Large Permutations

排列是指物体按某种顺序排列：扑克牌中的纸牌，线性聚合物中的原子，年表中的物理或历史事件。该奖项支持从数学角度理解排列的研究，特别是涉及大量对象的排列。在组合学和概率论的界面上，强大的新技术允许对排列集合的行为进行大规模研究。与其他大型组合对象打交道的经验使数学家们期望它们以某种可预测的方式运行。因此，当这种排列在自然界中出现，并且不具有预测的特性时，科学家们知道要寻找一个隐藏的解释。例如，这涉及到对染色体上基因顺序的重要性的研究，以及对大型随机排列的整体行为的了解可能会告诉遗传学家何时寻找一种模式。新的发现使得更多地了解大排列的本质成为可能。特别地，所有大小的排列空间都可以用极限对象来完成，这些极限对象是单位正方形上具有均匀边缘的概率度量。通过最大化某个熵积分，原则上可以找到描述几乎所有具有某些给定性质的排列的极限对象。提议者和他的合作者将使用分析和实验技术来利用这一变分原理。一个重要的目标，刚刚开始，是用两个固定的模式密度绘制排列的景观，并确定，对于每个可行点，相应的排列的数量和特征。

A permutation is an arrangement of objects in some order: cards in a deck, atoms in a linear polymer, physical or historical events in a chronology. This award supports research into the understanding of permutations from a mathematical perspective, in particular permutations involving large numbers of objects. Powerful new techniques at the interface of combinatorics and probability allow the large-scale study of the behavior of ensembles of permutations. Experience with other large combinatorial objects has led mathematicians to expect them to behave in certain predictable ways. As a result, when such permutations arise in nature and do not have the predicted properties, scientists know to seek a hidden explanation. For example, this involves the study of the significance of the order of genes on a chromosome, and the knowledge of the behavior of ensembles of large random permutations may tell geneticists when to look for a pattern.New discoveries have made it possible to learn much more about the nature of large permutations. In particular, the space of permutations of all sizes can be completed by limit objects that are probability measures on the unit square with uniform marginals. By maximizing a certain entropy integral, one can, in principle, find the limit object that describes nearly all permutations with certain given properties. The proposer and his collaborators will be using both analytic and experimental techniques to exploit this variational principle. One important objective, barely begun, is to plot the landscape of permutations with two fixed pattern densities and determine, for each feasible point, the number and character of the corresponding permutations.