This webpage is about the 2025 seminar on Continuous Logic and Free Probability. There are TEN lecturers from diverse universities to give related talks. Titles and abstracts are updated on this webpage.
Summary
2025/09/19, 10:00 a.m., Tencent Meeting
Hu Yuqi (UC Irvine) Reviewer: Wang Wei (SYSU)
Basic continuous model theory
I will discuss continuous logic and show how it can be used to extend Voiculescu's free entropy of microstates to model-theoretic types. I will also present some results and open problems related to the analogy between microstates and types.
在本报告中，我将探讨连续逻辑，并展示如何利用它将 Voiculescu 的自由微态熵推广至模型论类型。此外，我将介绍关于微态与类型类比方面的现有研究成果和未解之谜。
References:
[1] Isaac Goldbring (Ed.) Model Theory of Operator Algebras.
Dr. Hu discusses the basic concepts of continuous logic and free probability theory and their applications in model theory, with particular attention to the developement of free probability theory based on continuous logic, as well as proving a major theorem of Voiculescu's free entropy from the perspective of continuous logic. -- Wang Wei
2025/10/03, 9:30 a.m., Tencent Meeting Password: LQL2
Li Xiangchao (USTC) Reviewer: Xu Yuanyuan (CAS)
Central limit theorem for mesoscopic eigenvalue statistics of the free sum of matrices
This paper studies the central limit theorem (CLT) for mesoscopic eigenvalue statistics of the free sum of random matrices. Consider matrices of the form: \[H_N = A_N + U_N B_N U_N^*,\] where \(A_N, B_N\) are deterministic Hermitian matrices, and \(U_N\) is a Haar-distributed random unitary (or orthogonal) matrix. The focus is on linear eigenvalue statistics: \[\sum_{i=1}^N g\left(\frac{\lambda_i - E}{\eta}\right) - \mathbb{E}\sum_{i=1}^N g\left(\frac{\lambda_i - E}{\eta}\right)\] within the mesoscopic regime \(N^{-1} \ll \eta \ll 1\) in the regular bulk of the spectrum. The main result establishes convergence to a centered Gaussian random variable with variance: \[\frac{1}{2\beta\pi^2} \int_{\mathbb{R}}\int_{\mathbb{R}} \frac{(g(x_1) - g(x_2))^2}{(x_1 - x_2)^2} dx_1 dx_2 = \frac{1}{\beta\pi} \int_{\mathbb{R}} |\xi||\hat{g}(\xi)|^2 d\xi, \] where \(\beta = 2\) (unitary case) or \(\beta = 1\) (orthogonal case). The proof uses:
Characteristic function analysis of linear statistics.
Ward identities from Haar measure invariance.
Local laws for resolvents, e.g., \(|G_{ij}(z) - \frac{1}{a_i - \omega_B(z)}\delta_{ij}| \prec \frac{1}{\sqrt{N|\eta|}}\).
Analytic subordination properties of free additive convolution: \[F_{\mu_A}(\omega_B(z)) = F_{\mu_B}(\omega_A(z)), \quad F_\mu(z) = -1/m_\mu(z).\] Extensions include:
Orthogonal conjugation case (\(H_N = A + OBO^T\)).
Edge behavior with modified variance and bias terms.
This work connects to universality in random matrix theory and free probability, providing a unified framework for mesoscopic fluctuations.
The report mainly focused on the indepth discussion of random matrix theory on free sums. It explored the universality in the bulk and special behaviors at the edge. For the edge, analyzing its unique properties precisely, studying the relationship with other regions and how it behaves under Tracy-Widom limits are potential research directions.. -- Xu Yuanyuan
2025/10/19, 09:30 a.m., Zoom
Wang Haoming (SYSU) Reviewer: Koki Shimizu (TUS)
Matrix distributions under classical group actions
This work is based on the proper orthogonal decomposition or Karhunen-Lo'eve theorem for stochastic processes. Four canonical digonal forms \(T_{1}\), \(T_{1\frac{1}{2}}\), \(T_{2}\) and \(T_{3}\) are considered.
