Exploring new arrows in the BCGW-groupoid II

Coarse cohomology of a metric space is an invariant that measures the asymptotic behaviour of the space at infinity and the way in which uniformly large subspaces fit together. In this talk we tackle the problem of computing coarse cohomology in terms of standard better-known tools from classical algebraic topology.

The key point of the strategy we will follow is based on the observation that a metric space has three structures interacting in a compatible way: topology, bornology, and coarse structure. Here we shall present a cohomological approach to understand the relations between these three.

Lastly, classical cohomology theories such as Alexander-Spanier can be enriched with a cup product, and this additional structure allows one to tell more spaces apart. This will lead us to the Roe product, and we shall discuss how to compute it from the classical cup product.

The stability of leaves of a geometric structure is a classical problem: Given a geometric structure that induces a (singular) foliation on a manifold and a leaf of this foliation, when is the leaf preserved under deformations of the geometric structure? This question has been addressed for Lie algebroids and Poisson manifolds by M. Crainic and R. Fernandes, who found a cohomological obstruction to stability.

I will discuss how the question can be reformulated into an algebraic statement concerning a pair consisting of a differential graded Lie algebra and a differential graded Lie subalgebra, and give a cohomological obstruction to this abstract stability, which recovers the results by M. Crainic and R. Fernandes. Then, I will apply the result to obtain a stability criterion for leaves of Dirac structures in arbitrary Courant algebroids of split signature .

We will begin with a short reminder on double Lie groupoids and VB groupoids. Based on a paper by Mehta and Tang, we then discuss the nerve functor and the bar construction first in the more general case of double Lie groupoids before discussing the goal of my master's thesis: Using a description of VB groupoids as cochains to achieve more specific results in the VB groupoid case.

Dirac geometry, as is the case of many other important geometries, has its origin in Physics (particularly classical mechanics) and generalises the structures present in Symplectic and Poisson geometry. In particular, those Dirac structures can be shown to have an associated foliation leading to a normal form around points (à la Darboux or Weinstein). Moreover, by complexifying, we are able to describe a much larger class of structures, from the already mentioned symplectic and Poisson to (Generalised) Complex and CR structures, and, thus, we would also like to obtain some normal form results in this setting. Unfortunately, complex Dirac structures turn out to be much more complex and, even though a partial result for a normal form has been given by Agüero and Rubio (2022), there is still much to be understood. In this talk, we shall explore all of these facets of Dirac geometry as well as discuss on-going work on the ($L_{\infty }$-algebras encoding) deformations of certain complex Dirac structures and holomorphisation of (real-)analytic complex Dirac manifolds.

Poisson structures generalise symplectic geometry, and appear in diverse mathematical and physical contexts. In this talk, we focus on generic Poisson structures, exploring their definition, fundamental examples, and basic properties. A key distinction arises between even- and odd-dimensional manifolds: in even dimensions, generic Poisson structures correspond to log-symplectic structures, while in odd dimensions, their behaviour is more intricate and remains less understood. My PhD research is dedicated to studying generic Poisson structures in odd dimensions, and I will illustrate my current research directions through examples.