GSAS: Torsion Function and Analytic Perturbation of Convex Domains  

Torsion function is a classic concept in elasticity theory, which has been studied for well over a hundred years. Given a bounded planar domain, the gradient of the torsion function measures the torsional stress of the domain at each point. It is well-known that for convex domains, the torsional stress is maximized at a point on the boundary. However, what convex domain maximizes the torsional stress remains an open problem. The analysis of the torsion function has been traditionally done with functional or probabilistic techniques. In this talk, I will go through my attempt to study the torsion function using conformal geometry.