Four matrix normal distributions are introduced, extending the separable covariance \(\varPhi \otimes \varPsi\) with potentially variable-level (\(\varPsi\)) and/or sample-level (\(\varPhi\)) correlations.
The corresponding Wishart distribution, matrix \(t\)-distribution, matrix \(F\)-distribution are considered in the finte sample case. Several well-known results, including the non-central Wishart distribution and normal quadratic forms, now appear as corollaries.
Distributions of the largest and smallest roots in principle component analysis and analysis of variance are calculated, with discussion of future extensions to large sample studies.
This study extends the classical matrix distribution theory under the Gaussian model by classifying covariance structures into four forms. Generalized Wishart and related distributions are presented, and further developments are expected for the complex case and the elliptical model. Numerical computation for the distributions in the special cases obtained remains a topic for future work. -- Koki Shimizu
2025/10/31, 10:00 a.m., Tencent Meeting Password: R7YS
Continuous model theory and local geometry of Banach space
In this talk, I will talk about model theoretic properties of finite representability and super-weak compactness of bounded convex metric spaces, and give a model theoretic proof of a result by Xiaoling Chen and Lixin Cheng on Kuratowski measure of noncompactness.
Preprint.
钱进同学发现了程立新之前在Banach空间几何的工作和连续模型论之间的联系，利用连续模型论给出了程立新关于Kuratowski测度的结果，一个简洁的新证明。 -- 宋诗畅
2025/11/14, 10:00 a.m., Tencent Meeting Password: IF7G
Liu Han (Fudan) Reviewer: Liu Weihua (ZJU)
Asymptotic property C for certain wreath-like products of groups
In this paper, we present generalizations of some results on the asymptotic property C for wreath products. Specifically, we prove that certain wreath-like products admit asymptotic property C, thus providing some new examples for further studies.
Article To appear in Journal of Noncommutative Geometry.
本次报告中，刘晗博士深入阐述了她在粗几何领域的最新研究成果。报告从格罗莫夫（Gromov）定义的渐进维数等核心概念入手，介绍了其工作在几何群论中的相关背景和问题，并展示了如何通过度量几何的基本框架并结合Wreath-like Product构造方法构建了一类新型的、具有渐进性质C的群。 -- 刘伟华
2025/11/28, 10:00 a.m., Tencent Meeting Password: HDF1
Zhu Yue (CAS) Reviewer: Wang Haoming (SYSU)
The free probability approach to random matrices
This report offers an introduction to the powerful framework of free probability and its profound applications in random matrix theory. We will explore how the classical limit theorems, such as Wigner's semicircle law, find their natural explanation within this framework. The core concept of freeness - a non-commutative analogue of independence - will be introduced to explain the asymptotic behavior of large-dimensional random matrices. We will demonstrate how key tools like the R-transform and S-transform allow for the computation of limiting spectral distributions of sums and products of independent matrices through the operations of free additive and multiplicative convolution. This overview aims to illustrate why free probability has become the indispensable language for modern random matrix analysis.
Dr. Zhu Yue introduces basic concepts of random matrices and free probability, with emphasis on GOE, GUE, GSE, and semi-circular elements. In an additive random matrix model with rank one perturbation, she illustrates how to combine novel methods from free probability to handle this problem by using the R-transformation and S-transformation. -- Wang Haoming
2025/12/05, 10:00 a.m., Tencent Meeting Password: RSWL
Yu Tingzhou (Alberta) Reviewer: He Pingan (BNBU)
Quantitative estimates of the smallest singular value of random combinatorial matrices
In this report, we study \(n \times n\) random combinatorial matrices \(M\), where each row is chosen independently and uniformly from the set of binary vectors containing exactly \(d\) ones. We specifically investigate the smallest singular value, \(s_n(M)\), in the dense regime, defined by \(d = pn\) for a fixed constant \(p \in (0, 1/2]\). We establish a probabilistic upper bound for \(s_n(M)\). This result complements existing lower bounds of the order \(\Omega(n^{-1/2})\) and confirms that the smallest singular value is typically of the order \(n^{-1/2}\) in this dense setting. Additionally, we discuss quantitative results concerning the sparse regime. These results are joint work with Dongbin Li and Alexander Litvak.
