Oren Becker’s research while affiliated with University of Cambridge and other places

This paper is a journal counterpart to [5], in which we initiate the study of property testing problems concerning a finite system of relations E between permutations, generalizing the study of stability in permutations. To every such system E , a group Γ = Γ E is associated and the testability of E depends only on Γ (just like in Galois theory, where the solvability of a polynomial is determined by the solvability of the associated group). This leads to the notion of testable groups, and, more generally, Benjamini–Schramm rigid groups. The paper presents an ensemble of tools to check if a given group Γ is testable/BS-rigid or not.

We prove that every uniform approximate homomorphism from a discrete amenable group into a symmetric group is uniformly close to a homomorphism into a slightly larger symmetric group. That is, amenable groups are uniformly flexibly stable in permutations. This answers affirmatively a question of Kun and Thom and a slight variation of a question of Lubotzky. We also give a negative answer to Lubotzky’s original question by showing that the group \mathbb{Z} is not uniformly strictly stable. Furthermore, we show that \operatorname{SL}_{r}(\mathbb{Z}) , r\geq3 , is uniformly flexibly stable, but the free group F_{r} , r\geq2 , is not. We define and investigate a probabilistic variant of uniform stability that has an application to property testing.

This paper is a journal counterpart to our FOCS 2021 paper, in which we initiate the study of property testing problems concerning a finite system of relations E between permutations, generalizing the study of stability in permutations. To every such system E, a group $\Gamma=\Gamma_E$ is associated and the testability of E depends only on $\Gamma$ (just like in Galois theory, where the solvability of a polynomial is determined by the solvability of the associated group). This leads to the notion of testable groups, and, more generally, Benjamini-Schramm rigid groups. The paper presents an ensemble of tools to check if a given group $\Gamma$ is testable/BS-rigid or not.

Reduction is a fundamental computer science (CS) notion (Schwill, 1994). In solving reduction tasks one must think at the problem level, in order to recognize suitable correlations between problems. Thinking at the problem level involves recognition of declarative features of problems. This requires a high level of abstraction. Reduction is well apparent in the more advanced stages of CS studies, but it is also relevant at earlier stages. It may serve as an important means for practice and awareness of abstraction. We designed reduction tasks for the earlier tutoring and training of algorithms students. Our designs are illustrated with three creative tasks of different characteristics. Student solutions for each task are presented and discussed. The students demonstrated different levels of abstraction, insight and flexibility in solving the tasks.

We initiate the study of property testing problems concerning equations in permutations. In such problems, the input consists of permutations $\sigma_{1},\dotsc,\sigma_{d}\in\text{Sym}(n)$, and one wishes to determine whether they satisfy a certain system of equations E, or are far from doing so. If this computational problem can be solved by querying only a small number of entries of the given permutations, we say that E is testable. For example, when d=2 and E consists of the single equation $\mathsf{XY=YX}$, this corresponds to testing whether $\sigma_{1}\sigma_{2}=\sigma_{2}\sigma_{1}$. We formulate the well-studied group-theoretic notion of stability in permutations as a testability concept, and interpret all works on stability as testability results. Furthermore, we establish a close connection between testability and group theory, and harness the power of group-theoretic notions such as amenability and property $\text{(T)}$ to produce a large family of testable equations, beyond those afforded by the study of stability, and a large family of non-testable equations. Finally, we provide a survey of results on stability from a computational perspective and describe many directions for future research.

We prove that every uniform approximate homomorphism from a discrete amenable group into a symmetric group is uniformly close to a homomorphism into a slightly larger symmetric group. That is, amenable groups are uniformly flexibly stable in permutations. This answers affirmatively a question of Kun and Thom and a slight variation of a question of Lubotzky. We also give a negative answer to Lubotzky's original question by showing that the group $\mathbb{Z}$ is not uniformly strictly stable. Furthermore, we show that $\text{SL}_{r}(\mathbb{Z})$, $r\geq3$, is uniformly flexibly stable, but the free group $F_{r}$, $r\geq 2$, is not. We define and investigate a probabilistic variant of uniform stability that has an application to property testing.

In recent years, there has been a considerable amount of interest in stability of equations and their corresponding groups. Here, we initiate the systematic study of the quantitative aspect of this theory. We develop a novel method, inspired by the Ornstein–Weiss quasi-tiling technique, to prove that abelian groups are polynomially stable with respect to permutations, under the normalized Hamming metrics on the groups $\textrm{Sym}(n)$. In particular, this means that there exists $D\geq 1$ such that for $A,B\in \textrm{Sym}(n)$, if AB is $\delta$-close to BA, then A and B are $\epsilon$-close to a commuting pair of permutations, where $\epsilon \leq O\left (\delta ^{1/D}\right )$. We also observe a property-testing reformulation of this result, yielding efficient testers for certain permutation properties.

In recent years, there has been a considerable amount of interest in the stability of a finitely-generated group Γ with respect to a sequence of groups {Gn}n=1∞, equipped with bi-invariant metrics {dn}n=1∞. We consider the case Gn=U(n) (resp. Gn=Sym(n)), equipped with the normalized Hilbert-Schmidt metric dnHS (resp. the normalized Hamming metric dnHamming). Our main result is that if Γ is infinite, hyperlinear (resp. sofic) and has Property (T), then it is not stable with respect to (U(n),dnHS) (resp. (Sym(n),dnHamming)). This answers a question of Hadwin and Shulman regarding the stability of SL3(Z). We also deduce that the mapping class group MCG(g), g≥3, and Aut(Fn), n≥3, are not stable with respect to (Sym(n),dnHamming). Our main result exhibits a difference between stability with respect to the normalized Hilbert-Schmidt metric on U(n) and the (unnormalized) p-Schatten metrics, since many groups with Property (T) are stable with respect to the latter metrics, as shown by De Chiffre-Glebsky-Lubotzky-Thom and Lubotzky-Oppenheim. We suggest a more flexible notion of stability that may repair this deficiency of stability with respect to (U(n),dnHS) and (Sym(n),dnHamming).

In recent years, there has been a considerable amount of interest in stability of equations and their corresponding groups. Here, we initiate the systematic study of the quantitative aspect of this theory. We develop a novel method, inspired by the Ornstein-Weiss quasi-tiling technique, to prove that abelian groups are polynomially stable with respect to permutations, under the normalized Hamming metrics on the groups $\operatorname{Sym}(n)$. In particular, this means that there exists $D\geq 1$ such that for $A,B\in \operatorname{Sym}(n)$, if AB is $\delta$-close to BA, then A and B are $\epsilon$-close to a commuting pair of permutations, where $\epsilon\leq O(\delta^{1/D})$. We also observe a property-testing reformulation of this result, yielding efficient testers for certain permutation properties.