AF: Small: Mysteries of Geometric Arrangements

该奖项致力于计算几何和组合几何的界面研究。具体地说，相关问题涉及几何排列，也就是说，当一些几何形状叠加在一起时，在平面上形成的图案。在安排方面，可以提出数量惊人的计算问题。相反，相当多的组合问题等同于关于排列的陈述。在略微更多的技术细节上，私人投资将：--推动安排领域的最新技术，解决该学科中一些具有挑战性的未决问题；--开发具有独立兴趣并将在提案中列出的一组问题之外找到应用的一般技术；--以根本不同的方式看待经典问题，找到简单的解决方案，并揭示看似不相关的问题之间的联系；-将为处理几何对象的排列而开发的机器应用于一组全新的问题。PI将指导一名博士生和一名博士后，指导他们的研究活动，并让他们熟悉最新的组合几何和计算几何工具。将通过参加讲习班和科学会议来传播所取得的成果。研究生和本科生将被介绍到该学科目前的一些研究问题。

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.