该报告思路清晰、技术路线严谨，研究了\((n\times n)\)的随机组合矩，在稠密情形\((d=pn)\)下给出了最小奇异值的概率上界，补全了已有的下界；报告还讨论了关于稀疏情形的定量结果。这一结果在编码理论与数值稳定性分析、特定的数据分析等领域具有很大的应用潜力。 -- 何平安
2025/12/19, 10:00 p.m., Tencent Meeting
Zhang Yilong (Bonn) Reviewer: Will Johnson (Fudan)
Hrushovski construction in ordered fields
The Hrushovski construction is a variant of amalgamation methods. It was invented to construct new examples of strongly minimal theories. The method was later adapted to expansions of fields, including colored fields and powered fields. In this talk, I will present my attempt to apply the Hrushovski construction to ordered fields. I will construct an expansion of RCF by a dense multiplicative subgroup (green points). The construction induces a back-and-forth system, enabling us to study the dp-rank and the open core of this structure. I will also introduce my recent progress on powered fields, an expansion of RCF by "power functions" on the unit circle, and my plan to axiomatize expansions of the real field using the Hrushovski construction.
2026/01/02, 10:00 a.m., Tencent Meeting
Wang Leda (Yale) Reviewer: Chen Zaoli (USTC)
Approximate message passing algorithms for rotaionally invariant matrices
Approximate Message Passing (AMP) algorithms have seen widespread use across a variety of applications. However, the precise forms for their Onsager corrections and state evolutions depend on properties of the underlying random matrix ensemble, limiting the extent to which AMP algorithms derived for white noise may be applicable to data matrices that arise in practice. In this work, we study more general AMP algorithms for random matrices W that satisfy orthogonal rotational invariance in law, where W may have a spectral distribution that is different from the semicircle and Marcenko-Pastur laws characteristic of white noise. The Onsager corrections and state evolutions in these algorithms are defined by the free cumulants or rectangular free cumulants of the spectral distribution of W. Their forms were derived previously by Opper, Cakmak, and Winther using non-rigorous dynamic functional theory techniques, and we provide rigorous proofs.
References:
[1] Fan Zhou. Approximate Message Passing algorithms for rotationally invariant matrices. Ann. Stat. 50. (2022).
2026/01/16, 10:00 a.m., Tencent Meeting
Zou Guangyi (USTC) Reviewer: Chen Zaoli (USTC)
Edge statistics of random band matrices
Random band matrices are interpolation models between Wigner matrices and random Schr"odinger operators. When the bandwidth changes at different rates relative to the system size, the eigenvalues of band matrices exhibit two distinct behaviors: Wigner matrix eigenvalue statistics and Poisson eigenvalue statistics.
In this talk, we consider the edge statistics of band matrices and discuss the mechanism behind the transition between Wigner statistics and Poisson statistics of eigenvalues. Our discussion involves a type of Feynman diagram expansion and Feynman integral estimates. This discussion is based on collaborative work with Professor Dang-Zheng Liu (USTC), arxiv:2401.00492.
带状矩阵的边界谱分布
随机带状矩阵是 Wigner 矩阵和随机薛定谔算子之间的插值模型。当带宽相对系统大小以不同速率变化，带状矩阵的特征值会出现两种不同行为：Wigner 矩阵特征值统计和 Poisson 特征值统计。
本次报告我们考虑带状矩阵的边缘谱分布特征值 Wigner 统计 / Poisson 统计转变背后的机制。我们的讨论会涉及到一类 Feynamn 图展开和 Feynman 积分估计。这次讨论基于我和刘党政教授（USTC）的合作工作 arxiv:2401.00492