## Elements of contemporary mathematics

- Basic definitions in group theory: grupoid, semigroup, groups; general group classifications, subgroups, cosets, quotient groups, group products, conjugation, conjugacy classes, basic theorems (Cayley, Lagrange, rearrangement theorem etc.).
- Group homomorphism and isomorphism, isomorphism theorems.
- Vector spaces, simple sums and tensor products. Rings, algebras, Lie algebras, division algebras. Abelian, associative and alternative algebras. Ideals.
- Group action on a set, oribt, stabilizer.
- Representation of groups; reducibility, unitarity, unitary representations of finite and compact groups. Tensor product of representations, representations of algebras.
- Schur’s lemma.
- Tensor operators, Eckart–Wigner theorem.
- Manifolds: basic definions. Orientability, coordinates, tangent and cotangent space. Vectors and tensors on manifolds.
- Elements of topology: homeomorphisms, simplices, orientation, simplicial complexes, boundaries and cycles, homotopy groups, homologies.
- Differential forms on manifolds. External algebra, external products, closed and exact forms, cohomologies: symplicial and Cˇech cohomology.
- Integration of differential forms over chains, Stokes theorem, relation of homology and cohomology, de Rham complex.
- Action of groups on manifolds. Quotient spaces and orbifolds. Covering spaces.
- Lie groups: basic definitions and topological properties.
- Fibre boundles, principal and associated fibre bundle.
- Connection on fibre bundles, connection forms, curvature. Connection on vector bundles, local connection forms. Metric. Linear connection, torsion, Levi-Civita connection.
- The Lie algebra: generators, structure constants, Killing forms, adjoint representation, simple and semi-simple Lie algebras.
- Covering groups, the third Lie theorem, exponentiation of the Lie algebra.
- Representations of Lie algebras: the Cartan-Weyl matrix, weights and roots.
- Master formula for roots. Geometry of roots. Representation construction: the highest weight method. Dynkin coefficients.
- Cartan matrix, Dynkin diagrams, P-systems, reduction rules for Dynkin diagrams.
- Classification of simple Lie algebras.

Construction of simple Lie subalgebras with Dynkin diagrams.