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Embark on a comprehensive exploration of quantum integrability in this first installment of a five-part lecture series. Delve into the fundamental concepts and principles that form the foundation of this fascinating field of physics. Over the course of two and a half hours, gain insights into the mathematical frameworks and theoretical approaches used to study integrable quantum systems. Examine the historical context and development of quantum integrability, and discover its applications in various areas of physics. Engage with complex topics presented in an accessible manner, suitable for graduate students and researchers interested in expanding their knowledge of quantum mechanics and mathematical physics.