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

Event Date: Wednesday, July 9, 2025

Location: Aula Naranja, ICMAT, Madrid

Organised by: SYMCREA hub and ICMAT Hitchin-Ngo Lab

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

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 Mercado de San Miguel.

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.

09:30 - 10:15: Introduction to derived Lie n-groupoids by Chenchang Zhu

10:30 - 11:15: Introduction to derived Lie n-groupoids by Chenchang Zhu

11:15 - 11:45: Coffee break

11:45 - 12:30: Shifted Lagrangian structures on derived Lie n-groupoids by Miquel Cueca

12:30 - 14:00: Lunch

14:00 - 14:45: Shifted Lagrangian structures on derived Lie n-groupoids by Miquel Cueca

15:00 - 16:00: Continuing with higher differentiation: forms and the n=2 case by Alejandro Cabrera

19:30: Summer Tapas at Mercado de San Miguel

Introduction to derived Lie n-groupoids

Chenchang Zhu, 9:30h, 10:30h

Abstract: We will first give an explicit introduction to derived manifolds via d.g. manifolds of positive amplitude (according to Behrend-Xu-Liao), and their weak equivalences. Then we will introduce derived Lie n-groupoids, which should be considered as a model for higher derived differential n-stacks. We show how to put shifted symplectic on them and how these objects help to understand singular symplectic reduction. 

Shifted Lagrangian structures on derived Lie n-groupoids

Miquel Cueca, 11:45h, 14h

Abstract: Drawing on ideas from derived algebraic geometry, this talk introduces shifted Lagrangian structures in a smooth setting. Specifically, we employ derived Lie n-groupoids to model derived differentiable stacks. We then define shifted Lagrangian structures on derived Lie n-groupoids and explain how these structures can be applied in classical symplectic geometry. 

Continuing with higher differentiation: forms and the n=2 case

Alejandro Cabrera, 15:00h

Abstract: This talk will be complementary to one presented two days before in Barcelona but also self-contained, also based on joint work with M. del Hoyo. After reviewing our general novel approach to differentiation, we shall detail the process of differentiating differential forms from a higher Lie groupoid to elements of the Weil algebra of its higher Lie algebroid. We shall also mention an underlying algebraic description, a van Est isomorphism theorem for forms, and some cases of 2-groupoids related to Courant algebroids.