Initiative for an Enhanced Mathematics Education in Princeton

普林斯顿VIGRE项目-摘要。一般的哲学。普林斯顿大学的VIGRE项目是数学系和应用与计算数学项目（PACM）的合资项目；它旨在为本科生、研究生和博士后提供更广泛的研究和教学经验。在学科深度的基础上，我们希望培养教育的探索和广度，这在当前快速变化的研究世界和就业市场中越来越重要。为学生和博士后准备各种不同的教学任务和研究机会比以往任何时候都更重要。无论他们从事学术或非学术职业，所有学生和博士后都将受益于教学和演讲经验，以及与其他机构和学术界以外的数学世界的接触。不同的成分。VIGRE项目的组成部分涉及研究生培训、本科教育、博士后奖学金、课程创新和推广活动。有几个组件是垂直集成的，它们的各个方面可以归入几个标题。对于大多数项目，我们不区分数学和PACM学生、博士后或教师：VIGRE项目将借鉴并结合两个项目的优势。2A研究生培训。通过大多数研究生教育的过滤，研究生们认为数学已经变得越来越碎片化。这是数学中几乎所有学科的情况，不管它们是多么“纯粹”或“应用”。将研究生教育中遇到的不同组成部分整合成更全球化的观点通常留给博士后研究。在我们的项目中，我们希望协调一些举措，帮助我们的研究生进入他们的研究和教学生涯，让他们接触到更广阔的数学及其应用视野。所有学生，无论VIGRE是否支持，都将被鼓励参加这些活动。有些活动是VIGRE学生必须参加的。我们将举办针对研究生的研究研讨会，由教师、访客或研究生就开放问题（通常与应用程序相关）进行说明性演讲。在同伴问题研讨会上，感兴趣的研究生将研究开放问题（通常比家庭作业问题更复杂，但不是论文大小的问题）；他们将被邀请在研究研讨会上展示他们的成果。这一倡议已经从分析和应用系列开始，并且可以扩展到其他领域。每年秋天，我们将邀请非学术数学家（例如来自工业研究实验室、政府组织和金融界的数学家）到普林斯顿大学进行一系列演讲，针对我们的研究生和高级本科生，介绍学术界以外的数学。我们将鼓励学生在他们的研究生学习期间在另一个环境中至少做一次实习（可能但不一定是在夏天）。我们将把这些机会“组合”在一起，并继续努力更新它，增加新的可能性。VIGRE学生将至少在另一个研究机构（学术或非学术）进行一次暑期实习或另一次长期访问。在适当的情况下，我们将让研究生与个别研究实验室“合作研究”。这种合作通常从一个小型项目开始，完成后出版，并在上述研究研讨会系列上发表报告。在某些情况下，这些项目可以发展成博士论文。由研究生和本科生代表组成的学生讨论会委员会将组织针对学生的说明性演讲，演讲者可以是教师、校外人士或学生；委员会将有机会邀请以阐述能力著称的外部演讲者。本学期每周至少有一名学生演讲者和至少一名教师或客座演讲者，由本委员会邀请。研究生将参与本科生新课程和课程材料的开发（见下文）。VIGRE支持的学生将在培训期间至少一个学期教授一门课程的一条规则。二、本科教育。参加由学生主持的说明性讲座系列（见上文）。在非学术数学环境中的实习，类似于研究生实习（在某些情况下，与他们相结合）将被强烈鼓励，并将机会组合在一起。我们将为数学专业的学生设立一个本科数学实验室，并由研究生和教师参与。我们将收集一个问题和猜想的数据库，这些问题和猜想在数学上是有趣的，并且（具有适度的背景和机器）适合于数值实验和科学检验。本科项目（可用于必修的两篇大三论文中的一篇，或者，如果内容更丰富，可用于大四论文）将是实验（即数值）检验某些现象；学生将被鼓励进一步探索自己，以及尝试自己的手证明特殊情况。这将引导他们体验发现的兴奋（以及失败的痛苦，也是研究的一部分！）口头陈述的经验：参加PACM本科证书课程的学生将在每周一次的PACM证书研讨会上对他们的独立工作进行陈述；这些报告首先与太平洋医学学院的研究生一起排练和编辑。我们鼓励数学专业的学生在学生组织的说明性讲座系列中做类似的演讲（关于他们的毕业论文，或者可能是关于暑期研究经历或本科实验室的项目）。2 .博士后奖学金。VIGRE博士后将被鼓励在一个学期的奖学金期间访问另一个研究所，可能是其他资助（例如GOALI）。博士后将被要求为课程发展做出重大贡献（见下文），并将被鼓励参与创新本科课程的教学。VIGRE博士后将有机会教授研究生课程：要么是自己专业的高级课程，要么是他们感兴趣的研究生课程。课程与发展与检讨。我们要发展和加强拓宽和整合的数学新课程。下面是几个例子。新模块：本课程旨在通过指出具体的对象或问题，探讨潜在的数学思想和概念，来解释数学在我们社会中的重要性；它针对的是那些不主修数学、科学或工程的学生。该课程主要以独立的两周模块组织；有关当前存在的模块，请参阅http://www.princeton.edu/~matalive/。我们想开发其他主题的新模块。我们还将开发一门类似的课程，为数学专业的学生提供更丰富的数学经验和技能；本课程将再次在（较少）模块中进行结构，但它将更深入地探索每个模块。我们还想开发一门涵盖几个学期的综合分析课程。它将整合和密切联系几个领域，如偏微分方程，谐波分析，实与复分析，解析数论和概率论。本课程主要针对本科三、四年级的学生

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 http://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