I will discuss two related problems in the dynamics of complex fluids at low Reynolds number. In the first, we study numerically the regularity and mixing properties of the Oldroyd-B equations – a popular visco-elastic fluid-flow model -- for flow geometries characterized by driving vortices and organizing stagnation points. We find the emergence of singular structures in the flow as well as strong mixing dynamics characterized by both vortex destruction and persistence. We use the same flow geometry to study the temporal and spatial dynamics of elastic fibers moving in simple cellular flows. We identify a "stretch-coil" transition -- complementary to the coil-stretch transitions of polymer fluid flows -- above which the fiber can act as a spatially extended test particle, particularly through buckling at hyperbolic points, yielding complex transport properties across space. Bio: Michael Shelley is Professor of Mathematics and Neural Science at NYU’s Courant Institute of Mathematical Sciences. His research interests include fluid dynamics, computational physics, and numerical analysis.