Janusz Grabowski

Graded bundles in geometry and physics

• Talk 1: Graded bundles
  We start with showing that the multiplication by reals completely determines a smooth real vector bundle. Then, we consider a general smooth actions on the monoid of multiplicative reals on smooth manifolds. In this way homogeneity structures are defined. The vector bundles are homogeneity structures which are regular in a certain sense. It can be shown that homogeneity structures are manifolds whose local coordinates have associated degrees taking values in non-negative integers - graded bundles are born. A canonical example are the higher tangent bundles. We show also how to lift canonically homogeneity structures (graded bundle structures) to tangent and cotangent fibrations.
• Talk 2: Double structures and algebroids
  We define double graded bundles (in general n-tuple graded bundles) in terms of homogeneous structures. Classical examples are double vector bundles obtained from lifts, especially to $TE$ and $T^*E$ for a vector bundle $E$. We show the canonical isomorphism of double vector bundles $T^*E^*$ and $T^*E$. We define general algebroids (in particular, Lie algebroids) in terms of double vector bundle morphisms. We introduce also the concept of a Lie groupoid.
• Talk 3: Tulczyjew triples and geometric mechanics on algebroids
  Starting with the classical Tulczyjew triple involving $TT^*M$, $T^*TM$ and $T^*T^*M$, we define the triple associated with a general algebroid involving $TE^*$, $T^*E$ and $T^*E^*$. Using now Lagrangian and Hamiltonian functions we explain how to construct dynamics and Euler-Lagrange equations out of them. We end up with dynamics of strings as mechanics on double graded bundles and the Plateau problem.