Exploring new arrows in the BGW-groupoid

The infinitesimal counterpart of a Lie group(oid) is its Lie algebra(oid). I will show that the differentiation procedure works in any category with an abstract tangent structure in the sense of Rosicky. Mainly, I will construct the abstract Lie algebroid of a differentiable groupoid in a cartesian tangent category C with a scalar R-multiplication, where R is a ring object in C. Examples include differentiation of infinite-dimensional Lie groups, elastic diffeological groupoids, etc. This is joint work with Christian Blohmann.

By Lie's third theorem, a Lie algebra integrates to a Lie group if it is finite-dimensional. This otherwise fails; there are even Banach Lie algebras without an integrating Banach Lie group. However, these Lie algebras often integrate to elastic diffeological groups. In this talk, we will show how the tangent category of Banach (more generally convenient, in the sense of Frölicher, Kriegl, and Michor) manifolds embeds into the tangent category of elastic diffeological spaces, and how this embedding allows us to integrate certain Banach Lie algebras. This is joint work-in-progress with Christian Blohmann.

Differentiable stacks describe geometric spaces more general than manifolds: orbifolds, classifying spaces and many moduli spaces can be expressed in their framework. The cohomology (with real coefficients) of a differentiable stack can be interpreted as a generalisation of equivariant cohomology, meaning that in special cases, there are efficient models, the Weil and Cartan models, to compute it. In my PhD project, I have studied how to use and adapt these existing methods to compute differentiable stack cohomology for a certain class of differentiable stacks that are related to proper and regular Lie groupoids. In my talk, I will survey the progress of my research with respect to both results and difficulties that have emerged.

The character plays an important role in the representation theory of finite groups. In this talk, I will introduce the notion of 2-character of 2-representations of a finite 2-group $\mathcal{G}$. The conjugation invariance implies that the 2-characters can be viewed as objects in the Drinfeld center $\mathcal{Z}_1(\mathbf{Vect}_{\mathcal{G}})$. I will also introduce a topological quantum field theory (TQFT) point of view on the 2-characters and show that they are Lagrangian algebras in $\mathcal{Z}_1(\mathbf{Vect}_{\mathcal{G}})$. If time permits, I will also discuss the orthogonality of 2-characters, which categorifies the classical orthogonality of characters. This talk is based on arXiv:2404.01162, joint with Mo Huang and Zhi-Hao Zhang.

After a reminder on Lie groupoids, stacks, and simplicial Lie n-groupoids as background, I will describe shifted symplectic Lie n-groupoids based on the article https://arxiv.org/pdf/2112.01417 by Miquel Cueca and Chenchang Zhu. This will lead me to talk about the current goals in my master thesis: (1) Writing down a proof for the invariance of shifted symplectic structures under Morita equivalence. (2) Constructing a shifted symplectic model for the classifying stack BG that uses the free loop group LG instead of based loops.

Principal bundles play a fundamental role in differential geometry, topology, and gauge theory, providing a framework for understanding symmetry and connections on manifolds. However, when extending to higher categorical settings, new structures, such as principal 2-bundles, or principal n-bundles emerge. These higher bundles allow us to capture more intricate geometric and information. By extending the classical theory, higher principal bundles provide a richer framework for understanding symmetries and fields, which are crucial in areas like string theory and the study of higher structures. We will discuss various definitions of higher principal bundles in its different incarnations. We will also provide a way of generalizing all of them in a single framework. We will also touch up on some ongoing projects to understand the notion of higher connection on these principal bundles.

Recently, Konrad Waldorf started a project aiming at using the language of vector 2-bundles to describe cohomology classes in Tate K-theory, an example of elliptic cohomology. This project is inspired, among other things, by the work of Brylinski, who showed that one can construct classes in Tate K-theory of a manifold $X$, by looking at vector bundles over its loop space $LX$ together with an action by the Virasoro group which is compatible with the action by order preserving diffeomorphisms of $S^1$ on $LX$. In my talk, I'll briefly present my tasks within this project and what results we expect in the end.

It is well known that the collection of linear frames of a smooth $n$-manifold $M$ defines a principal $GL(n)$-bundle over $M$ (called the frame bundle); more generally, this construction makes sense for any vector bundle over $M$. Conversely, any principal bundle together with a representation induces an associated vector bundle; these processes establish therefore a correspondence between vector bundles on one side, and principal bundles with representations on the other side. In this short talk, I will quickly recall the notions mentioned above and sketch how to associate to any given vector bundle groupoid (VB-groupoid) a diagram of Lie groupoids and principal bundles, together with the action of a (strict) Lie 2-groupoid $GL(l, k)$; this will lead to the general notion of a principal bundle groupoid (PB-groupoid) and to the correspondence between VB-groupoids and PB-groupoids. This is joint work with Alfonso Garmendia.

I will sketch in a few minutes the interactions occurring in between my different research fields.