报 告 人：Janusz Grabowski， Institute of Mathematics, Polish Academy of Sciences
Meeting ID: 869 9431 4172
报告题目1(Title 1)：Graded bundles
报告时间(Beijing Time)：Feb 22, 2021, 16:00-17:00
报告摘要(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.
报告题目2(Title 2)： Double structures and algebroids
报告时间(Beijing Time)：Feb 23, 2021, 16:00-17:00
报告摘要(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.
报告题目3(Title 3)：Linearization of graded bundles and weighted structures
报告时间(Beijing Time)：Feb 24, 2021, 16:00-17:00
报告摘要(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.
报告题目4(Title 4)：Tulczyjew triples and geometric mechanics on algebroids
报告时间(Beijing Time)：Feb 25, 2021, 16:00-17:00
报告摘要(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.
Janusz Grabowski，波兰科学院数学研究所教授，J. Geom. Mech.杂志编委，从事Poisson几何与数学物理的研究，在Compos. Math., J. Reine Angew. Math. J. Differential Equations, Math. Z.等杂志上发表130余篇高水平学术论文，被引用1100余次。