些青少年提供数学课程和辅导。b)哥伦比亚周六科学荣誉计划（SHP），这是一个长期存在的，高度选择性的计划，面向在数学和科学方面有特殊天赋的高中生。该提案将允许SHP聘请高中数学教师作为SHP提供的数学课程的“教学伙伴”，目标是使他们能够在自己的学校提供类似的课程。此外，计划资助的学生和博士后也可参与创新数学新课程的设计和教学。该项目由数学科学司、MPS多学科活动办公室和初等、中等和非正规教育司资助。

9810750PinkhamThe main goal of the Columbia University VIGRE award is to offer mathematical training that is better aligned with contemporary scientific needs and job opportunities for mathematicians. These require a more cohesive teaching of traditional branches of mathematics, with an exposure at an early stage to other scientific disciplines and to non-academic environments. Thus a central feature of this proposal is a restructured "core curriculum", for the first year of the Ph.D., which will provide all new math graduate students with a broad, yet tightly interwoven introduction to the basic themes and techniques of modern mathematics. The material ranges from classical topics to state-of-the-art developments, and cuts across the traditional subdivisions of mathematics into algebra, analysis and topology. It is covered in a series of one-year courses, with large overlaps, where the same ideas occur repeatedly in different guises. Besides lectures, the courses all incorporate round table discussions led by advanced graduate students and postdoctoral fellows supported by the proposal.The project focuses on three areas in the mathematical sciences, in which Columbia has particular strength: 1) core mathematics, with an emphasis on geometry; 2) the mathematics of theoretical physics; 3) probability, with an emphasis on the mathematics of finance. In each area, we offer (a) a restructured curriculum and research activities reflecting cutting edge developments in the subject. (b) joint postdoctoral-Ritt Assistant Professor positions structured to provide better support for young mathematicians beginning their careers as teachers and researchers. These positions will have half the teaching load of the ordinary Ritt Assistant Profressor positions in the Department. (c) internship positions in allied institutions such as other departments or financial or industrial firms. (d) increased emphasis at all levels on pedagogical issues in mathematics, much of this through collaboration with the mathematics department of Teacher's College. (e) improved funding for graduate students in order to shorten time to degree. (f) improved research activities and funding for mathematics undergraduates. This will be done through Columbia University's I.I. Rabi scholarship program, which already has an established track record of promoting undergraduate reseach in science and mathematics, in particular through a summer internship program allowing Rabi scholars to join active research groups and participate in the actual process of mathematical and scientific investigation and discovery. (g) seminars and other activities designed to increase the amount of interaction between students at different levels and faculty at different stages of their careers.Mathematics in the urban environment of New York will be an important aspect of our efforts. It will include outreach to both mathematics high school teachers and high school students in New York City, with an emphasis on disadvantaged students and teachers working in schools with a high percentage of disadvantaged students. Here we will work in collaboration with two well-established and nationally recognized Columbia University programs: a) the Double Discovery Center (DDC), founded in 1965 at Columbia University, a youth service agency serving over 1,000 at-risk high school students in New York City. The undergraduate and graduate students supported by this proposal will collaborate with the DDC on math courses and tutorials for these youths. b) the Columbia Saturday Science Honors Program (SHP), a long-standing, highly selective program for high school students who have exceptional talent in mathematics and the sciences. This proposal will allow the SHP to hire high school math teachers as "teaching partners" in the mathematics courses offered by SHP, with the goal of enabling them to offer similar courses in their own schools. It will also allow students and postdocs supported by the proposal to participate in the design and teaching of innovative new mathematics courses.This project is being funded by the Division of Mathematical Sciences, the MPS Office of Multidisciplinary Activities and the Division of Elementary, Secondary, and Informal Education.