AF: Small: Mysteries of Geometric Arrangements

该奖项致力于研究计算几何和组合几何的接口。 具体地说，相关的问题涉及几何排列，即当许多几何形状彼此重叠时在平面上形成的图案。 大量的计算问题都可以用排列的方式来解决。 相反，相当数量的组合问题等价于关于排列的陈述。 PI将专注于这两者之间的相互作用。在技术细节方面，PI将：-推动安排领域的最新技术，解决该主题中一些具有挑战性的开放问题;-开发具有独立兴趣的通用技术，并将发现应用远远超出提案中列出的问题集;--以完全不同的方式看待经典问题，找到简单的解决方案，揭示看似无关的问题之间的联系;--将为处理几何物体的排列而开发的机器应用于一系列全新的问题。PI将指导一名博士生和一名博士后研究员，指导他们的研究活动，并用组合和计算几何的最新工具指导他们。 将通过参加讲习班和科学会议传播所取得的成果。 研究生和本科生将被介绍给一些当前的研究问题的主题。

This award is devoted to the studies on the interface of computational and combinatorial geometry. Specifically, the relevant problems involve geometric arrangements, which are patterns formed, say, in the plane, when a number of geometric shapes are overlaid on top of each other. A surprising number of computational problems can be cast in terms of arrangements. Conversely, a significant number of combinatorial problems are equivalent to statements about arrangements. The PI will focus on the interplay of these two.In slightly more technical detail, the PI will:-- push the state of the art in the area of arrangements, solving some of the challenging open problems in the subject;-- develop general techniques which will be of independent interest and which will find applications well beyond the set of problems listed in the proposal;-- by looking at classical problems in a radically different way, find simple solutions and reveal connections between seemingly unrelated questions;-- apply the machinery developed for dealing with arrangements of geometric objects to an entirely new set of problems.The PI will supervise a PhD student and a postdoctoral fellow, guiding them in their research activities and acquainting them with the latest tools of combinatorial and computational geometry. The results obtained will be disseminated by participation in workshops and scientific conferences. Graduate and undergraduate students will be introduced to a number of current research problems in the subject.