# Topological Methods in Field Theories

### Introduction

This is an elective course of the Master Degree in Mathematics, and it contributes to the educational training in Mathematical Physics. All lectures are given in English, and they are delivered during the second semester of each academic year, from March to June. It consists of 5 hours per week of front lectures, accounting for 8 University Credit Units (CFUs) of the Master Degree in Mathematics.

### Aims

The course aims at providing the basic notions, methods and techniques for a geometric and topological approach to fundamental aspects of classical field theory, with particular emphasis on aspects of vortex dynamics, magnetic fields, superfluidity, and defect theory in geometric optics and condensed matter.

### Programme

The course programme focusses on two complementary parts; the first part is of general character, and it is dedicated to the presentation of the fundamental equations of field theory; the second part is dedicated to the presentation of specific, more advanced topics that are at the core of contemporary research.

- I Part. Fundamentals of potential theory, Green’s identities, fluid flows and diffeomorphisms, kinematic transport theorem and conservation laws, decomposition of fluid motion, Euler’s equations, vorticity transport equation, Helmholtz’s conservation laws, Biot-Savart law, Maxwell’s equations, ideal magnetohydrodynamics, magnetic helicity, perfect and non-perfect analogous Euler’s flows, Navier-Stokes equations, energy dissipation.
- II Part. Kelvin’s correction of Green’s identity for multiply connected regions, elements of knot theory, fluid dynamic interpretation of Reidemeister moves, inflexional configuration and twist energy, linking number and self-linking, derivation of linking number from magnetic helicity, writhing and total torsion helicity, magnetic energy relaxation, ground-state energy spectra, hydrodynamics interpretation of Gross-Pitaevskii equation, topological defects in condensates, change of topology of flux tubes and physical surfaces due to reconnection processes, knot polynomial invariants, measures of topological complexity.

### Reference material

- Lecture notes provided by the lecturer.
- Topics taken from: Ricca, R.L. (2009)
*Lectures on Topological Fluid Mechanics.*Springer-CIME Lecture Notes in Mathematics**1973**. Springer-Verlag. Heidelberg.

### Examination

A list of examination questions taken from the course topics is given to the student. Four questions are taken from this list, and they are posed to the student for discussion. The final mark is expressed in thirtieths.

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