The Centre de Recerca Matemàtica is pleased to invite you to the Inaugural Colloquium of the CRM–María de Maeztu Chair of Excellence 2025, featuring Professor Chenchang Zhu (University of Göttingen). This event marks the beginning of her stay at CRM as Chair of Excellence, a distinction awarded to internationally renowned researchers whose work aligns with the centre’s strategic research areas. Professor Zhu’s contributions in higher category theory, differential geometry, and mathematical physics open new perspectives at the crossroads of geometry and quantum theory. Join us to celebrate the launch of this prestigious appointment.

Higher structures in symplectic geometry

Almost four hundred years ago, at the other end of the Mediterranean, Galileo observed that the fastest descent curve for a little ball is not the straight line! However to solve this problem rationally,  he foresaw that a ”higher” mathematics would be needed. Six decades later, J. Bernoulli posed this challenge to the mathematical world, prompting Newton’s overnight solution. The higher mathematics that emerged to solve such problems—and, ultimately, all of classical mechanics—was calculus.  

This journey through mechanics and geometry continued with Lagrange’s formulation of mechanics, Noether’s theorem on symmetries and conservation laws, and the rise of modern symplectic geometry. In this setting, Alan Weinstein’s insight—”everything is Lagrangian”—has shaped our understanding of phase space, reduction, and quantization. However, just as “higher mathematics” was needed for Galileo’s challenge, contemporary problems in symplectic reduction and singularities call for a new, more powerful framework.

In this talk, I will introduce higher and derived structures in differential geometry, inspired by Grothendieck’s derived algebraic geometry, which provide a natural language to resolve singularities and allow shifts of symplectic structure. The physical nature behind these higher structures, I believe is the dimension of our universe being 3+1 dimensional. In another word, higher and derived geometry may serve as a mathematical language to describe topological quantum field theories (TQFTs) and sigma models with greater clarity.

To illustrate this approach, I will introduce key ideas in higher and derived differential geometry and discuss their application to Marsden-Weinstein symplectic reduction, demonstrating how this modern perspective refines classical techniques and opens new directions in symplectic geometry and mathematical physics, based on work joint with Cueca, Dorsch and Sjamaar.