EWM-EMS Summer School: Symmetry and Symmetry Breaking in Sweden

This school is devoted to the study of symmetry and symmetry-breaking phenomena in nonlinear partial differential equations (PDEs), with a particular emphasis on two powerful analytical tools: the group theoretical approach and the Ljapunov–Schmidt reduction. Symmetry plays a central role in the analysis of PDEs, as symmetric solutions often correspond to lower energy states and benefit from improved compactness properties in invariant functional spaces. This perspective, which originated from Palais’ principle of symmetric criticality, has become fundamental for constructing solutions with prescribed symmetries and gaining insight into their energy levels.

At the same time, symmetry breaking is a key mechanism underlying the emergence of complex structures and patterns in physical and geometric problems. A variety of methods—variational, topological, bifurcation, perturbative, and reduction techniques—can be used to detect and analyze such phenomena. In particular, the interplay between symmetry and symmetry breaking is often revealed through bifurcations from symmetric branches, leading to the construction of multiple solutions with fewer symmetries than the original problem. Among these methods, the Ljapunov–Schmidt reduction provides a robust framework for detecting bifurcating solutions and has evolved into a sophisticated tool for constructing concentrating solutions with prescribed symmetric profiles.

Through a series of lectures by Mónica Clapp and Monica Musso, leading experts in this research area, alongside invited talks, the school aims to offer participants a comprehensive introduction to these methods, their theoretical foundations, and their wide-ranging applications in geometry and physics.