Trends in Mathematics

This volume collects papers based on lectures given at the XL Workshop on Geometric Methods in Physics, held in Białowieża, Poland in July 2023. These chapters provide readers an overview of cutting-edge research in infinite-dimensional groups, integrable systems, quantum groups, Lie algebras and their generalizations and a wide variety of other areas. Specific topics include:

• Yang-Baxter equation
• The restricted Siegel disc and restricted Grassmannian
• Geometric and deformation  quantization
• Degenerate integrability
• Lie algebroids and groupoids
• Skew braces

Geometric Methods in Physics XL will be a valuable resource for mathematicians and physicists interested in recent developments at the intersection of these areas.

This contributed volume provides an integrated perspective on modern mathematical and computational techniques for addressing complex problems in networks, control systems, learning, and game theory. It encompasses state-of-the-art research from a diverse range of disciplines, including dynamical systems, stochastic analysis, optimization, game theory, machine learning, and transportation theory. Particular emphasis is placed on connecting rigorous theoretical developments with real-world applications.

This volume is organized into twelve chapters. The first part (Chapters 1-6) addresses the optimal transportation problem in traffic networks. The second part (Chapters 7-11) investigates Markov chains with memory by examining the geometric, algebraic, dynamical, and ergodic properties of quadratic (polynomial) stochastic operators associated with cubic stochastic hypermatrices. Finally, Chapter 12 illustrates how methods from stochastic analysis and dynamical systems can be applied to lattice models of statistical mechanics defined on a Cayley tree.

Intended to serve as a comprehensive reference for academics and researchers, this volume may also benefit practitioners and anyone interested in exploring the interplay among these fields.