This webpage is about the 2026 seminar on Integral Geometry and Special Function. There are EIGHT lecturers from diverse universities to give related talks. Titles and abstracts are updated on this webpage.
Summary
2026/10/16, 10:00 a.m., Tencent Meeting Password: CQDW
Huang Zhangkai (SYSU) Reviewer: Liu Hui (FZU)
RCD Spaces: Background, Motivation, and Recent Developments
RCD spaces provide a framework for extending Ricci curvature lower bounds to nonsmooth metric measure spaces. We will introduce the background and motivation of the theory through its connections with optimal transport and limits of Riemannian manifolds. We will then survey key structural results and recent developments.
The talk provided a clear and accessible introduction to RCD spaces, highlighting their geometric motivation, fundamental definitions, and connections to classical Riemannian geometry. It was informative and helpful for understanding the basic ideas of synthetic Ricci curvature. -- Liu Hui
2026/10/23, 10:00 a.m., Tencent Meeting Password: BHNE
Hu Jie (JZU) Reviewer: Huang Yi (NJNU)
Laguerre (q-Laguerre) Weight Recurrence and Geometric Theory of Painleve equations
In this presentation, the connection between recurrence coefficients of classical and semi-classical orthogonal polynomials and solutions of the Painlevé equations is studied. We consider two classes of weight functions by geometric theory of Painleve equations. For the Laguerre weight, using Sakai's geometric theory of Painlevé equations, we reduce certain recurrence satisfied by associated recurrence coefficients to one of the canonical discrete Painlevé equation. For the deformed q-Laguerre weight, using Sakai's geometric theory of Painlevé equations, we present the identification procedure how the q-Laguerre weight recurrence is equivalent to the one of the discrete Painlevé equation.
胡洁的报告围绕 Laguerre 型及变形 \(q\)-Laguerre 权函数，介绍了正交多项式递推系数与离散 Painlevé 方程之间的联系。对于 Laguerre 情形，报告利用 Sakai 几何理论，将相关非线性递推关系化为标准离散 Painlevé 方程。对于变形 \(q\)-Laguerre 情形，其特色在于揭示次数演化对应于 \(A_5^{(1)}\) 型曲面上两个标准离散 Painlevé 动力学的复合。这些工作展示了几何方法在识别递推关系的可积结构、建立不同方程之间显式联系方面的作用。报告最后还介绍了关于八次 (octic) Freud 权的在研工作及阶段性结果。 -- 黄益
2026/11/06, 10:00 a.m., Tencent Meeting
Wang Haoming (NKU) Reviewer: Zhang Jiyuan (SCUT)
A pushforward probability density function on Grassmannians from the Wishart law
In this paper, the exact joint density of the top \(k\) sample eigenvectors is derived for any \(p\times p\) population covariance matrix \(\varSigma\), sample size \(n > p -1\), and \(1 \le k \le p\). Using symmetric functions such as zonal polynomials and a dual summation identity by I. G. Macdonald, the result is expressed as a series in terms of determinants of differential operators. A matrix Kummer transformation converts the resulting local alternating expansion into a globally absolutely convergent series over the entire positive definite matrix space. For \(k=2\), the ordered eigenvalue integrals reduce to the Gauss hypergeometric function \({}_2F_1\) with a simple closed form when \(n=p+1\). Previously, explicit formulas were only available for a scalar matrix \(\varSigma = \sigma^2 I_p\) or a general matrix \(\varSigma\) with \(p=2\) or \(k=1\), obtained by T. W. Anderson and T. Sugiyama, respectively. The frame law thereby induces an exact probability density on Grassmannians. This is the joint work with Koki Shimizu.
