Knowledge and understanding: geometric foundations of classical field theory. Applying knowledge and understanding: use of abstract techniques for modelling phenomena in fundamental physics. Autonomous judgment: being able to independently determine which elements of abstract formalism to apply to different problems of physical origin. Communication skills: being able to explain the ideas underlying mathematical formalism and describe physical systems in abstract terms. Learning skills: creatively applying formalism to physical problems encountered in fundamental physics and being able to consult advanced texts

Basics of group theory and differential geometry.

The course consists of an introduction to the mathematics of classical field theories, particularly Yang-Mills theories coupled to matter. It is aimed at mathematicians and physicists. The aim of the course is to discuss some of the mathematical structures underlying fundamental physics. Contents: Elements of differential geometry. Lie groups and Lie algebras Fiber bundles and gauge theories. Clifford algebras. Pin and Spin groups and their representations. Spin structures. Dirac operators. Classical lagrangians and symmetries. Classical solutions. Advanced topics. NOTE: the course should be taken during the FIRST year of the master program, in mathematics and in physics. It is activated during the second year due to an administrative error, but it can be included in the curriculum during the first year (eventually by using the printed form).

Geometry, Topology and Physics, M. Nakahara Mathematical Gauge Theory, Mark J.D. Hamilton Modern Differential Geometry for Physicists, C. Isham.

Elements of differential geometry: manifolds, differential forms, and tensors. Lie groups and Lie algebras. Representations. Vector bundles and principal bundles. Geometric formulation of gauge theories. Clifford algebras, spin groups, and spinors. Classical Lagrangians: Klein-Gordon, Yang-Mills, Dirac. Symmetries. Semiclassical solutions: instantons. Advanced topics

Blackboard lessons, consisting of the exposition of theoretical content, theorems, demonstrations, and examples.

Any changes to the methods described here, which become necessary to ensure the application of the safety protocols related to the COVID19 emergency, will be communicated on the websites of the Department of the Study Program.

Seminar on advanced topics agreed upon with the instructor during which the candidate must be able to respond to questions on the course syllabus. To pass the exam the student has to show an understanding of the course's topics and has to be able to apply them concretely during the exam. To reach the maximum mark, the student has to go deep in the chosen exam topic and show an excellent understanding of the topics discussed in the course.