Quantitative geometry.
نویسنده
چکیده
Rather than describing a traditional field of mathematics, “quantitative geometry” refers to a modern way of thinking about geometry and its applications that occur in a wide range of mathematical disciplines. The overarching theme here is that certain geometric problems can only be formulated after the introduction of auxiliary quantitative parameters and asking for their asymptotic or approximate behavior. This leads to meaningful questions that do not necessarily have counterparts in classical “qualitative” geometric investigations, a viewpoint that allows one to uncover deep and useful structural information that appears only in the quantitative regime. Quantitative reasoning allows one to tame complicated objects and phenomena that are unwieldy if one insists on exact measurements. It is often the case that a rich yet structured picture emerges only if one asks the correct questions by allowing for controlled errors. The term quantitative geometry was coined as an attempt to formulate the widespread understanding of many researchers worldwide that in recent years numerous natural questions have emerged that mandate the study of geometric problems from an approximate perspective. This trend is sometimes dictated by application areas in which exact computation is infeasible yet approximate measurements are of crucial importance, but it arises most often as a natural modern development of traditional mathematical disciplines, where quantitative refinements of classical investigations are necessary to probe deeper and fully understand basic mathematical objects.
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عنوان ژورنال:
- Proceedings of the National Academy of Sciences of the United States of America
دوره 110 48 شماره
صفحات -
تاریخ انتشار 2013