Reporter: Janusz Grabowski, Institute of Mathematics, Polish Academy of Sciences

Report location: Zoom

https://us02web.zoom.us/j/86994314172?pwd=UUxPMktKdHErZ3ZKMFdsQ2h2eHd4Zz09

Meeting ID: 869 9431 4172

Passcode: Euler

School contact: Sheng Yunhe shengyh@jlu.edu.cn

Title 1 of the report: Graded bundles

Reporting time (Beijing Time): Feb 22, 2021, 16:00-17:00

Abstract: We start with showing that the multiplication by reals completely determines a smooth real vector bundle. Then we consider a general smooth action 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.

Title 2 of the report: Double structures and algebroids

Reporting time (Beijing Time): Feb 23, 2021, 16:00-17:00

Abstract: 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 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.

Title 3: Linearization of graded bundles and weighted structures

Reporting time (Beijing Time): Feb 24, 2021, 16:00-17:00

Abstract: We consider weighted structures which are geometric structures with a compatible homogeneity structure, for instant weighted Lie groupoids and weighted Lie algebroids which are natural generalizations of VB-groupoids and VB-algebroids. We introduce also the functor of linearization of graded bundles. Linearizing subsequently a graded bundle of degree n we arrive at n-tuple vector bundle. Those n-tuple vector bundles can be characterized geometrically, so that we obtain an equivalence of categories.

Title 4 (Title 4): Tulczyjew triples and geometric mechanics on algebroids

Reporting time (Beijing Time): Feb 25, 2021, 16:00-17:00

Abstract: 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 out of them, also in constrained cases, and Euler-Lagrange equations. We end up with mechanics on Lie algebroids with higher order Lagrangians.

Brief introduction of the speaker:

Janusz Grabowski, professor at the Institute of Mathematics of the Polish Academy of Sciences, editor of J. Geom. Mech., engaged in Poisson geometry and mathematical physics, at Compos. Math., J. Reine Angew. Math. J. Differential Equations, Math. Z More than 130 high-level academic papers have been published in magazines, and have been cited more than 1,100 times.