The study of vortex rings in incompressible 3D fluids dates back to Kelvin and Helmholtz in the mid 1800's. In 1906, Da Rios and Levi-Civita gave a formal derivation of a geometric flow for filaments of infinitely small cross section and arbitrary shape. This flow is now widely called the binormal curvature flow. In the talk, I will first review and then present recent results on stability estimates for the filament flow, and their application to so-called Schrodinger maps. 

Braids are beautiful objects in low dimensional topology. They can be seen likewise as tangles in the 3-ball or as elements of the mapping class group of the punctured disc or as automorphisms of free groups.We start by recalling the construction of the HOMFLYPT invariant for tangles , the Niesen-Thurston classification of diffeomorphisms of the punctured disc and the growth rate of  automorphisms of  free groups.
We present then our machinery for constructing 1-cocycles which produce HOMFLYPT invariants for 1-parameter families of tangles. It turns out that they contain information about the geometry of braids. There will be lots of examples in the talk.