Simons Foundation

Thought experiments have played an important role in figuring out the laws of physics. For the unification of quantum mechanics and gravity, where the phenomena take place in extreme regimes, they are even more crucial. Hawking’s 1976 paper “Breakdown of Predictability in Gravitational Collapse” presented one of the great thought experiments in the history of physics, arguing that black holes destroy information in a way that requires a modification of the laws of quantum mechanics. Skeptics for years failed to poke holes in Hawking’s argument, but concluded that if quantum mechanics is to be saved then our understanding of spacetime must break down in a radical way.

In biological systems, there are striking examples where complicated structures (i.e., the bacterial ribosome) can spontaneously assemble, driven by specific interactions between the components. But how can systems be designed to have this property? Recent technological  advances have created the opportunity for making technologically relevant systems that self assemble, using strands of DNA or objects coated with DNA. We will use these systems as inspiration to formulate theoretical models to understand how self assembly works in these systems, through theory, numerical simulation and experiment — and start to speculate as to whether resulting principles might be useful for unravelling the rules of biological self-assembly.

Integral equation methods play an important role in the numerical simulation of electromagnetic scattering. They are easy to employ in complex geometry and impose the desired radiation conditions at infinity without the need for artificial numerical boundaries. Two of the obstacles faced by current forward simulation tools are “low-frequency breakdown” and the lack of easy to use high order quadrature rules for complicated surfaces. In this talk, I will review the relevant background material, discuss a new mathematical formalism for scattering from perfect conductors and briefly describe a new quadrature technique that yields easily implementable high order rules for singular and weakly singular integrals. The scheme, denoted QBX (quadrature by expansion) is compatible with fast hierarchical algorithms such as the fast multipole method.