Abstract: In optimization theory, one clear dividing line between "easy" and "hard" problems is convexity. In convex optimization problems, all local optima are global optima, which can be found by efficient computational algorithms. By contrast, nonconvex problems can have highly oscillatory landscapes, and one must typically use local optimization techniques or black-box approaches. Nanophotonic design problems, and many design problems across physics, reside squarely in the latter category of nonconvex optimization problems.

Or do they? I will show that there is a surprising amount of mathematical structure hidden in the typical differential equations of physics, and that this structure enables new connections to modern techniques in convex optimization. The key differential-equation constraints can be transformed to infinite sets of local conservation laws, which have a structure amenable to quadratic and semidefinite programming. This approach can lead to global bounds ("fundamental limits") for many design problems of interest, and potentially to dramatically new approaches to identifying designs themselves.

Spectral (frequency) degrees of freedom have further structure to be exploited. Specific to electromagnetic scattering, I will describe a new positive-definite-oscillator construction of scattering matrices that leads to convexity-based methods for identifying fundamental limits across any bandwidth of interest.

Throughout I will emphasize novel applications where we utilize these techniques, including: minimal-thickness perfect absorbers, scaling laws for analog photonics, speed limits in quantum optimal control, and a theory of the ultimate limits of near-field radiative heat transfer.

Bio: Owen Miller is an Asst. Prof. of Applied Physics and Physics at Yale. His research interests center around developing large-scale computational and analytical design techniques for discovering novel structures and new phenomena in nanophotonics. He is the recipient of AFOSR and DARPA young investigator awards, as well as the Yale Graduate Mentor award.