A Summer Tapas Invitation to Higher Structures in Symplectic and Poisson Geometry I

Event Date: Monday, July 7, 2025

Location: Sala de Graus, EPSEB, UPC, Barcelona

Organised by: SYMCREA hub and ICMAT Hitchin-Ngo Lab

A Summer Tapas Invitation to Higher Structures in Symplectic and Poisson Geometry II information here

https://www.crm.cat/a-day-of-higher-structures-in-symplectic-and-poisson-geometry-with-summer-tapas-at-upc-barcelona/

A one-day thematic gathering at the crossroads of Symplectic and Poisson Geometry, with a taste of Higher Structures — both mathematical and culinary. This informal minicourse features inspiring talks, in-depth sessions, and open discussions over tapas at SYMCREA.

Higher Structures in Symplectic and Poisson Geometry arise naturally when one seeks to understand complex interactions between geometry, algebra, and physics beyond classical frameworks. They encompass categorified notions such as Lie ∞-algebroids, higher groupoids, and derived stacks, which extend the foundational ideas of symplectic and Poisson structures to incorporate layers of homotopical or homological information. These structures are not only powerful tools for encoding deformation theory and quantisation but also offer a unifying language bridging differential geometry, topological field theory, and mathematical physics. In the context of this minicourse, we explore how physical intuition and categorical sophistication jointly illuminate the rich landscape of symmetries and constraints governing geometric systems.

11:00 - 11:45: From higher Lie groupoids to higher Lie algebroids through cochains by Alejandro Cabrera

11:45 - 12:30: Group Photo + Coffee break

12:30 - 13:45: Minicourse, Part I: Higher Structures by Chenchang Zhu

14:00 - 15:30: Tapas lunch at SYMCREA + Informal Discussions

15:30 - 16:45: Minicourse, Part II: Higher Structures by Miquel Cueca

17:00 - 17:45: Closing talk by Mario Garcia-Fernandez

From higher Lie groupoids to higher Lie algebroids through cochains

Alejandro Cabrera, 11:00h

Abstract: This talk is based on joint work with Matias del Hoyo. The idea is to explain an explicit, direct, and rigorous construction of the higher Lie algebroid underlying any higher Lie groupoid. We identify a key ideal of cochains which has the information of what differentiation means in simplicial and geometric terms. This part of the results entails two main theorems: a normal form one and one characterizing the quotient by the ideal, also yielding a novel generalization of the van Est differentiation map. Finally, we give our third theorem which is a generalization to the higher context of the classical "van Est isomorphism theorem" in cohomology.

Minicourse, Part I: Higher Structures - Introduction to derived n-groupoids

Chenchang Zhu, 12:30h

Abstract: Derived n-groupoids are models for derived higher stacks in differential geometry. To explain this, we need to explain first what are derived structures and what are n-groupoids.   We will first introduce the easiest derived manifolds, namely quasi-smooth manifolds, and easiest Lie n-groupoids, namely Lie groupoids. They can already help us to model singular symplectic reduction explicitly. We will see this explicit model. 

However, to really see the stacks behind, we need to understand what their Morita equivalences are. We  thus introduce the concept of Lie n-groupoids via Kan complexes and their Morita equivalence. 

For the most general derived n-groupoids and their Morita equivalence, we leave it to the second lecture.

Minicourse, Part I: Higher Structures - m-Shifted Symplectic Lie n-Groupoids

Miquel Cueca, 15:30h

Abstract: In this talk, I will recall Getzler's definition of shifted symplectic structures on higher Lie groupoids and exhibit several relevant examples related to (higher) Courant algebroids. 

Parallel spinors and higher geometry

Mario Garcia-Fernandez, 17:00h

Abstract: In this talk I will present a (mostly conjectural) picture, relating parallel spinors on a compact spin manifold and derived stacks. This relationship arises when considering connections with totally skew-symmetric torsion H, satisfying a Bianchi-type identity dH = \delta, along with their corresponding parallel spinors. The talk will mainly focus on the case that the manifold is 2n-dimensional and H = 0, in relation to Calabi-Yau metrics. Based on joint work with Alfredo llosa Lazo, Roberto Téllez Dominguez, and Luis Álvarez Cónsul.