Mathematical Sciences: Small Geometry Project

一项涉及12名学生的本科生研究经验项目将获得该奖项的支持。学生们将在7位指导老师的指导下工作10周。其他资金将为额外的学生提供支持，使该小组的总人数达到20人左右。该补助金将继续资助过去三年成功推行的类似规模的计划。虽然许多参与者还没有毕业，但至少有7人正在攻读数学和计算机科学的研究生课程。该计划的结构使每个学生与两个小组一起工作。他们最初关注几个问题，直到其中一个成为最有希望的问题。主题是几何的，尽管标题中的术语SMALL是首字母缩略词，代表第一组顾问的首字母，而不是用作形容词。典型的问题领域将包括涉及结理论的问题。学生将考虑如何分析结的交叉可以提供关于它们的一般分类的信息。这些思想与图论中的其他思想相关，因为某些结点可以嵌入图中。人们想要描述这样的图。其他主题包括被称为贝蒂数的拓扑特征的计算，在柯西-黎曼结构的微分几何中出现的向量值厄米形式，以及在解析数论中对等差数列的除数问题的研究。学生将从新英格兰地区招募。我们将努力在夏季结束后继续进行研究。威廉姆斯学院招收的学生尤其如此。预计该项目将产生几篇可发表的论文，一些学生将有机会在专业会议上展示他们的工作。

A Research Experiences for Undergraduates project involving 12 students will be supported by this award. The students will work for ten weeks under the supervision of seven faculty advisors. Other funds will provide support for additional students bringing the total size of the group to about 20. The grant renews funding of successful projects of similar size carried out during the past three years. Although many of the participants have not yet graduated, at least seven are known to be enrolled in graduate study in mathematics and computer science. The program is structured so that each student works with two subgroups. They focus on several problems initially until one emerges as the most promising. The themes are geometric although the term SMALL in the title is an acronym representing the initials of the first group of advisors and is not used as an adjective. Typical problems areas will include questions involving knot theory. Students will consider how an analysis of the crossings of knots can provide information about their general classification. These ideas are related to others in graph theory in that certain knots can be embedded in graphs. One would like to characterize such graphs. Other topics include the calculation of topological characteristics called Betti numbers, vector-valued Hermitian forms arising in the differential geometry of Cauchy-Riemann structures and work in analytic number theory on the divisor problem for arithmetic progressions. Students will be recruited from the New England area. Efforts will be made see that the research continues after the summer ends. This will be particularly true of the students who are recruited from Williams College. It is expected that several publishable papers will result from the project and that some students will have the opportunity to present their work at professional meetings.