Abstracts for Higher Geometric Structures along the Lower Rhine XIV, March 12 - 13, 2020

I will discuss the correspondence between differential characters and higher principal $S^1$-bundles with connection, via the integration of Lie algebroids. The main point of the talk is that one needs to consider an "homological" integration, different from the standard "homotopic" integration. This is true already for ordinary principal $S^1$-bundles where the relevant integration is the so called genus integration of the prequantum algebroid, introduced in joint work with Contreras.

DelikArt Bonn

Koszul duality between Lie algebras and cocommutative coalgebras constructed by Hinich is the basis for formal deformation theory, at least in characteristic zero.  In this talk I explain, following Manetti, Pridham and Lurie, how Koszul duality, combined with Brown representability theorem from homotopy theory leads to representability of a formal deformation functor up to homotopy. Sometimes a formal deformation functor has a `noncommutative structure', meaning that it is defined on a suitable homotopy category of associative algebras.

To a Poisson manifold $(M,\pi)$ one can naturally associated a cohomology called Poisson cohomology. Although Poisson cohomology is important for questions such as linearization and deformations of poisson structures, it is in general quite difficult to compute. In this talk we look at the Poisson structure on the dual of a Lie algebra $(g, [~,~])$. We look at the relation of Poisson cohomology with the linearization problem and outline the general ideas behind the calculations for the case of $\mathrm{sl}_2(\mathbb{C})$. This is based on joint work with Ioan Marcut.