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Abstract: Colloids, particles with diameters of nanometres to micrometres, form the building blocks of many of the materials around us, and are widely studied both to understand existing materials, and to design new ones. One challenge in simulating such particles is they are âstickyâ: the range over which they interact attractively, is often much shorter than their diameters, so the SDEs describing the particlesâ dynamics are stiff, and take a long time simulate up to the timescales of interest. I will introduce methods aimed at accelerating these simulations, which simulate instead the limiting equations as the range of the attractive interaction goes to zero. In this limit a system of particles is described by a diffusion process on a collection of manifolds connected by so-called âstickyâ boundary conditions. A canonical example is a (root-2) reflecting Brownian motion that is sticky at the origin, whose forward and backward equations are identically f_t = f_{xx} with sticky boundary condition f_{x} = kf_{xx} at x=0, where k>0. I will show that discretizing such an equation in space, rather than in time, gives a numerical method that is orders of magnitude faster than resolving the short-range potential directly, at least in Euclidean spaces. Furthermore, I will introduce a Monte-Carlo method to sample a probability density on a manifold, which can handle arbitrarily large timesteps. Combining these two methods is future work that is hoped to yield more efficient simulations of attractive colloidal particles, though I will point out some ongoing challenges arising from singularities in the geometry of the particlesâ configuration space. (The first part of the talk is joint work with Nawaf Bou-Rabee, and the second part with Jonathan Goodman.)