MAGIC070: Singularities in symplectic and contact spaces

Course details


Autumn 2010
Monday, October 4th to Friday, December 17th


Live lecture hours
Recorded lecture hours
Total advised study hours


11:05 - 11:55


Caustics and wave fronts of optic rays were studied since 17th century. However only recently the general theory of singularities of projections of submanifolds in symplectic and contact spaces related the properties of optical caustics and wave fronts with other problems in differential equations, geometry, mathematical physics, variational problems. This theory which is mainly due to V.Arnold is based on the deep relations of critical points of functions with invariants of Lie algebras, Hamiltonian mechanics, algebraic topology and differential geometry.

This area serves also as a corner stone of modern symplectic and contact topology.


General courses in real and complex calculus, diferential geometry


  • Symplectic spaces: examples cotangent bundle, space of extremals of variational problem, Liouville tori.
  • Contact spaces : projectivised cotangent bundle, jet space, hypersurfaces in symplectic spaces symplectisation and contactisation.
  • Submanifolds of symplectic spaces: Weinstein and Givental theorems, Lagrangian submanifolds, isotropic, symplectic submanifolds.
  • Legendre submanifolds of contact spaces.
  • Lagrangian bundles, Lagrangian projections, caustics, Legendre projections, wave fronts, examples from differential geometry.
  • Legendre transformation, dual surfaces.
  • Local singularities: generating functions for Legendre and Lagrange germs. Space of germs of functions, classification of function germs singularities.
  • Mozer�s homotopy method.
  • Malgrange�s preparation theorem.
  • Versality of families of functions.
  • Stability of Lagrange, Legendre projections.
  • Global properties of Lagrange manifolds: Arnold-Maslov class. Four vertex theorem for a plane curve and its generalizations



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