Princeton VIGRE program - Abstract. General Philosophy. The VIGRE program at Princeton is a joint venture of the Mathematics Department and the Program in Applied and Computational Mathematics (PACM); it aims at exposing undergraduate students, graduate students, and postdoctoral fellows to a broader research and teaching experience. On top of disciplinary depth, we want to foster educational exploration and breadth-qualities that are increasingly important in the current rapidly changing research world and job market. It is more important than ever to prepare students and postdoctoral fellows for a variety of different teaching tasks and research opportunities. Whether they proceed to academic or non-academic careers, all student and postdoctoral fellows will benefit from experience in teaching and exposition, and from contacts with the mathematical world at other institutions and outside academia. The different components. Our VIGRE program has components that pertain to graduate training, to undergraduate education, to postdoctoral fellowships, to curriculum innovation, and to outreach activities. Several components are vertically integrated and have aspects that fit under several headings. For most initiatives, we do not distinguish between Mathematics and PACM students, postdocs, or faculty: the VIGRE program will draw on and combine the strengths of the two programs. 2A Graduate training. Mathematics, as perceived by graduate students through the filter of most graduate educations, has become increasingly fragmented. This is the case for virtually all disciplines within mathematics, regardless of how "pure" or "applied" they are. Integration of the different components encountered during graduate education into a more global view is typically left to postdoctoral studies. In our program we want to coordinate initiatives that will help launch our graduate students into their research and teaching careers, exposing them to a broader view of mathematics and its application s. All students, whether VIGRE supported or not, will be encouraged to participate in these activities. Some of the activities will be required for VIGRE students. We shall run research seminars aimed at graduate students, featuring expository presentations (by faculty members, visitors or graduate students) about open problems, often related to applications. In a companion problem seminar, interested graduate students will work on open problems (typically more complex than a homework problem but not a thesis-size problem); they will be invited to present their results to the research seminar. This initiative has started already with an Analysis and Application series, and can be extended to other fields. Every Fall, we shall bring non-academic mathematicians (e.g. from industrial research labs, from government organizations, and from the financial world) to Princeton for a series of presentations, aimed at our graduate and advanced undergraduate students, about mathematics outside academia. We shall encourage students to do at least one internship (probably but not necessarily during the summer) in another environment, during their graduate studies. We shall put together a "portfolio" of such opportunities, and continue to work on updating it and adding new possibilities. VIGRE students will spend at least one summer internship or another extended visit at another research institute, academic or not. Where appropriate, we shall involve graduate students in "partnerships in research" with individual research labs. Such collaborations will typically start with work on a mini-project, leading to a publication after completion, as well as a presentation at the research seminar series mentioned above. In some cases, these projects could grow into Ph.D. theses. A student colloquium committee, consisting of graduate students together with an undergraduate representation, will organize expository talks, aimed at students, in which speakers can be faculty or outsiders, or students ; the committee will be given the means to invite outside speakers known for their expository skills. Every week during the term there will be at least one student speaker and at least one faculty or visiting speaker, invited by this committee. Graduate students will be involved in the development of new courses and course material for undergraduates (see also below). VIGRE supported students will teach one precept of a course for at least one full semester during their training. 2B Undergraduate education. Participation in the student-run expository talks series (see above). Internships in non-academic mathematical environments, similar to the graduate internships (and in some cases, coupled to them) will be encouraged strongly, and a portfolio of opportunities will be put together. We shall set up an undergraduate laboratory in mathematics for math majors, with involvement by graduate students and faculty. We shall collect a database of problems and conjectures which are mathematically interesting and (with modest background and machines) amenable to numerical experimentation and scientific examination. The undergraduate project (which could be used for one of the two required junior papers, or, if more substantial, for the senior thesis) would be to examine experimentally (i.e., numerically) some phenomenon; students will be encouraged to probe further on their own, as well as to try their hand at proving special cases. This will lead them to experience the thrill of discovery (as well as the agony of defeat, part of research as well!). Experience in oral presentation: students enrolled in the PACM undergraduate certificate program will make a presentation about their independent work to the weekly PACM certificate seminar; these presentations are first rehearsed and edited with PACM graduate students. Math majors will be encouraged to make a similar presentation (about their senior thesis, or possibly about a summer research experience or a project in the undergraduate lab) to the student-run expository talks series. 2C Postdoctoral Fellowships. VIGRE postdocs will be encouraged to visit another institute during one term of their fellowships, possibly with other funding (e.g. GOALI). Postdoctoral fellows will be called upon to contribute substantially to course development (see below), and will be encouraged to participate in the teaching of innovative undergraduate courses. VIGRE postdoctoral fellows will be given the opportunity to teach graduate courses: either advanced courses in their own specialty, or, if they are interested, introducing graduate courses. 2D Curriculum and development and review. We want to develop and enhance new courses in mathematics that broaden and integrate. The following are a few examples. We expect to develop others as well: New modules for Math Alive: This course was developed to explain the importance of mathematics in our society by pointing at concrete objects or issues and probing into the underlying mathematical ideas and concepts; it is aimed at students who will not major in mathematics, science, or engineering. The course is organized in largely independent two-week modules; for presently existing modules see www.princeton.edu/~matalive/. We want to develop new modules with other themes. We will also develop a similar course tailored to the greater mathematical experience and skills of math majors; this course will be structured again in (fewer) modules, but it will explore each in greater depth. We also want to develop an integrated analysis course that will cover several semesters. It will integrate and closely interrelate several fields, such as partial differential equations, harmonic analysis, real and complex analysis, analytic number theory and probability theory. This course will be primarily directed to third- and fourth-year undergraduate majors and b
