This volume collects papers based on lectures given at the XLII Workshop on Geometric Methods in Physics, held in Białystok, Poland in June-July 2025. These chapters provide readers an overview of cutting-edge research in quantum field theories, infinite-dimensional groups, integrable systems, noncommutative geometry, and a wide variety of other areas. Specific topics include:

• Twisted Araki–Woods Algebras
• Monoidal actions in Lie-algebraic context
• Introductory course to quantum formalism in low dimensions
• Gelfand triplets in Quantum Mechanics
• Riemann Hypothesis in terms of the continuous Newton flow 

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