置换与作用迹中的约束稳定性

置换间关系的可测试性

Testability of relations between permutations

该研究首次探究置换间关系的属性测试问题，定义关联图编码可测试性信息，基于图扩张性质给出可测试/不可测试判定准则，还将置换稳定性概念纳入可测试性框架并梳理未来方向。

局部Hilbert--Schmidt稳定性

Local Hilbert--Schmidt stability

该研究引入局部Hilbert--Schmidt稳定性概念，给出非剩余有限群的相关例子，为顺从群提供特征标判别准则，研究其灵活类似物并证明相关结果，还证明无限sofic/hyperlinear性质(T)群不满足对应局部稳定性，强化了经典结论并回答了公开问题。

不变迹单纯形与相对性质(T)

Invariant trace simplices and relative property (T)

本文研究了群作用下不变迹单纯形的Bauer性质，证明了在相对性质(T)和ergodic性条件下，该单纯形具有Bauer性质，并应用于多种群C*-代数和交叉积。

Tracial stability for C*-algebras

We consider tracial stability, which requires that tuples of elements of a C*-algebra with a trace that nearly satisfy the relation are close to tuples that actually satisfy the relation. Here both "near" and "close" are in terms of the associated 2-norm from the trace, e.g., the Hilbert-Schmidt norm for matrices. Precise definitions are stated in terms of liftings from tracial ultraproducts of C*-algebras. We completely characterize matricial tracial stability for nuclear C*-algebras in terms of certain approximation properties for traces. For non-nuclear $C^{\ast}$-algebras we find new obstr…

Constraint metric approximations and equations in groups

We introduce notions of a constraint metric approximation and of a constraint stability of a metric approximation. This is done in the language of group equations with coefficients. We give an example of a group which is not constraintly sofic. In building it, we find a sofic representation of free group with trivial commutant among extreme points of the convex structure on the space of sofic representations. We consider the centralizer equation in permutations as an instance of this new general setting. We characterize permutations $p\in S_k$ whose centralizer is stable in permutations with r…

酉群与矩阵代数约化积的迹范数刚性

Trace-norm rigidity for reduced products of unitary groups and matrix algebras

研究有限维酉群和矩阵代数的迹度量约化积间同态，证明酉群相关稳定性变体等，表明乘积形式同构由特定方式诱导，还得到坐标识别及自同构群的刚性与分类结果，矩阵代数在更一般背景下有类似刚性结果。

A converse to the Grace--Walsh--Szegő theorem

We prove that the symmetrizer of a permutation group preserves stability of a polynomial if and only if the group is orbit homogeneous. A consequence is that the hypothesis of permutation invariance in the Grace-Walsh-Szegő Coincidence Theorem cannot be relaxed. In the process we obtain a new characterization of the \emph{Grace-like polynomials} introduced by D. Ruelle, and prove that the class of such polynomials can be endowed with a natural multiplication.

Stability for product groups and property $(τ)$

We study the notion of permutation stability (or P-stability) for countable groups. Our main result provides a wide class of non-amenable product groups which are not P-stable. This class includes the product group $Σ\timesΛ$, whenever $Σ$ admits a non-abelian free quotient and $Λ$ admits an infinite cyclic quotient. In particular, we obtain that the groups $\mathbb F_m\times\mathbb Z^d$ and $\mathbb F_m\times\mathbb F_n$ are not P-stable, for any integers $m,n\geq 2$ and $d\geq 1$. This implies that P-stability is not closed under the direct product construction, which answers a question of B…

B型Iwahori-Hecke代数的同调稳定性

Homological stability for Iwahori-Hecke algebras of type B

证明了B型Iwahori-Hecke代数族的同调稳定性，将同调与相关Tor群等同，推广了Hepworth关于A型的结果。

Stability of approximate group actions: uniform and probabilistic

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 f…