Research Group in Global Analysis and Applications

12th International Summer School in Global Analysis and Applications
August 20th - August 24th, 2007,

Summer School Programme

The course will cover the new, equivariant approach to the classical method of moving frames, and its many applications.

Topics to be covered (time permitting)

  1. Quick review of Lie groups, Lie algebras, Maurer--Cartan forms, structure equations.
  2. Jet bundles, prolongation, contact forms; the variational bicomplex; symmetries of differential equations; calculus of variations.
  3. Moving frames for Lie groups; invariants, including differential invariants, joint invariants, and joint differential invariants; invariantization; recurrence formulae and differential invariant algebras.
  4. Equivalence and symmetry of functions and submanifolds; signatures.
  5. Applications in differential geometry, classical invariant theory, and image processing.
  6. The invariant bicomplex; invariant variational problems, invariant submanifold flows, integrable systems and recursion operators; signature evolution.
  7. Invariant numerical methods; multi-space; geometric integration.
  8. Lie pseudo-groups; Maurer--Cartan forms and structure equations; moving frames; differential invariants; applications to partial differential equations.

The aim of the lectures is to present foundations of the variational calculus on fibred spaces, and to introduce participants of the course to research topics in this field of global analysis. Applications (variational principles in higher order mechanics and field theory) are also discussed.


  1. Differential forms on jet manifolds, differentiation and integration, odd forms
  2. Variational functionals on fibred spaces, Lepage forms (incl. the Cartan form), first variation formula, extremals, Noether's symmetries, local variational functionals
  3. The variational sequence, local and global inverse problem of the calculus of variations for source forms and differential equations, the Helmholtz variationality conditions
  4. Differential invariants and natural Lagrange structures (covariant variational principles), symmetries and conservation laws
  5. Example: The geometric structure of the Hilbert variational functional for the Einstein equations (principal Lepage form, conservation laws, regularity)