Target group |
Master’s Programme in Theoretical and Computational Methods is responsible for the course.
Modules where the course belong to:
**TCM300 Advanced Studies in Theoretical and Computational Methods**
**PAP300 Advanced Studies in Particle Physics and Astrophysical Sciences** Optional for:
- Study Track in Particle Physics and Cosmology
The course is available to students from other degree programmes. |

Timing |
The recommended (but not compulsory) time for having completed the course would be by the end the first year of the master programme.
The course may be offered in the autumn or spring term or both, during the first period of the term. |

Learning outcomes |
After the course, the student will be familiar with basic concepts of group theory, group representation theory, and topology. The student can identify different common groups, study if their representations are reducible, irreducible or not, and knows why the theory of groups and their unitary representations is important in quantum physics of systems with various symmetries. The student also understands distinctions between nonhomeomorphic topological spaces and understands the use of topological invariants (such as homotopy groups) in their classification. |

Completion methods |
The course is lectured as contact teaching, but it is also possible to pass the course by studying it independently (e.g. by a final exam), if so agreed with the lecturer. In the course form, the completion is based on a final exam and weekly homework performance. |

Prerequisites |
There are no formal prerequisites for this course, but it is recommended to be familiar with linear algebra, differential and integral calculus, and (partial) differential equations. It is also helpful to know basic physics such as classical mechanics, electrodynamics, some quantum mechanics, and theory of special relativity. |

Contents |
Group theory: finite groups, continous groups, conjugacy classes, cosets, quotient groups
Representation theory of groups: complex vector spaces and representations, symmetry tranformations in quantum mechanics, reducible and irreducible representations, characters
Topology: topological spaces, topological invariants, homotopy, homotopy groups |

Study materials and literature |
The course follows "Mathematical methods of physics III", lecture notes by E. Keski-Vakkuri, C. Montonen and M. Panero. Supplementary reading is listed in the lecture notes. The students are encouraged to actively search for additional supplementary material from the Web (e.g., from Wikipedia and other such pages.) |

Activities and teaching methods in support of learning |
Weekly contact lectures, independent work of the student including solving weekly homework problem sets. The solutions to the problem sets will be submitted weekly, graded by the teaching assistant and discussed in weekly exercise sessions. In these sessions the students may also discuss and get tutoring for next week's homework. |

Assessment practices and criteria |
The grade is determined in a way agreed in the beginning of the course. |