Groups, Manifolds, and Complexes

在科学中，一个相当重要的问题是关于模拟感兴趣系统的方程的解的稳定性。问题的一般形式是：给定一个近似解，是否也必然存在一个接近近似解的精确解？这个问题已经在许多方面得到了深入的研究，答案取决于具体的系统和“近似解”和“近似”的确切含义。“目前的研究项目开发了这样一个理论的方程组的排列。这种情况是特别感兴趣的两个原因：首先，它可以被解释为一个“本地可测试性”的问题，在计算机科学的极大兴趣的主题。也就是说，它是一系列问题的一部分，这些问题具有以下特点：给定了一个非常大的字母串，人们想通过只知道其中几个字母来检查它的性质。第二个原因是，这个项目所研究的问题与抽象代数中的一些深层概念密切相关。因此，这是一个非常跨学科的项目，涉及数学和计算机科学的几个领域的混合。更详细地说，该项目涉及群论中的属性测试。属性测试是计算机科学中的一个研究方向，它寻找方法来检查给定对象的属性，比如一个由0和1组成的长向量，只看它的一小部分随机位。下面的问题就是这种类型。给定一个由集合T上的对称置换群的元素X和Y构成的字w（X，Y），假设对于两个置换A和B，w（A，B）保持T的一些随机选择的元素不变。是否可以推断w（A，B）是单位元，或者至少A和B接近于一对排列A'和B'，其中w（A '，B'）是单位元？事实证明，这个问题的答案只取决于（通常是无限的）群的结构，群的生成元和关系的表示由这些词给出。这暗示了群论中的可测性（或稳定性）概念。该项目的主要目标是开发一套工具，以确定哪些组是可测试的，哪些是不可测试的。该项目还将探索与舒适性，Kazhdan的财产（T）和sofic集团的联系。

A question of considerable importance in science concerns the stability of solutions to equations that model systems of interest. A general form of the question asks: Given an approximate solution, is there necessarily also an exact solution that is close to the approximate solution? This question has been studied intensively in many directions, with answers depending on the specific system and the exact meaning of "approximate solution" and "close." The current research project develops such a theory for equations in sets of permutations. This case is of special interest for two reasons: Firstly, it can be interpreted as a "local testability" question, a subject of great interest in computer science. Namely, it is part of a series of problems of the following flavor: a very large string of letters is given, and one wants to check its properties by knowing only a few of the letters. A second reason is that it turns out that the question under study in this project is intimately related to some deep concepts in abstract algebra. It is thus a project of a very interdisciplinary nature, involving a blend of several areas of mathematics and computer science.In more detail, the project deals with property testing within group theory. Property testing is a direction of research in computer science that looks for methods to check a property of a given object, say a long vector of 0's and 1's, by looking only at a small number of random bits of it. The following question is of this type. Given a word w(X,Y) built from elements X and Y of the symmetric group of permutations on a set T, assume that for two permutations A and B, w(A,B) leaves unchanged some randomly selected elements of T. Can it be inferred that w(A,B) is the identity, or at least, that A and B are close to a pair of permutations A' and B' for which w(A',B') is the identity? It turns out that the answer to this question depends only on the structure of the (usually infinite) group whose presentation by generators and relations is given by these words. This suggests a concept of testability (or stability) in group theory. The main objective of this project is to develop a set of tools to determine which groups are testable and which are not. The project will also explore connections with amenability, Kazhdan's Property (T), and sofic groups.