*Recent Progress in Topological Fluid Dynamics * (Ravello, 2017)

Course contents (6 hours):

- Topological interpretation of helicity
- Coherent structures and topological fluid mechanics
- Diffeomorphisms and topological equivalence
- Kinetic and magnetic helicity of flux tubes
- Gauss linking number
- Călugăreanu invariant and geometric decomposition

- Vortex knots dynamics and momenta of a tangle
- Localized Induction Approximation (LIA) and Non-Linear Schroedinger (NLS)
equation
- Integrable vortex dynamics and LIA hierarchy
- Torus knot solutions to LIA
- Linear and angular momentum in terms of signed area information

- Magnetic knots and groundstate energy spectrum
- Magnetic relaxation
- Topology bounds the energy
- Inflexional instability of magnetic knots
- Constrained minimization of magnetic energy
- Groundstate energy spectra of magnetic knots and links
- Bending energy spectra: Magnetic vs. elastic systems

- Topological transition of soap films
- Seifert surfaces and soap films
- The Plateau problem: the catenoid
- Topological transition from 1-sided to 2-sided surface
- Local analysis of the twisted saddle catastrophe
- Geometry and energy considerations during transition

- Helicity change under reconnection: the GPE case
- Reconnection and change of helicity
- Application to solar cornal loops
- Writhe conservation under reconnection
- Cascade of quantum vortex links under Gross-Pitaevskii equation
- Iso-phase surfaces as Seifert surfaces
- Evolution of iso-phase surfaces of minimal area

- Topological decay measured by knot polynomials
- Knot polynomials as new tools in turbulence research
- Polynomial skein relations
- Adapted polynomials as new invariants of fluid mechanics
- Vortex knot cascade detected by monotonically decreasing sequence of polynomial numerical values
- HOMFLYPT as best quantifier of topological complexity