Mathematical Sciences: Small Geometry Project

由REU Site奖资助的八名学生将参与研究几个当前数学感兴趣的主题。该奖项延续了威廉姆斯学院在1988年进行的一个非常成功的试点项目，将为其扩展和取代不再可用的种子基金提供资源。标题中的术语几何并没有涵盖学生要研究的所有主题，尽管它确实提供了一个重点。几何问题包括面积最小化网络的研究，这些问题具有直接和重要的实际应用。特别感兴趣的是对这种网络的奇点结构和有向最小化网络的结构的分析。也将在纽结理论和属于拓扑学范畴的双曲3-流形的三角剖分方面进行工作。问题涉及具有14个或更少交叉点的所有纽结的计数(那些具有13个或更少交叉点的纽结已被编目)，以及新纽结和环多项式与双曲纽结或环补的双曲体积之间的关系。其他研究将涉及并行处理的可视化编程系统，探索和图和乘积图之间的关系-设计比目前已知的更有效的标记算法，以及在存在某些障碍的情况下试图评估平面部分的可见性时出现的计算几何问题。最后一个主题对堡垒的设计有有趣的应用。该奖项将为一个更大的努力的一部分提供支持，该努力涉及多达17名学生和6名教职员工。

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.