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"VARNA 2002":
4th International Conference on Geometry, Integrability and Quantization ; Sts Constantine and Elena, Bulgaria, June 5-16, 2002


Infinite Dimensional Lie Groups and Applications in Mathematical Physics
by
Rudolf Schmid


  • Lecture 1: Infinite dimensional Lie Groups: General definitions, finite versus infinite di, Lie groups, examples, applications of abelia gauge groups to Maxwell's equations, loop groups;


  • Lecture 2: Diffeomorphism groups: Algebraic structure, geometric structure, Lie group structure, Lie algebra, exponential map, ILH Lie groups, gauge groups;


  • Lecture 2a: BRST Symmetries: classical field theory (QED,QCD), gauge symmetries, chiral symmetries, BRST symmetries, Chevalley-Eilenberg cohomology, local differential forms, anomalies, Wess-Zumino consistency condition, g-symplectic structures, solution of consistency condition;


  • Lecture 3: Subgroups of diffeomorphism groups and applications: Volume preserving diffeos (fluid dynamics), symplectomorphisms (plasma physics), contact transformations , global hamiltonian vector fields, quantomorphisms (geom. quantization), gauge transformations (quantum field theory);


  • Lecture 4: Lie group of Fourier integral operators: Basic definitions and properties. Exact sequence of groups, local section, smoothness of manifold structure, smoothness of group operations, ILH Lie group structure ;


  • Lecture 5: Non-compact manifolds. Applications: KdV- equation, Topological Euler equations, non-homogeneous Euler equations, Pseudodifferential operators and quantization. References;