Approximate Isomorphism of Metric Structures

Uniform Roe coronas

A uniform Roe corona is the quotient of the uniform Roe algebra of a metric space by the ideal of compact operators. Among other results, we show that it is consistent with ZFC that isomorphism between uniform Roe coronas implies coarse equivalence between the underlying spaces, for the class of uniformly locally finite metric spaces which coarsely embed into a Hilbert space. Moreover, for uniformly locally finite metric spaces with property A, it is consistent with ZFC that isomorphism between the uniform Roe coronas is equivalent to bijective coarse equivalence between some of their cofinite…

Logic for metric structures and the number of universal sofic and hyperlinear groups

Using the model theory of metric structures, I give an alternative proof of the following result by Thomas: If the Continuum Hypothesis fails then there are power of the continuum many universal sofic groups up to isomorphism. This method is also applicable to universal hyperlinear groups, giving a positive answer to a question posed by Thomas.

关于对称群的度量约化积的自同构群

On automorphism groups of metric reduced products of symmetric groups

在OCA和马丁公理下，研究对称群度量约化积的同构，刻画其自同构群。

Unstable classes of metric structures

We prove a strong non-structure theorem for a class of metric structures with an unstable pair of formulae. As a consequence, we show that weak categoricity (that is, categoricity up to isomorphisms and not isometries) implies several versions of stability. This is the first step in the direction of the investigation of weak categoricity and weak stability of metric classes.

度量值域的近似AKE原理

An Approximate AKE Principle for Metric Valued Fields

本文基于连续逻辑研究度量值域，提出在投影线上的一种近似AKE原理，描述了等价性与余域和值群的关系，并证明了度量值差域在任意特征下无模型伴。

Games and Scott sentences for positive distances between metric structures

We develop various Ehrenfeucht-Fra\"ıssé games for distances between metric structures. We study two forms of distances: pseudometrics stemming from mapping spaces onto each other with some form of approximate isomorphism, and metrics stemming from measuring the distances between two spaces isometrically embedded into a third space. Using an infinitary version of Henson's positive bounded logic with approximations, we form Scott sentences capturing fixed distances to a given space. The Scott sentences of separable spaces are in $\mathcal{L}_{ω_1ω}$ for 0-distances and in $\mathcal{L}_{ω_2ω}$ f…

An approximate Herbrand's theorem and definable functions in metric structures

We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite-dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable functions in Hilbert spaces expanded by a group of generic unitary operators and Hilbert spaces expanded by a generic subspace. We also show how Herbrand's theorem can be used to characterize definable functions in some absolutely ubiquitous structures from classical logic.

关于连续模型论中$\mathrm{C}^*$-代数超积的同构问题

On the isomorphism problem for ultraproducts of $\mathrm{C}^*$-algebras in continuous model theory

本文探讨了$\mathrm{C}^*$-代数超积的同构问题，证明在否定连续假说下，存在两个元素等价的无限维单位$\mathrm{C}^*$-代数，其超积不相等，展示了模型论与连续假说的关系。

关于度量近似子群

On metric approximate subgroups

本文将离散近似子群的 Lie 模型定理推广到双不变度量群，在有限覆盖条件下证明度量近似子群可由典范 Lie 群解释，并推出有界指数群中的近子群结论。

A Model Existence Theorem for Infinitary Formulas in Metric Spaces

We prove a Model Existence Theorem for a fully infinitary logic for metric structures. This result is based on a generalization of the notions of approximate formulas and approximate truth in normed structures introduced by Henson and studied in different forms by Anderson and Fajardo & Keisler. This theorem extends Henson's Compactness Theorem for approximate truth in normed space structures to infinitary formulas.