Understanding Geometric Arrangements: Unions and Beyond

建议：CCF-0830691机构：纽约理工大学PI：Aronov，Boris标题：了解几何排列：并集和超越研究中心是一类涉及对象并集的几何问题。在组合几何和计算几何中大量研究的各种问题都可以用这些术语来描述。在研究组合学和算法之间联系的重要经验的基础上，研究人员探索对象并集问题中出现的组合问题，以开发新的、简单和强大的组合工具，并改进现有的工具。通过研究，研究人员强调了以下主题：开发新的、更强大的工具来处理组合问题。发现“经典”问题的更容易的证据和简化的方法，使学生和非专家更容易获得研究成果，并最终与其声称的受益者--应用领域的从业者--更相关。此外，它们很有可能促进该领域的进一步进展。

Proposal: CCF- 0830691Institution: Polytechnic University of New YorkPI: Aronov, BorisTitle: Understanding Geometric Arrangements: Unions and Beyond ABSTRACTThe research centers on a class of geometric problems involving unions of objects. A variety of much-studied problems in combinatorial and computational geometry can be cast in these terms. Building on significant previous experience investigating the connections between combinatorics and algorithmics, the investigators explore combinatorial problems arising in the union-of-objects questions in order to develop new, simple, and powerful combinatorial tools and to refine those already available. Refining the techniques developed so far, the work expands the scope of problems to encompass questions of optimization in overlays of objects.Throughout the investigation, the researchers emphasize the following theme: Developing new, more powerful tools for dealing with combinatorial problems. Discovery of easier proofs of and streamlined approaches to ``classical'' problems makes research results significantly more accessible to students and non-specialists and ultimately more relevant to its claimed beneficiaries, the practitioners in the application areas. Moreover, they are very likely to foster further progress in the field.