Vision and Objectives of the Columbia Mathematics VIGRE Grant
Today, the importance of mathematics is recognized perhaps more than at any other time. The last few decades have shown repeatedly how deep mathematical structures are essential to the very formulation of physical laws. For instance, in the early part of this century, Einstein's work on the gravitational force relied crucially on Riemannian geometry, and more recent developments in particle physics and superstring theory have furthered the physics/geometry interface by utilizing concepts from modern algebra, topology, complex analysis, differential geometry, and algebraic geometry. In fact, various developments in supersymmetric field theory and string theory have helped to solve long standing mathematics problems as well as launch new areas of mathematical investigation. Moreover, with remarkable developments such as the emergence of topology in DNA networks and in the formation of complex chemical chains, geometrical ideas have also begun to penetrate other natural sciences. In the financial world, the theory of finance has become increasingly mathematical. And there is a strong demand for qualified mathematics teachers nationwide.
Mathematics training has not however evolved accordingly. Young math Ph.D.'s are usually very specialized, and already have difficulty interacting with their peers in pure mathematics, let alone in other scientific fields. This places them at a disadvantage in the marketplace, especially outside of academe, where flexibility and a "can do" attitude are prime assets. To a large extent, this can be traced to the archaic graduate program prevailing at most universities, where algebra, analysis, and topology are taught as distinct subjects to students at a key formative stage. The curriculum has also not evolved, so that the gap between MA requirements and Ph.D. thesis topics is widening steadily. The outcome is an excessive time-to-degree, a fragmented and compartmentalized knowledge of the subject, and a lack of a global viewpoint. At the undergraduate level we see the same impatience with the traditional mathematics curriculum and a marked increase in the number of joint math majors.
A central feature of this grant is a more encompassing teaching of the traditional branches of mathematics, with an exposure at an early stage to other scientific disciplines. In particular, a restructured "core curriculum" will provide all new math graduate students -- independently of their ultimate specializations -- with a broad, yet tightly interwoven introduction to the basic themes and techniques of modern mathematics. The material is distilled from the most classical topics to state-of-the-art developments, and cuts across the traditional subdivisions. It is covered in a series of six one-year courses, with large overlaps, and where the same ideas occur repeatedly in different guises. Besides lectures, the courses all incorporate round-table discussions led by advanced graduate students supported by the grant.
The core curriculum is common to all mathematics students, and allows specialization in any direction. The core curriculum, the more advanced courses, and especially the seminars provide ample opportunities for graduate students, postdocs, and senior faculty to interact. The stress in all activities is on round-table discussions and open problems, so as to maximize joint efforts between participants.
The program is committed to making its activities as widely accessible as possible, through live video conferencing and by storing the more innovative or advanced courses/seminars on video tapes.
Finally, our program is also committed to exploring outreach to many groups, including
- The large pool of most-gifted students in the New York area, through Columbia's Saturday Science Honors Program (SHP);
- New York City high school students from underrepresented groups, through the Double Discovery Center;
- New York City public high school teachers, who will serve as ``teaching partners" in SHP;
- undergraduates from the Northeast potentially interested in teaching careers, in conjuction with the Mathematics Education Program at Teachers College;