Alex Lubotzky “Stability and Testability of Permutations’ Equations“ | The Vinberg Lecture

The Vinberg Lecture Series: Lecture 1 (Oct 5, 2021) _______ Alex Lubotzky (The Hebrew University of Jerusalem): Stability and Testability of Permutations’ Equations Abstract: Let A and B be two permutations in Sym(n) that ``almost commute’’ -- are they a small deformation of permutations that truly commute? More generally, if R is a system of words-equations in variables X = {x_1, ..., x_d} and A_1, ..., A_d are permutations that are nearly solutions; are they near true solutions? It turns out that the answer to this question depends only on the group presented by the generators X and relations R. This leads to the notions of ``stable groups’’ and ``testable groups’’. We will present a few results and methods which were developed in recent years to check whether a group is stable or testable. We will also describe the connection of this subject with property testing in computer science, with the long-standing problem of whether every group is sofic, and