A wide range of physics problems classically formulated as elliptic PDEs may be equivalently cast as boundary integral equations (BIE) of the second kind. Key advantages of this approach feature dimensionality reduction and improved conditioning; they may be best leveraged by developing fast algorithms for the evaluation and solution of these integral equations. We present a general framework to accelerate the solution of such boundary integrals using the Quantized Tensor Train (QTT) decomposition as an approximate compression and inversion scheme. We demonstrate that using QTT-based preconditioners for a Krylov subspace method leads to a fast, memory-efficient and robust solver. Computational costs and storage of computing and updating the QTT inverse are extremely modest O(log N), and it can be applied in O(N logN) time. Finally, we introduce a recently developed BIE formulation for rigid body Stokes flows, given prescribed forces and torques. We discuss the use of QTT and other fast algorithms in this context, and discuss applications to particle sedimentation, low Reynolds number swimming and magnetorheological flows. Speaker(s): Eduardo Corona (University of Michigan)