Aims

According to the educational objectives of the Course, the taught material aims to provide students with the basic notions regarding the definitions and the fundamental results for a geometric and topological approach to the study of classical field theory, with particular emphasis on classical vortex dynamics, ideal magnetohydrodynamics and quantum hydrodynamics. The course aims to provide also the necessary competences to understand and use standard techniques and the demonstration methods involved in the theory, as well as the capabilities to use them to solve exercises and tackle problems.

The expected outcomes include: 

• the knowledge and understanding of the fundamental definitions and statements, as well as of the basic strategies of proof in classical geometric and topological field theory; the knowledge and understanding of some key examples where theory is fully applied;
• the ability to recognize the role that concepts and techniques from a geometric and topological approach play in various areas of mathematics (such as vortex theory, ideal magnetohydrodynamics, quantum fluids), and in the mathematical modelling of physical situations (vortex dynamics, relations between energy and complexity, topological defect production, knotting and linking); skills to apply the basic concepts to the elaboration of practical examples and to the solution of posed questions; the ability to communicate and explain in a clear and precise manner both the theoretical aspects of the course as well as their applications to specific situations, possibly in analogous but different contexts.

Contents

Part I. Fluid flows and diffeomorphisms, Green's identities, conservation theorems,  Euler's equations,  Helmholtz's conservation laws,  Navier-Stokes equations, ideal magnetohydrodynamics, magnetic helicity.

Part II. Elements of knot theory, torus knot solutions to LIA, Gross-Pitaevskii equation, topological defects, helicity and linking numbers, measures of topological complexity.

Detailed program

The course is divided into two parts, the first being of introductory and general character, the second focused on more specific topics of current research.

Part I. Fluid flows and diffeoporphisms, Green's identities, Kelvin's correction for multiply connected domains, kinematic transport theorem, conservation theorems, decomposition of fluid motion, Euler's equations, vorticity transport equation, Helmholtz's conservation laws, Biot-Savart law, Navier-Stokes equations, energy dissipation, Burgers' steady vortex solution, Maxwell's equations, ideal magnetohydrodynamics, magnetic helicity, perfect and non-perfect analogous Euler's flows.

Part II. Localized induction approximation (LIA), elements of knot theory, fluid dynamic interpretation of Reidemeister moves, inflexional configuration and twist energy,  torus knot solutions to LIA, fluid dynamics interpretation of Gross-Pitaevskii equation, topological defects, helicity and linking numbers, writhe, twist, knot polynomial invariants, measures of topological complexity.