Explore the fascinating world of phase transitions across physics and computer science in this comprehensive lecture. Delve into the universal nature of phase transitions, from classical examples like water state changes to their occurrence in neuroscience, biology, financial markets, and information processing systems. Discover the principle of minimum thermodynamic potential, mean-field models of imitation effects, and the Ising model for ferromagnetism. Investigate phase transitions in combinatorial optimization problems, including the Hamiltonian path and graph coloring. Examine the fundamental limits of communication through channel capacity and error-correction techniques. Gain insights into perceptron learning for classification and the broader implications of phase transitions in complex systems.