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数学学院、所2021年系列学术活动(第10场):Janusz Grabowski, Institute of Mathematics, Polish Academy of Sciences

发表于: 2021-02-18   点击: 

报 告 人:Janusz Grabowski, Institute of Mathematics, Polish Academy of Sciences

报告地点:Zoom

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

Meeting ID: 869 9431 4172

Passcode: Euler

校内联系人:生云鹤 shengyh@jlu.edu.cn


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.


报告人简介:

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。