Many physical phenomena feature interfaces that minimize their surface area, much like the effects of surface tension. Within this framework, minimal surfaces and the Ginzburg-Landau model of phase transitions, while widely applied in various research fields, are predominantly practical phenomenological models. 

This prompts an exploration into alternative mathematical frameworks exhibiting similar macroscopic features. 

Emerging initially from this mindset, nonlocal minimal surfaces have later proven to be rich and profound geometric entities. We've recently discovered that these surfaces exhibit enhanced regularity compared to classical minimal surfaces. These better regularity properties position them as prime candidates for the application of variational methods. I'll discuss these insights in an accessible manner, highlighting their intriguing implications in the realms of minimal surfaces and phase transitions.