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Learn advanced concepts in Yang-Mills theory through this comprehensive seminar from the Institute for Advanced Study, exploring the mathematical foundations and physical applications of non-Abelian gauge theories that form the backbone of the Standard Model of particle physics. Delve into the geometric structure of gauge fields, connection forms, and curvature tensors while examining how Yang-Mills equations describe the dynamics of fundamental forces including the strong and weak nuclear interactions. Investigate the topological aspects of gauge theory, including instantons, monopoles, and the role of fiber bundles in understanding gauge transformations and their physical significance. Explore the mathematical techniques essential for working with Yang-Mills theory, including differential geometry on principal bundles, Lie group theory, and the calculus of variations applied to gauge field Lagrangians. Examine the quantum aspects of Yang-Mills theory, including path integral formulation, gauge fixing procedures, and the challenges of non-Abelian gauge theory quantization that led to developments in BRST symmetry and Faddeev-Popov ghost fields. Analyze the mass gap problem and confinement phenomena in Yang-Mills theory, connecting to some of the most profound unsolved problems in theoretical physics and mathematics, including the Clay Institute's Millennium Prize problem for Yang-Mills existence and mass gap.
