Miquel Cueca (University of Göttingen)

Title: Shifted Symplectic Higher Lie Groupoids

Abstract:  The AKSZ construction in TQFT showed the relevance of symplectic Q-manifolds and their Lagrangian Q-submanifolds. In the first lecture, I will introduce them and provide several fundamental examples in mathematics and physics. Then, we will study its prequantization and end with the deformation theory of Lagrangian Q-submanifolds.

In the following two lectures, we will study their integration and introduce the shifted symplectic higher Lie groupoids defined by Getzler. Our first main goal is to produce examples and show their relevance in Poisson geometry. As a bypass, the examples will help us to clarify Getzler's definition and present an alternative characterization.

In the last lecture, I will focus on the shifted Lagrangian structures on higher Lie groupoids. I will provide its definition, show how they encode several moment map constructions, and explain the differentiation of higher Lie groupoids.

João Nuno Mestre (University of Coimbra)

Title: Deformations of Lie Groupoids and Related Structures

Abstract:  In this mini-course we will:

1. discuss deformations of Lie groupoids, and present the cohomology which controls them; explain applications to rigidity and normal form results;

2. study properties of this cohomology, such as Morita invariance and vanishing results for proper groupoids;

3. relate deformation cohomologies of Lie groupoids with those of Lie algebroids (a Van Est Theorem), and of examples such as Lie group actions and foliations;

4. discuss deformations of related objects, which may include Lie groupoids morphisms, symplectic or holomorphic groupoids, VB-groupoids, and others.

The mini-course is based on joint works with Cristian Cárdenas, Marius Crainic, Ivan Struchiner, Luca Vitagliano.

Luca Accornero (MPIM Bonn)

Title: Chern-Weil Characteristic Classes for Principal Groupoid Bundles

Abstract: The classical Chern-Weil map provides a connection-curvature description of cohomological invariants for principal group bundles. Motivated by studying geometric structures on manifolds in terms of principal groupoid bundles, we investigate a generalization of Chern-Weil theory to principal actions of Lie groupoids with a multiplicative connection. This is joint work in progress with Ivan Struchiner and Mateus de Melo.

Kadri Berktav (Bilkent University)

Title: Shifted Contact Structures on Derived Stacks

Abstract: In this talk, we outline our program for the development of shifted contact structures in the context of derived algebraic geometry. We start by recalling some key notions and results from derived algebraic/symplectic geometry. Next, we discuss shifted contact structures on derived Artin stacks and report our results regarding their local theory, together with some examples/future directions.

Janusz Grabowski (IMPAN, Warsaw)

Title: G-groupoids, Jacobi Algebroids, and Lifts of Jacobi Structures

Abstract: G-groupoids and algebroids are Lie groupoids and Lie algebroids being simultaneously principal G-bundles such that G acts by Lie groupoid/Lie algebroid isomorphisms. A particularly interesting is the case of G being the multiplicative group R* of invertible reals which plays a prominent rôle in contact and Jacobi geometry.

We will present results describing the structure of G-groupoids/algebroids, especially in the case of R*-algebroids, called in the literature Jacobi or Kirillov algebroids. We will also discuss tangent and cotangent lifts of principal actions. There are much larger families of natural tangent and cotangent lifts of principal bundles in the case of G = R* than what we have for a general Lie group G. They can be used for lifting Jacobi structures and associating Jacobi algebroids with Jacobi bundles.

Niels Kowalzig (University of Rome Tor Vergata)

Title: Brackets and Cobrackets on Cohomology and Homology

Abstract: We explain how not only grafting but also degrafting of (planar) trees, that is, a vertical composition or decomposition in an operadic or cooperadic sense, leads to the construction of brackets or cobrackets. When a multiplication or a comultiplication element is added, one furthermore obtains a product or coproduct of trees that induce a cohomology or homology theory on which the structure of a Gerstenhaber algebra or coalgebra can be observed, which corresponds to the one naturally appearing in deformation quantisation resp. Poisson geometry.

Wilmer Smilde (University of Illinois at Urbana-Champaign)

Title: Moduli of Geometric Structures and Algebroids

Abstract: There is a very interesting connection between finite type moduli problems of geometric structures (coframes) and Lie algebroids, observed by Bryant and worked out by Fernandes and Struchiner. In short, a finite-dimensional moduli problem for a G-structure gives rise to a certain Lie algebroid. Integrations of the Lie algebroid correspond to special types for families of solutions to the moduli problem. The stack of the simply connected complete solutions is represented by a certain groupoid integrating the Lie algebroid. 

This leads to the question of what the correct “algebroid” is for infinite type moduli problems for geometric structure. In this talk, I will propose an algebroid-like object, that we call a Bryant algebroid, that describes the moduli problem infinitesimally. I will explain the connection between PDEs with symmetries and show, through examples, how this object can lead to better understanding of the moduli problem. This is joint work with Rui Loja Fernandes.