Non-Commutative Probability and Related Topics 2026

Reference No.	2026a002
Type/Category	Grant for General Research-Workshop（Ⅱ）
Title of Research Project	Non-Commutative Probability and Related Topics 2026
Principal Investigator	Nobuhiro ASAI（Aichi University of Education・Professor）
Research Period
Keyword(s) of Research Fields	Noncommutative (quantum) probability, Random matrix, Mathematical physics, Orthogonal polynomials/Special functions, Quantum walk, Learning theory
Abstract for Research Report	Noncommutative (quantum) probability theory has developed through deep interactions with many areas of mathematics, including mathematical physics, operator algebras, probability theory, representation theory, combinatorics, function theory, quantum walks, and the theory of orthogonal polynomials and special functions. At the workshop held in 2025, researchers in noncommutative probability and related fields gathered to exchange ideas and discuss recent developments. These discussions reaffirmed the effectiveness of interdisciplinary dialogue and suggested that several concrete issues for future research are beginning to emerge. Several talks illustrated the potential of such interactions. In particular, a talk by a researcher in integrable systems reported an extension of quantum walk models to higher-dimensional settings using methods from algebraic combinatorics and multivariable orthogonal polynomials. This work suggested new connections between noncommutative probability, integrable systems, combinatorics, and function theory. Another presentation on free probability from a representation-theoretic perspective highlighted the role of algebraic symmetries underlying probability distributions and moment structures, pointing toward the possibility of a more unified theoretical understanding. Many talks also emphasized the roles of special functions and orthogonal polynomials, reflecting the rich function-theoretic structures inherent in noncommutative probability. The present workshop builds on the discussions and outcomes of the 2025 meeting. The workshop planned for 2026 will focus on distributions, limit theorems, and moment structures in noncommutative probability. At the same time, it will explore connections with quantum walks, random matrix theory, integrable systems, representation theory, and the theory of special functions and orthogonal polynomials. By comparing combinatorial and function-theoretic methods developed in these areas, we aim to clarify common structures and differences and thereby further refine the theoretical framework of noncommutative probability. To continue and further develop the interdisciplinary discussions initiated in 2025, the 2026 workshop will encourage speakers to discuss not only their research results but also the background and motivations of their work. This will facilitate the sharing of open problems and future research directions. The workshop is also expected to stimulate new collaborations and provide young researchers with opportunities to broaden their perspectives. Building on discussions with researchers from industry at the 2025 workshop, we will continue exploratory discussions on possible conceptual connections between noncommutative probability theory and information science. Although immediate applications are not the primary objective, such dialogue may help establish a foundation for future interdisciplinary developments. Through this workshop, we aim to strengthen the research network formed in 2025 and deepen ongoing discussions in this area, thereby contributing to the sustained development of noncommutative probability and its related fields.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Nobuhiro ASAI（Aichi Univesity of Education・Professor） Yuki Ueda（Hokkaido University of Education・Associate Professor） Etsuo Sagawa（Yokohama National University・Professor） Fumio Hiroshima（Kyusyu University・Professor） Hiroaki Yoshida（Josai University/ Ochanomizu University・Professor）