AIM Workshop: Symmetry-breaking of optimal shapes

This workshop, sponsored by AIM and the NSF, investigates Shape Optimization problems for which symmetry-breaking is believed to occur: the energy functional is radially symmetric and yet its minimizer fails to enjoy full symmetry. Unlike in many other problems, the optimal shape is not the ball. The ball does solve many natural shape optimization problems for geometric and physical models, including the classical isoperimetric inequality and inequalities for capacity and the first eigenvalue of the Laplacian under Dirichlet, Robin or Neumann boundary conditions. Thus it becomes particularly fascinating to investigate problems where the ball is NOT the best, and to determine the optimal shape in those situations.

Topics: The main topics for the workshop are: Topic 1: Examples where the ball is non-optimal. Topic 2: Techniques for identifying optimal shapes: shaking, tensorization, geometric flows, first or second order arguments to prove the optimal shape must be a polygon or polytope, free boundaries analysis. Topic 3: Notable conjectures including Polya's capacity problem, Neumann spectral problems, problems from convex geometry, isodiametric capacity maximizers, Alexandrov's conjecture for intrinsic diameter.