Abstract

Govind Menon (Brown)
Abstract: We will discuss the interplay between some ideas from random matrix theory and deep learning. Specifically, we present a geometric perspective on classical constructions in random matrix theory and then adapt these ideas to training dynamics in deep learning. The talks are aimed at graduate students and young researchers and will include several open problems.

Jake Mundo (Brown) 
Abstract: The multivariate Bessel processes are stochastic processes on Weyl chambers which are parametrized by a multiplicity function k on a root system. We will construct these processes from diffusion processes on certain spaces of matrices, making use of the geometry of the orbits of the adjoint action of a Lie group on its Lie algebra. The construction applies for arbitrary positive k, and generalizes to a broader algebraic setting the construction of arbitrary-beta Dyson Brownian motion from a diffusion process in the space of Hermitian matrices.

Satoshi Yabuoku (Fukuoka)
Abstract: We consider the non-Hermitian matrix-valued stochastic process whose entries evolve as independent complex Brownian motions, this process is a natural time-dependent model of Ginibre ensemble in random matrix theory, because of the non-normality of matrices, its eigenvalue processes should be treated together with their eigenvector-overlap processes, here, the eigenvector-overlap processes are defined by the products of the inner products of corresponding right and left eigenvectors, and they determine the quadratic variations of eigenvalues, although right and left eigenvectors are defined only up to scaling, eigenvector-overlaps are invariant under such transformations, and hence they are uniquely determined, in our previous study, we derived SDEs for the eigenvector-overlap processes, however, the joint system of eigenvalues and eigenvector-overlaps is not closed by itself, hence, we need to take into account additional factors in this system, in this talk, we introduce generalized eigenvector-overlaps processes, which naturally arise in the quadratic variations of eigenvector-overlap processes, then, we explain the extended system consisting of eigenvalues and eigenvector-overlaps with these generalized eigenvector-overlaps, and this is based on the joint work with Syota Esaki (Oita University), Makoto Katori (Chuo University) and Jacek Ma¥lecki (Wroc¥law University of Science and Technology). 

Makoto Katori (Chuo) 
Abstract: A critical lattice system in two dimensions is conjectured to give a conformal field theory (CFT) in a scaling limit. On the other hand, Schramm--Loewner evolution (SLE) was introduced as a scaling limit of a critical domain-interface. An expected interrelation between them was called the SLE/CFT correspondence. Now it would be more appropriate to call the relation a coupling rather than correspondence. Since the Gaussian free field (GFF) is a probabilistic avatar of the CFT of massless free bosons, we believe that the SLE/CFT-correspondence is realized as a coupling between SLE and GFF. In our recent work, we have shown that coupling between chordal (resp.~radial)multiple SLE and GFF occurs if and only if the chordal (resp. radial) multiple SLE is driven by the usual Dyson Brownian motion (BM) (resp.~circular Dyson BM). The Dyson BM for parameter $\beta=1,2,4$ are eigenvalue processes of dynamical random matrices. At the level of stochastic differential equations (SDEs), it is natural to extend the parameter to $\beta>0$ as it is interpreted as the inverse temperature for stochastic log-gases. However, once we step back and ask why this generalization to $\beta>0$ is a right direction, we find that the origin of general $\beta$-Dyson BM is quite subtle.Recently, Huang--Inauen--Menon (2023) settled this subtlety by showing that the general $\beta$-Dyson BM is certainly an eigenvalue process of a dynamical random matrix. Otherwise, if the Dyson BM does not have to be an eigenvalue process, our work could be regarded as providing a natural origin of $\beta$-Dyson BM in terms of the SLE/GFF-coupling. In the present talk, I will discuss possible further work in this direction. As upgraded versions of the hydrodynamic limits of log-gasesdescribed by inviscid Burgers-like equations, we are interested in the deterministic limits of the corresponding multiple SLEs (the dynamical laws of large numbers), when the number of SLE curves tends to infinity. I expect that the recent work by Healey--Menon (2025) on the Dyson superprocess will give a hint to include fluctuation in some scaling limits.  

This talk is based on the joint work with Shinji Koshida (Aalto Univ.), Chizuru Soukejima (Chuo Univ.), and Raian Suzuki (Chuo Univ.).　

Hirofumi Osada (Chubu)
Abstract: TBA

Atsushi Katsuda (Kyushu)
Abstract: We explain the basic idea behind extending Floquet–Bloch theory from the abelian to the nilpotent setting (i.e., non‑abelian discrete Fourier analysis) and its application to the long‑time asymptotics of heat kernels on the corresponding covering spaces. If time permits, we also discuss the current strategy for further generalization to polycyclic groups (that is, discrete groups realizable as lattices in solvable Lie groups).

Takuya Murayama (Kobe)
Abstract: In non-commutative probability several concepts of “independence” have been considered, and in this talk we focus on monotone additive processes, or non-commutative stochastic processes with monotonically independent increments. I report that, as is the case with other independences, we have an analogue to the L\’evy-Khintchine formula that characterizes the marginal distribution of such a process. Moreover, this characterization is formulated by a generalization of the Loewner differential equation, which is famous not only in its origin, complex analysis, but in probability theory as well in relation to SLE. This talk is based on the joint work with Takahiro Hasebe (Hokkaido University) and Ikkei Hotta (Yamaguchi University).　

Yuzuru Inahama (Kyushu)
Abstract: In this talk we study a large deviation principle of Freidlin-Wentzell type for pinned hypoelliptic diffusion measures associated with a natural sub-Laplacian on a compact sub-Riemannian manifold. To prove this large deviation principle, we use rough path theory, manifold-valued Malliavin calculus and quasi-sure analysis.

Poster presenter

Raian Suzuki (Chuo)
Yano Shota (Tokyo)

Hirofumi Shiba (ISM)  

Rathindra Nath Karmakar (Kyushu)

Shuhei Shibata (Kyushu)

Karin Ikeda (Kyushu)

Son Kyungho (Kyushu)
Junpei Otsuka (Tsukuba)