Symplectic Geometry Seminar Outline

  Mathematics & Computer Science Department
  Northern Kentucky University
  Seminar leader 1994-96: Kevin Kirby
  Participants: Leon Arriola, Dan Curtin, Maria Falbo-Kenkel, Don Krug,                 
                Kristin Pfabe, Gail Wells, Steve Wilkinson

• From Lagrangian to Hamiltonian mechanics
• Trajectories of dX/dt = J grad H and level curves of H
• Linearization; Symplectic and Hamiltonian matrices
• Poisson brackets (of functions on phase space)
• Liouville's Theorem

• Gaussian optics: translation & refraction
• The thin lens; conjugate planes; telescopic systems
• Theorem: Matrices are optical iff symplectic
• The eikonal function

• Gauging field theories
• Conformal gravity

• Groups preserving bilinear forms: O(n) vs. Sp(2n)
• Lie algebras as tangent spaces of Lie groups

• Vectors and algebraic k-forms
• Duality: sharps and flats
• Wedge products, interior products
• The canonical symplectic form on V x V*
• From symplectic vector spaces to symplectic manifolds
• The canonical symplectic form on a cotangent bundle