Interactions Among Probability, Group Theory, Graph Theory, and Ergodic Theory

概率、群论、图论和遍历理论之间的相互作用

• 批准号: 1007244
• 负责人: Russell Lyons
• 金额: $ 30.32万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007244&HistoricalAwards=false
• 关键词: Interactions; Among; Probability; Group; Theory

It is proposed to deepen various connections among the mathematical areas of probability, group theory, graph theory, and ergodic theory. Mainly this will be achieved by probabilistic thinking about questions that arise in other areas. In group theory, the PI will work on the question of whether every group is sofic. The PI discovered with Aldous that a probabilistic setting leads to a wider framework for this question and suggests a new approach to it. In graph theory, probabilistic thinking leads to new questions and results involving inequalities for finite graphs. Namely, one often finds when counting combinatorial objects that a subgraph contains fewer of them than the whole graph. But if the subgraph is based on fewer vertices, then one ought to normalize the counts to reflect this. The PI has some partial results in this direction and proposes to find more. In ergodic theory, questions involve graphings, colorings, and factors. In fact, it may be that these investigations will lead to progress in the theory of percolation on groups. Combining several of these areas are very natural processes that are related (by previous work of the PI) to algebraic invariants known as ell-2-Betti numbers. Therefore, they suggest ways of resolving an important open question about these Betti numbers.In the 19th century, Cayley introduced graphs (networks) to represent the algebraic objects known as groups. It is always desirable to have finite approximations to infinite objects, and the same holds for infinite groups.Gromov and Weiss suggested a way to use finite networks for this purpose.If one can actually succeed in making such approximations for all groups, then this would resolve a host of important conjectures in a variety of fields of mathematics. The PI proposes to continue work on this question.Inequalities are important in most areas of mathematics. The field of graph theory and combinatorics contains many inequalities, often of the form that certain graphs contain the most (or the fewest) possible objects of a certain type among all graphs in a given class. The PI will develop novel inequalities of this type, which are inspired by a probabilistic viewpoint.Topology is the study of the shape of things. One tool is to count the number of holes of various dimensions. But if the whole space of interest is infinite, then the number of holes is often either zero or infinity. It turns out that there is a more informative way to count holes, and previous work of the PI has shown how it is related to random combinatorial objects.Therefore, the PI will attempt to use his new random objects to resolve an important open question about this hole counting.

建议加深概率、群论、图论和遍历理论等数学领域之间的各种联系。 这主要是通过对其他领域出现的问题进行概率思考来实现的。 在群论中，PI 将研究每个群是否都是 sofic 的问题。 PI 与 Aldous 一起发现概率设置可以为这个问题带来更广泛的框架，并提出了一种新的解决方法。在图论中，概率思维引发了涉及有限图不等式的新问题和结果。 也就是说，在计算组合对象时，人们经常会发现子图包含的组合对象少于整个图。但是，如果子图基于较少的顶点，那么应该对计数进行标准化以反映这一点。 PI 在这方面取得了一些部分成果，并建议寻找更多成果。 在遍历理论中，问题涉及图形、着色和因子。事实上，这些研究可能会导致群体渗透理论的进步。 将其中几个区域组合起来是非常自然的过程，与称为 ell-2-Betti 数的代数不变量相关（通过 PI 之前的工作）。 因此，他们提出了解决有关这些贝蒂数的重要开放问题的方法。 在 19 世纪，凯莱引入了图（网络）来表示称为群的代数对象。 人们总是希望对无限对象进行有限近似，对于无限群也是如此。格罗莫夫和韦斯提出了一种使用有限网络来实现此目的的方法。如果人们真的能够成功地对所有群进行这种近似，那么这将解决数学各个领域中的许多重要猜想。 PI 建议继续研究这个问题。不等式在数学的大多数领域都很重要。图论和组合学领域包含许多不等式，通常的形式是某些图在给定类的所有图中包含某种类型的最多（或最少）可能的对象。受到概率观点的启发，PI 将开发出此类新颖的不等式。拓扑学是对事物形状的研究。一种工具是计算各种尺寸的孔的数量。但如果整个感兴趣的空间是无限的，那么孔的数量通常为零或无穷大。事实证明，有一种信息更丰富的方法来计算孔洞，并且 PI 之前的工作已经展示了它与随机组合对象的关系。因此，PI 将尝试使用他的新随机对象来解决有关此孔洞计数的重要悬而未决的问题。