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 course is offered in the autumn term, during II period. 
Learning outcomes 
After the course, the student will be familiar with basic concepts of calculus on differentiable manifolds and Riemannian geometry, which are mathematical tools used in physics e.g. in the contexts of general relativity and gauge field theories. The student will also be familiar with basics of Lie algebra representation theory, which is used e.g. in particle physics and condensed matter theory. The student can work with differential forms, express metrics in different coordinates and compute metric tensors of general relativity. He also understands basic representations of Lie algebras used e.g. in the theory of strong interactions. 
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 
The student should be familiar with and understand the concepts of the course Mathematical Methods of Physics IIIa. In addition 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 
Differentiable manifolds and calculus on manifolds: differentiable manifolds, manifolds with boundary, differentiable maps, vector fields, 1form fields, tensor fields, differentiable map and pullback, flow generated by a vector field, Lie derivative, differential forms, Stokes' theorem
Riemannian geometry: metric tensor, induced metric, connections, parallel transport, geodesics, curvature and torsion, covariant derivative, isometries, Killing vector fields
Semisimple Lie algebras and representation theory: SU(2), roots and weights, SU(3), introduction to their most common unitary irreducible representations 
Study materials and literature 
The course follows "Mathematical methods of physics III", lecture notes by E. KeskiVakkuri, 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. 
