This page displays the program schedule and talk titles with detailed abstracts and programme summary here. Moreover, there is a list of project collaborations undertaken during the program. All 45-minute talks consist of 40 minutes for the speaker’s presentation followed by 5 minutes for questions. Participants are encouraged to follow up with deeper discussions during free time. Legend: Statistical mechanics of log-gases Machine learning and statistical inference (Free) probability theory Orthogonal polynomials and asymptotic analysis Integrable systems including Painlevé equations Week 1 Schedule Week 2 Schedule Talk Titles Please find detailed abstracts here. You may click talk titles to download slides. Week 1 Gernot Akemann Recent advances in non-Hermitian random matrix theory Zhenyu Liao Random matrix theory for deep learning: Opportunities and challenges Arno Kuijlaars Riemann-Hilbert problems Folkmar Bornemann Fredholm determinants and Painlevé transcendents: A pragmatist’s perspective Benoît Collins Recent developments around strong convergence for random matrices Sung-Soo Byun Recent progress on free energy expansions of two-dimensional Coulomb gases James Mingo Real infinitesimal free independence Thomas Wolfs Multiple orthogonal polynomial ensembles of derivative type Peter Forrester On and around large x,N, small k expansions for log-gases and random matrices Jacobus Verbaarschot Non-Hermitian random matrix theories and integrability Dong Wang Asymptotics of biorthogonal polynomials related to Muttalib–Borodin ensemble and Hermitian random matrix with external source Satya N. Majumdar Dynamically emergent strong correlations via stochastic resetting Gregory Schehr Higher-order cumulants of linear statistics in Coulomb and Riesz gases Daria Tieplova Information-theoretic reduction of deep neural networks to linear models in the over-parametrized proportional regime Dan Dai The multiplicative constant in asymptotics of higher-order analogues of the Tracy–Widom distribution Masahiko Ito Trigonometric and elliptic Selberg integrals Yuanyuan Xu Optimal decay of eigenvector overlap for non-Hermitian random matrices Youyi Huang Cumulant structures of entanglement entropy over Hilbert–Schmidt ensemble Lu Wei Entropic cumulant structures of random state ensembles Zhigang Bao Law of fractional logarithm for random matrices Giorgio Cipolloni Universality of the spectral form factor Week 2 Shi-Hao Li Multiple skew orthogonal polynomials and applications Fei Wei On the moments of the derivative of CUE characteristic polynomials inside the unit disc Justin Ko On the phase diagram of extensive-rank symmetric matrix denoising beyond rotational invariance Lun Zhang Asymptotics for the noncommutative Painlevé II equation Leslie Molag Universality for fluctuations of counting statistics of random normal matrices Meng Yang Planar orthogonal polynomials and their applications Nicholas Witte Orthogonal and Symplectic Integrals via modulated 2j-k bi-orthogonal polynomial systems. Beyond Painlevé? Dang-Zheng Liu Edge statistics for random band matrices Daniel M. George Third order cumulants of products Noriyoshi Sakuma Generalized Meixner-type free gamma distributions Guido Mazzuca Discrete and continuous Muttalib-Borodin process Aron Wennman An equivariant Weierstrass theorem Matthias Allard Correlation functions between singular values and eigenvalues Mathieu Yahiaoui Random winding numbers for determinantal curves from non-Hermitian matrix random fields Linfeng Wei Skewness of von Neumann entropy over Bures–Hall random states Yeong-Gwang Jung Spectral analysis of q-deformed unitary ensembles with the Al-Salam–Carlitz weight Eui Yoo Three topological phases of the elliptic Ginibre ensembles with a point charge Yong-Woo Lee Large deviations for the number of real eigenvalues of the elliptic GinOE Collaborative Projects The mysterious multilinear structure of asymptotic expansions in classical random matrix ensembles People: Folkmar Bornemann, Peter Forrester, Lun Zhang, Nicholas Witte Random Haagerup inequalities for random matrices People: Benoît Collins, Giorgio Cipolloni, James Mingo, Yuanyuan Xu Description: In operator algebra, a Haagerup inequality states that for a non-commuting polynomial of given degree r, in non-commuting variables, its operator norm and its l2 norm are comparable, up to a polynomial factor in the degree r. Such inequalities are true at the level of random matrices but the proof is very