Applications are closed
for this workshop

Symmetry-breaking of optimal shapes

June 17 to June 21, 2024

at the

American Institute of Mathematics, Pasadena, California

organized by

Dorin Bucur, Almut Burchard, Richard Laugesen, and Antoine Henrot

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.

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.

This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

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