Eberlein oligomorphic groups

寡态自同构群的混合恒等式

Mixed identities for oligomorphic automorphism groups

本文研究可数关系结构的寡态自同构群的混合恒等式，给出其不承认不含特定常元的混合恒等式的充分条件，并通过多个例子证明许多情形下不存在非奇异混合恒等式。

Unitary representations of oligomorphic groups

We obtain a complete classification of the continuous unitary representations of oligomorphic permutation groups (those include the infinite permutation group $S_\infty$, the automorphism group of the countable dense linear order, the homeomorphism group of the Cantor space, etc.). Our main result is that all irreducible representations of such groups are obtained by induction from representations of finite quotients of open subgroups and moreover, every representation is a sum of irreducibles. As an application, we prove that many oligomorphic groups have property (T). We also show that the G…

具有弱虚数消去的寡态群的同构问题

The Isomorphism Problem for Oligomorphic Groups with Weak Elimination of Imaginaries

本文证明在弱虚数消去条件下，实数相等关系是寡态群拓扑同构复杂性的严格下界，并将模型论方法用于置换群同构重构及无限维一般线性群自同构描述。

寡态群、其自同构群及其同构的复杂性

Oligomorphic groups, their automorphism groups, and the complexity of their isomorphism

本文研究Roelcke预紧的置换群子群，证明其内自同构群在自同构群中闭，且在外自同构群完全不连通局部紧；并给出方法证明两类寡态群的同构关系光滑，且其自同构群拓扑同构于寡态群，外自同构群为profinite。

Coarse groups, and the isomorphism problem for oligomorphic groups

Let $S_\infty$ denote the topological group of permutations of the natural numbers. We study the complexity of the isomorphism relation on classes of closed subgroups $S_\infty$ in the setting of Borel reducibility between equivalence relations on Polish spaces. Given a closed subgroup $G$ of $S_\infty$, the coarse group $\mathcal M(G)$ is the structure with domain the cosets of open subgroups of $G$, and a ternary relation $AB \sqsubseteq C$. If $G$ has only countably many open subgroups, then $\mathcal M(G)$ is a countable structure. Coarse groups form our main tool in studying such closed s…

寡态群与张量范畴

Oligomorphic groups and tensor categories

本文为寡态群引入测度概念，构造刚性张量范畴及其阿贝尔包络，推广了Deligne插值范畴，并首次得到正特征下超指数增长的半单pre-Tannakian范畴，核心贡献是建立寡态群上的积分理论。