The contents of the course change depending on the knowledge and the interests of students. The following is a list of topics studied in the last editions of the course, only a part of this will be covered.
·Lie Groups ·The method of characteristics ·The geometrical meaning of a differential equation ·Symmetry and reduction of a differential equation ·Symmetry in variational problems ·Lax systems ·Integrable systems ·Vector fields and forms ·Differential equations in Pfaff form ·Symmetry of differential equations in the Pfaff formalism ·Frobenius theorem for vector fields ·Frobenius theorem for forms ·Gauge theories and the Higgs mechanism
Prerequisites for admission
It will be assumed the student knows the material covered in the B.Sc. ("Laurea Triennale") program
P.J. Olver, Application of Lie groups to differential equations, Springer 1986 P.J. Olver, Equivalence, invariants, and symmetry, Cambridge 1995 H. Stephani, Differential equations, Cambridge 1989
Some lecture notes will also be available
Assessement methods and criteria
The final examination consists of three parts: a written exam, an oral exam and a lab exam.
- During the written exam, the student must solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve problems in XXX. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take N midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal. - The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems regarding XXX in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them. -The lab exam consists in developing a project, which will be assigned in advance by the professor. The project will be presented by the student during the oral exam. The lab portion of the final examination serves to assess the capability of the student to put a problem of XXX into context, find a solution and to give a report on the results obtained.
The complete final examination is passed if all three parts (written, oral, lab) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.