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

建议深化概率论、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将解决是否每个群都是SOFIC的问题。PI和Aldous一起发现，概率设置导致了这个问题的更广泛的框架，并建议了一种新的方法。在图论中，概率思维导致了涉及有限图的不等式的新问题和新结果。也就是说，当计算组合对象时，人们经常发现一个子图包含的组合对象比整个图少。但是，如果子图基于较少的顶点，则应该将计数归一化以反映这一点。PI在这个方向上有一些局部结果，并建议找到更多。在遍历理论中，问题涉及图形、颜色和因子。事实上，这些研究可能会导致关于群的渗流理论的进步。其中几个区域的组合是非常自然的过程，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，他们提出了解决关于这些Betti数的一个重要的未决问题的方法。在19世纪，Cayley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的对象有有限的近似，对无限的群也是如此。Gromov和Weiss提出了一种使用有限网络来实现这一目的的方法。如果一个人真的能成功地对所有的群进行这样的逼近，那么这将解决各种数学领域中的许多重要猜想。国际数学联合会建议继续在这个问题上工作。在大多数数学领域，不等式是很重要的。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特定类型的对象最多(或最少)。PI将在概率观点的启发下发展出这类新的不等式。拓扑学是对事物形状的研究。一种工具是计算不同尺寸的孔的数量。但如果整个感兴趣的空间是无限的，那么空穴的数量通常要么是零，要么是无穷大。事实证明，有一种更有信息量的方法来计数空洞，而PI的前人的工作已经证明了它是如何与随机组合对象相关联的，因此，PI将尝试使用他的新的随机对象来解决关于这种空洞计数的一个重要的公开问题。

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.