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Mathematical Methods for Modern Physics

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 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: Fluid flows and Lagrangian diffeomorphisms; fundamentals of potential theory and Green’s theorems; multiply-connected domains and Riemann’s cuts; conservation laws and topological invariants; Euler’s equations; Helmholtz’s equations; Navier-Stokes equations; Maxwell equations; solid angle interpretation of the induction law; analogies between ideal magneto-hydrodynamics and Euler’s equations.
  • II Part: physical knot theory; Gauss’ linking number; Călugăreanu’s self-linking, writhe and twist number; construction of knotted fields; energy relaxation for magnetic knots and links; hydrodynamic interpretation of the Gross-Pitaevskii equation; defect theory and Kleinert’s multi-valued gauge theory; Seifert surfaces and minimal surfaces; change of topology and reconnection mechanisms; relationships between structural complexity and energy.

Reference material

  • Lecture notes handed out during the course.
  • 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 is handed out at the end of the course. Usually, four questions are taken from this list and they are posed to the student for his discussion. The final mark is expressed in thirtieths.

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