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 nonnegative 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 ntuple 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 EulerLagrange equations out of them.
We end up with dynamics of strings as mechanics on double graded bundles and the Plateau problem.