hard. Namely, it relies on strong convergence. We believe there should be a direct proof (perhaps using concentration of measure or other RMT techniques). This is not only a natural question, this would also allow to reprove in a new conceptual way strong convergence in many cases, and hopefully, with better bounds. Infinitesimal free probability and Wigner ensembles People: James Mingo, Yuanyuan Xu, Anas Rahman, Daniel Muñoz George Description: Infinitesimal free probability has been able to recover, through combinatorial methods, differential equations characterising 1/N corrections to eigenvalue densities of classical matrix ensembles. We wish to see if these methods can be extended to obtain equivalent results for more general Wigner ensembles. Discrete notions of derivative type People: Thomas Wolfs, Mario Kieburg, Jiyuan Zhang, Peter Forrester, James Mingo Description: So far, mainly two notions of polynomial ensemble of derivative type (also called Pólya ensemble) have been considered: a multiplicative one, due to Kieburg-Kösters (2016), and an additive one, due to Kuijlaars-Róman (2019). They typically appear in connection to the squared singular values of products of invertible random matrices and the eigenvalues of sums of Hermitian random matrices. The main goal of this project is to develop discrete notions of derivative type that are compatible with certain non-intersecting path models induced by random tilings. Such developments will deepen our understanding of these models as ensembles of derivative type typically have useful properties, e.g., their kernel has a double integral representation, opening up the road for asymptotic analysis, and they posses certain (de)composition properties. Special interest goes to the discrete (multiple) orthogonal polynomial ensembles associated with the (multiple) Kravchuk, Hahn and q-Racah polynomials, but other polynomials from the (multiple) (q-)Askey scheme are considered as well. After an appropriate notion of derivative type has been set up, the goal is to obtain a double integral representation for the correlation kernel and to describe the (de)composition properties of such models. The latter is deeply connected to developing the surrounding finite free probability theory. A simple diffusion model for free cumulants People: Gregory Schehr, Peter Forrester, James Mingo Description: I will describe a simple diffusion model where a particle performs Brownian motion with a diffusion constant that changes randomly over time. Specifically, it switches between values drawn from a given distribution at a fixed rate. In the long-time limit, it turns out that the cumulants of the position grow linearly with time and are directly related to the free cumulants of the underlying distribution of diffusion constants. This is a joint work with M. Guéneau and S. N. Majumdar, https://arxiv.org/abs/2501.13754. The reason why free cumulants arise in this simple diffusion model remains open. Combinatorics of random matrix moments People: Anas Rahman, Daniel Muñoz George, Norm Do, James Mingo Description: It is known that the moments Tr(Mk) of matrices M can be understood as weighted sums over partitions of graphs obtained by identifying vertices of k-gons according to said partitions. In particular, moments of the Gaussian orthogonal ensemble (GOE) can be represented by pairwise identifying polygon edges to form locally orientable ribbon graphs. It has recently been shown that the 1/N (N being matrix size) correction to the GOE moments are given by counts of non-crossing annular pairings. These should be in bijection with ribbon graphs of demi-genus one. We would like to elucidate a bijection and extend it to the case of general Wigner ensembles. We will also be discussing combinatorics of certain neural networks, which correspond to graphs with edges of fixed multiplicity supporting cyclic walks. Bergman polynomials on a half-lemniscate People: Arno Kuijlaars, Aron Wennman Description: Johansson and Viklund (arXiv:2309.00308) have a result about the asymptotic behavior of the partition function for a Coulomb gas on a Jordan domain. In this project we would like to verify and possibly even extend their result for the special case of a half-lemniscate. The problem can be phrased in terms of planar orthogonal polynomials, and these turn out to be multiple orthogonal on an interval. The aim is to perform the steepest descent analysis of the associated Riemann Hilbert problem.