Dear all, we are pleased to announce the next appointment of the joint UNIPI-SNS Geometric Analysis Seminar. This week the speaker is Davide Parise, from Imperial College London. Thursday, April 16, 16:30, Aula Riunioni - Dipartimento di Matematica - UNIPI Title: From phase transitions to minimal submanifolds in low codimensions Abstract: Minimal submanifolds—critical points of the area functional—play a central role in geometric analysis, with deep connections to differential geometry, topology, and the calculus of variations. A major breakthrough in their existence theory came with Almgren’s min–max framework in the 1960s, which provides a powerful but technically intricate approach to constructing minimal submanifolds in great generality. In recent years, a new PDE-based perspective has emerged, inspired by models from phase transitions and superconductivity. The guiding principle is to realize minimal submanifolds as limits of nodal sets of critical points of elliptic functionals. In this talk, I will first give an overview of this variational–PDE framework. Using ongoing series of works with collaborators as examples, I will show how it can offer new conceptual insights and, in some settings, greater flexibility than classical min–max techniques. In codimension one, I will focus on the Allen–Cahn functional and explain how it can be used to construct minimal hypersurfaces with free boundary, i.e. meeting the ambient boundary orthogonally. We will then move to higher codimensions (specifically codimensions 2 and 3), where the theory is much less developed. Here, I will discuss the abelian and non-abelian Yang–Mills–Higgs functionals as natural higher codimension analogues and present recent progress in this setting. I will highlight other successes of this theory, and point to some open problems. The content of this talk is based on joint works with Martin Li, Lorenzo Sarnataro, Alessandro Pigati, and Daniel Stern.