Advanced Mathematical Methods

Advanced Mathematical Methods

Contact: S. Lazzarini & F. Piazza

Group theory (20 Hrs)

  • Groups: Discrete or Continuous, Finite or Infinite
  • Representing Group Elements by Matrices
  • Group Theory in the Microscopic World
  • Linearization: Lie algebras
  • Symmetry in the Laws of Physics: conserved quantities (Noether)
  • SU(2): Double Covering of SO(3) and the Spinors
  • Isospin and the Discovery of a Vast Internal Space: The Eightfold Way of SU(3)

Differential geometry (20 Hrs)

  • Differential forms
  • Application to electromagnetism / gradients, curl and divergence
  • Differential Calculus on Manifolds
  • Vector and covector fields / Differentiating tensors / Exterior calculus / Physical applications / Covariant derivatives / Metric manifolds
  • Fibre Bundles: When differential geometry and group theory meet each other.

References
[1] M.  Stone and  P. Goldbart ”Mathematics for Physics A guided tour for graduate students”, Pimander-Casaubon (2008), http://www.goldbart.gatech.edu/PostScript/MS PG book/bookmaster.pdf
[2] A. Alastuey, M. Magro, P. Pujol , ”Physique et outils mathématiques : méthodes et exemples”, EDP sciences (2008)

[3] A. Zee, « Group Theory in a Nutshell for Physicists », Princeton University Press, 2016