We study Langevin dynamics with noise projected onto the directions orthogonal to an isometric group action. This mathematical model is introduced to shed new light on the effects of symmetry on stochastic gradient descent for over-parametrized models. Our main result identifies a novel form of implicit regularization: when the initial and target density are both invariant under the group action, Langevin dynamics with projected noise is equivalent in law to Langevin dynamics with isotropic diffusion but with an additional drift term proportional to the negative log volume of the group orbit. We prove this result by constructing a coupling of the two processes via a third process on the group itself, and identify the additional drift as the mean curvature of the orbits.

We investigate the utility of a mathematical framework based on discrete geometry to model biological and synthetic self-assembly. Our primary biological example is the self-assembly of icosahedral viruses; our synthetic example is surface-tension-driven self-folding polyhedra. In both instances, the process of self-assembly is modeled by decomposing the polyhedron into a set of partially formed intermediate states. The set of all intermediates is called the configuration space, pathways of assembly are modeled as paths in the configuration space, and the kinetics and yield of assembly are modeled by rate equations, Markov chains, or cost functions on the configuration space. We review an interesting interplay between biological function and mathematical structure in viruses in light of this framework. We discuss in particular: (i) tiling theory as a coarse-grained description of all-atom models; (ii) the building game-a growth model for the formation of polyhedra; and (iii) the application of these models to the self-assembly of the bacteriophage MS2. We then use a similar framework to model self-folding polyhedra. We use a discrete folding algorithm to compute a configuration space that idealizes surface-tension-driven self-folding and analyze pathways of assembly and dominant intermediates. These computations are then compared with experimental observations of a self-folding dodecahedron with side 300 μm. In both models, despite a combinatorial explosion in the size of the configuration space, a few pathways and intermediates dominate self-assembly. For self-folding polyhedra, the dominant intermediates have fewer degrees of freedom than comparable intermediates, and are thus more rigid. The concentration of assembly pathways on a few intermediates with distinguished geometric properties is biologically and physically important, and suggests deeper mathematical structure.