Mathematical Sciences: Small Geometry Project

八名学生支持这个REU网站奖将 参与研究了当前数学领域的几个主题， 兴趣 继续开展非常成功的试点项目， 由威廉姆斯学院在1988年，这个奖项将提供资源 用于其扩展和更换种子基金， 不再可用。 标题中的术语几何并不涵盖所有的主题 学生们的调查，虽然它确实提供了一个 专心点 几何问题包括面积最小化的研究 这些网络具有直接和重要的实际意义， 应用. 特别令人感兴趣的将是对 这种网络的奇点结构和 定向最小化网络 工作也将在结 理论和双曲三维流形的三角剖分， 福尔斯属于拓扑学范畴。 问题涉及 枚举所有结有14个或更少的交叉（那些 13个或更少的已被编目）和关系 在新的纽结和链环多项式与双曲多项式之间， 双曲纽结或环补的体积。 其他研究将涉及可视化编程系统， 并行处理，探索和之间的关系， 图表和产品图表-设计更有效的标签 算法比目前已知的，和计算几何 在试图评估部分的可见性时出现的问题 飞机在某些障碍物的存在。 最后一个话题 在堡垒的设计中有着有趣的应用。 该奖项将为更大的努力提供部分支持 涉及多达17名学生和6名教师。

Eight students supported by this REU site award will be involved in researching several topics of current mathematical interest. Continuing a highly successful pilot project conducted by Williams College in 1988, this award will provide resources for its expansion and for the replacement of seed funds which are no longer available. The term geometry in the title does not cover all the topics to be investigated by the students, although it does provide a focus. Geometric problems include study of area minimizing networks which have immediate and important practical applications. Of particular interest will be an analysis of the structure of singularities of such networks and the structure of directed minimizing networks. Work will also be done in knot theory and on triangulations of hyperbolic 3-manifolds which falls within the province of topology. Questions involve the enumeration of all knots having 14 or fewer crossings (those with 13 or fewer have been catalogued) and the relationship between the new knot and link polynomials and the hyperbolic volume of hyperbolic knot or link complement. Other research will involve visual programming systems for parallel processing, explorations of the relation between sum graphs and product graphs - devising more efficient labeling algorithms than those currently known, and computational geometry problems which arise in trying to asses visibility of portions of the plane in the presence of certain obstacles. The last topic has interesting applications to the design of fortresses. The award will provide support for part of a larger effort involving up to 17 students working with six faculty members.