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Explore the vast landscape of conformal field theory through this comprehensive lecture delivered at the International Congress of Basic Science 2025, where renowned physicist Hirosi Ooguri examines the fundamental structures, mathematical frameworks, and theoretical implications that define this crucial area of theoretical physics. Delve into the intricate relationships between conformal symmetries and quantum field theories, understanding how these mathematical constructs provide powerful tools for analyzing critical phenomena in statistical mechanics and string theory. Investigate the classification schemes and organizational principles that help navigate the complex terrain of conformal field theories, from minimal models to more sophisticated constructions. Learn about the connections between conformal field theory and other areas of mathematics and physics, including algebraic geometry, representation theory, and quantum gravity. Examine specific examples and case studies that illustrate key concepts while gaining insights into current research directions and open problems in the field. Discover how conformal field theory serves as a bridge between different areas of theoretical physics and mathematics, providing a unified framework for understanding diverse physical phenomena from phase transitions to black hole physics.
