Abstracts for Workshop "Higher Geometric Structures along the Lower Rhine XVII", May 2 - 3, 2024

We will explore higher categorical structures in symplectic geometry by taking some not entirely rigorous but everthemore instructive shortcuts to the algebraic implications of the analysis of pseudoholomorphic curves. Please come prepared to engage in collective inquiry on questions such as

• Why do pseudoholomorphic disks generate $A_\infty$ algebras?
• What is $A_\infty$ algebra anyways?
• What 2- or $\infty$-categorical structures arise from interpreting string diagrams as pseudoholomorphic curves? 

Given a geometric structure which induces a foliation on a manifold and a leaf of this foliation, one can ask when the leaf is preserved under deformations of the geometric structure. For Poisson structures and Lie algebroids, this question was addressed by Marius Crainic and Rui Loja Fernandes, and they showed that a leaf is stable when a certain cohomology group vanishes. I will give a general approach to such questions in terms of the L-infinity-algebra governing the deformations of the geometric structure, and an L-infinity-subalgebra.

Studying the blow-down map in cohomology in the context of real projective blow ups of Lie algebroids can be used to gain a better insight into, or even compute, Lie algebroid cohomologies, which we discuss in this talk. The key observation is that the pullback via the blow-down map is an isomorphism when restricted to flat Lie algebroid forms, which leads to the consideration of formal Lie algebroid cohomology.