Report title: Ideal systems and characteristic classes

Reporter: Prof. Madeleine Jotz Lean University of Göttingen, Germany

Reporting time: 16:00-17:00, January 7, 2021

Report location: Join Zoom Meeting

https://uni-goettingen.zoom.us/j/97020051076?pwd=RitLdUQ5blZFSjZvSm94Z3ExTnI2QT09

Meeting ID: 970 2005 1076 Passcode: 658281

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

Report summary:

This talk describes the Atiyah class of a Lie pair (due to Laurent-Gengoux, Stiénon and Xu) and defines analogously the Atiyah class of an infinitesimal ideal system. The latter objects are considered the right notion of ideal in the context of Lie algebroids. The Atiyah class of a Lie pair turns then out to be an obstruction to the existence of an ideal system (à la Mackenzie and Higgins) on the Lie pair. Then we will discuss representations up to homotopy of Lie algebroids on graded vector bundles, in particular the adjoint representation up to homotopy of a Lie algebroid. The graded trace of the powers of the curvature of a connection up to homotopy induces characteristic classes of graded vector bundles. This yields obstructions to the existence of a representation up to homotopy on a graded vector bundle. As an immediate consequence, we find an obstruction to an infinitesimal ideal system in a Lie algebroid A over M, in terms of the Pontryagin classes of the underlying vector su bbundles of TM and of A.

Brief introduction of the speaker:

Madeleine Jotz Lean, a professor at the University of Göttingen, Germany, is engaged in the research of differential geometry. He has published many articles in Math. Ann., Tran. AMS, IMRN, J. Math. Pures Appl. and J. Symplectic Geom. Has important influence internationally.