Vertical Integration of Research and Education in Mathematics at Columbia University

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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.