Acquisition of operational capabilities in the use of geometric instruments in the physical field. At the end of the course the student will be able to: -recognize the role of geometry in different physical models; -provide examples; -apply techniques of differential geometry to problems in physics; -analyze a physical model from the geometric viewpoint; -compare different geometric models in order to select the most suitable one; -create new models
Prerequisiti
Basic knowledge of algebra, geometry, real analysis in several variables, mechanics, electromagnetism
Metodi didattici
Frontal lessons
Verifica Apprendimento
Solution of written exercises proposed by the teacher + oral exam At the end of the course, the docent will propose some exercises to be solved at home. It will be evaluated as first the ability to frame the problem in the right context, the correctness of the procedure, the correctness of the mathematical passages and the ability to provide an answer to the problem. It follows an oral discussion in order to evaluate the general theoretical knowledge of the student.
Contenuti
1 Multilinear algebra Tensors. Tensors in physics. The external algebra. 2 Representations of finite groups General theory. Characters. Irreducible representations. 3 Tensors and symmetric group Introduction. Schur functors. Examples. 4 Introduction to differential geometry Differential varieties. Fiber bundles. Differential forms. 5 Lie Groups and Lie algebras Lie groups. Simple Lie algebras. Representations of simple Lie algebras. Representations of sl(2). 6 Classification of simple Lie algebras Cartan algebra and weights. Roots and Dynkin diagrams. Real simple algebras. 7 Representations of simple Lie algebras Weights: integrality and symmetries. The highest weight. Irreducible representations of sl(n). The case n=3. 8 Orthogonal Groups Bilinear and quadratic forms. Orthogonal groups. Quaternions. The group SO(2m). The group SO(2m+1). 9 Spin representations Clifford algebras. The spin group. Spin representations of so(n). 10 Geometric structures Fiber bundles. Geometry and Lie groups. Connections. Covariant derivatives. Curvature. Riemannian structure. 11 Dynamical theory of symmetries Matter fields. Gauge fields. Yang-Mills theories. External symmetries, and relativity. 12 Scalar and spinoral fields Particles and the Poincaré group. Klein-Gordon equation. Dirac equation. The SO(1,3) group. Spin 1 fields. Fields and gravity. 13 Geometry of the Standard Model of particles and GUT The field content in the Standard Model. Constructing the Standard Model. Spontaneous symmetry breaking. The mass of neutrino. The seesaw mechanism. GUTs.