2026/11/20, 10:00 a.m., Tencent Meeting
Zhang Jiyuan (SCUT) Reviewer: Haoming Wang (NKU)
Biorthogonal ensembles of derivative type
We prove that biorthogonal ensembles on the real line with a specific derivative structure admit an explicit correlation kernel of double contour integral form. We will demonstrate that this expression is a valuable starting point for asymptotic analysis and that our class of biorthogonal ensembles admits a large variety of limit kernels, by proving that two new classes of limit kernels can occur. The first type is a deformation of the hard edge Bessel kernel which arises in polynomial ensembles describing the eigenvalues of the sum of two random matrices, while the second type arises for Muttalib-Borodin type deformations of polynomial ensembles.
2026/12/04, 10:00 a.m., Tencent Meeting
Huang Yi (NJNU) Reviewer: Chen Jianhua (NCU)
Beyond the solution of the Maxwell conjecture
The Maxwell conjecture is a 150-year-old idea in electrostatics stating that a system of n point charges in space can produce at most \((n-1)^2\) non-degenerate equilibrium (critical) points where the electric field cancels out to zero. The Maxwell Conjecture has been proven false in July 2026 by Arathoon, Ball, and Kvalheim. In this talk we shall discuss further interesting problems beyond their solution.
2026/12/11, 10:00 a.m., Tencent Meeting
Wang Huaming (AHNU) Reviewer: Hu Jie (JZU)
Biased random walk with traps
We consider a \(\lambda\)-biased random walk on the deterministic infinite tree \(\mathcal{T}=\{(i,j):i\ge0,\,0\le j\le i\}\), where \(\{(i,0):i\ge 0\}\) is the and, for each \(i\ge1\), \(\{(i,j):1\le j\le i\}\) is a attached to \((i,0)\). Traps cause long waiting times, leading to behavior distinct from simple random walks. The walk is recurrent for \(\lambda\ge1\) and transient for \(0<\lambda<1\). In the transient regime it is sub-ballistic: it goes to infinity with zero speed, and its distance from the root grows logarithmically, i.e.,\[\liminf_{n\to\infty}\frac{|X_n|}{\log n}=\frac{1}{\log(1/\lambda)},\quad\limsup_{n\to\infty}\frac{|X_n|}{\log n}=\frac{2}{\log(1/\lambda)},\quad\text{a.s.}.\] We also reveal a sharp contrast between spatial and temporal regeneration. Let \(C(n)\) be the number of cutpoints among the first \(n\) backbone vertices, and \(M(N)\) the number of cut times up to \(N\). Then \[\lim_{n\to\infty}\frac{C(n)}{n}= 1-\lambda,\quad \lim_{N\to\infty}\frac{M(N)}{\log N}=(1-\lambda)\log(1/\lambda),\quad\text{a.s.},\] that is, cutpoints occur with positive linear density while cut times grow only logarithmically.
2026/12/18, 10:00 a.m., Tencent Meeting
Lyu Zhiyang (USTC) Reviewer: Li Qiongling (NKU)
Extremal Length Limit Formulas along Teichmüller Rays and Their Applications
Walsh established a limit formula for extremal length along Teichmüller rays. We obtain a complementary limit formula and generalize it to a class of continuous, compatibly additive, positively homogeneous functions on the space of measured foliations. As applications, we establish limit formulas for the Hubbard-Masur correspondence and for the energy and Hopf differentials of harmonic maps between finite-type hyperbolic surfaces. This talk is based on the joint works with Yi Qi and Kento Sakai.
2027/01/08, 10:00 a.m., Tencent Meeting
Kang Mingyang (NKU) Reviewer: Jack Chou (Minnesota)
A solution to Butler's positivity conjecture
Let \(\lambda, \mu, \nu\) be distinct partitions such that \(\lambda, \mu \subset \nu\) and \(|\nu/\lambda| = |\nu/\mu| = 1\). We prove Butler's positivity conjecture posed in 1994: the expansion of the Macdonald intersection polynomial \[\frac{T_{\lambda} \tilde{H}_{\mu}(X; q, t) - T_{\mu} \tilde{H}_{\lambda}(X; q, t)}{T_{\lambda} - T_{\mu}}\] in terms of the Schur function basis has coefficients in \(\mathbb{Z}_{\geq 0}[q, t]\).