Tuesday, 16th of December, 13:15 to 14:00 in HG03.085.
Luca Pignatelli
Why soap films are smarter than calculus.
(Abstract)
Dip a wire frame into soapy water, and it instantly solves a complex area-minimization problem that stumped mathematicians for over a century. Why did it take us so long to catch up? Because classical calculus of variations assumes surfaces are well-behaved, smooth graphs. Unfortunately, soap films are topological rebels: they branch, they intersect, and they form singularities that break simple functional minimization. In this colloquium, we will trace the history of Plateau's Problem, moving from the limitations of classical analysis to the modern advances of Geometric Measure Theory. We will discuss why we had to redefine what a surface is, introducing concepts like Currents and Varifolds to handle the orientation and multiplicity issues that arise in physical films. Finally, we will see how the framework of (M,ε,δ)-minimal sets provides the rigorous structure necessary to prove Plateau's empirical laws.