A Day of Higher Structures in Symplectic and Poisson Geometry, with Summer Tapas at UPC Barcelona

The event “A Summer Tapas Invitation to Higher Structures in Symplectic and Poisson Geometry I”, held on July 7, 2025, at EPSEB–UPC, featured talks by Alejandro Cabrera, Chenchang Zhu, Miquel Cueca, and Mario Garcia-Fernandez, offering a rich and informal exploration of higher structures in geometry over lectures, discussions, and tapas.

The event “A Summer Tapas Invitation to Higher Structures in Symplectic and Poisson Geometry I“, held on July 7, 2025, at the Sala de Graus, EPSEB, UPC in Barcelona, was organized by SYMCREA, an excellence unit at the Universitat Politècnica de Catalunya. SYMCREA is led by Eva Miranda and Marta Mazzocco, both affiliated researchers at the Centre de Recerca Matemàtica (CRM). This one-day thematic gathering offered an immersive exploration of higher structures, blending rigorous mathematics with informal discussions — and a flavorful touch of local cuisine.

The day opened with a talk by Alejandro Cabrera (UFRJ), titled “From Higher Lie Groupoids to Higher Lie Algebroids through Cochains”. Cabrera presented a direct and explicit construction of the higher Lie algebroid associated with any higher Lie groupoid. At the heart of his approach was the identification of a key ideal of cochains that captures the notion of differentiation in both simplicial and geometric terms. His results included a normal form theorem, a characterization of the quotient by this ideal, and a generalization of the classical van Est isomorphism theorem — all contributing to a deeper understanding of the cohomological structure of higher groupoids.

After a group photo and a coffee break, Chenchang Zhu (CRM–María de Maeztu Chair of Excellence) led the first part of the minicourse “Higher Structures – Introduction to Derived n-Groupoids”. She introduced participants to derived n-groupoids as models for derived higher stacks in differential geometry. Starting with quasi-smooth manifolds and Lie groupoids, she gradually built up to the concept of derived stacks. Her exposition of Kan complexes and Morita equivalence provided a solid foundation for understanding how higher categorical structures model singularities and symmetries in geometric contexts.

Following a lively tapas lunch at SYMCREA, where mathematical conversations continued in a relaxed and convivial setting, the afternoon resumed with Miquel Cueca’s (KU Leuven) session “m-Shifted Symplectic Lie n-Groupoids”. Cueca revisited Getzler’s definition of shifted symplectic structures and illustrated their relevance through examples involving higher Courant algebroids. His talk highlighted how these structures serve as a bridge between abstract homotopical geometry and concrete physical models.

The day concluded with a closing talk by Mario Garcia-Fernandez (ICMAT), titled “Parallel Spinors and Higher Geometry”. He explored a conjectural framework linking parallel spinors on compact spin manifolds with derived stacks — a relationship that emerges when considering connections with totally skew-symmetric torsion satisfying a Bianchi-type identity. Focusing on the case of 2n-dimensional manifolds with vanishing torsion, he discussed connections to Calabi–Yau metrics. The talk opened new perspectives on the role of higher structures in spin geometry.

This first edition of A Summer Tapas Invitation successfully combined deep mathematical insight with an open and collaborative atmosphere. From foundational constructions to speculative frameworks, the event showcased the power of higher structures to unify ideas across geometry, algebra, and physics — all while celebrating the joy of shared discovery over good food and conversation.