Programme

Schedule

Lectures

• Multimomentum Maps and Classical Field Theory
  Mark J. Gotay, Pacific Institute for the Mathematical Sciences, University of British Columbia, Vancouver, Canada
  
  Description: Momentum maps are familiar tools for the study of symmetries in classical mechanics and symplectic geometry. I will explain how they can be usefully generalized to classical field theory (CFT) in the context of "multisymplectic geometry." The corresponding "covariant" or "multimomentum maps" play crucial roles in understanding the structure of CFTs. Indeed, we will see that the vanishing of the multimomentum map $J$ for the gauge group of a CFT gives rise to initial value constraints and that $J$ can be wielded to present the evolution equations in adjoint form (which displays, in the clearest and most concise way, the interplay between the dynamics, the constraints, and the gauge ambiguity of a system). We will also show that multimomentum maps for diffeomorphism groups provide a solution to the problem, dating back to the turn of the last century, of defining suitable stress-energy-momentum tensors for given collections of fields. A recurrent theme is thus that the multimomentum map encodes essentially all the dynamical information carried by a given classical field theory : its Hamiltonian, its initial value constraints, its gauge freedom, and even its stress-energy-momentum tensor.
  
  In this lecture series I will lightly cover the basics of the multisymplectic approach to CFT (including appropriate versions of Noether's Theorems), and then illustrate, with the aid of an extended example, how they lead to fundamental insights into the foundations of physical theories.
  
  
  
• The Noether Theorems: From Noether to Ševera
  Yvette Kosmann-Schwarzbach, Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, Palaiseau, France
  
  Description: The two theorems proved by Emmy Noether in 1918 have had many "descendants." We shall explain their original formulation and sketch their surprising history. Then we shall describe several generalizations of the first theorem (the geometry of partial differential equations, symmetries and conservation laws for non-variational equations). Introducing Lie algebroids and Courant algebroids will permit us to survey recent developments in Lagrangian and Hamiltonian dynamics.
• • In Noether's words: "Invariant Variational Problems" (1918).
• • The geometry of partial differential equations.
• • Generalizations of Noether's first theorem.
• • Bihamiltonian structures.
• • Lie algebroids and Courant algebroids in the Lagrangian and Hamiltonian formalisms.
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  Lecture Notes: http://www.math.polytechnique.fr/cmat/kosmann/various.html