AF: Small: Mysteries of Geometric Arrangements

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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.