Quadratic forms and groups

Description

Subject objectives: 1. Study the structure of Euclidean and Hermitian vector spaces. 2. Study orthogonal, symplectic, and Hermitian geometries. 3. Classify quadratic forms. 4. Understand tensors and their basic applications. 5. Understand the basic notions of group theory. 6. Introduce the basic notions of solvability and group actions, and apply them in different contexts. 7. Present Sylow’s theorems and apply them to the study of finite groups. Contents: 1. Bilinear forms (6 lecture hours and 6 laboratory hours). 1.1. Euclidean and Hermitian spaces. Sylvester’s criterion. 1.2. Orthogonality and orthogonal projection. 1.3. Self-adjoint and normal endomorphisms. Real and complex spectral theorem. 1.4. Normal and unitary endomorphisms. Structure theorem for isometries. 2. Quadratic forms and symplectic forms (3 lecture hours and 3 laboratory hours). 2.1. Quadratic forms. Classification. 2.2. Symplectic forms. Classification. 3. Tensors and tensor algebra (5 lecture hours and 5 laboratory hours). 3.1. Covariant and contravariant tensors. 3.2. Exterior product. 3.3. Tensor algebra and exterior algebra. 3.4. Tensor product of vector spaces. 4. Basic notions of group theory and examples (7 lecture hours and 8 laboratory hours). 4.1. Subgroups and Lagrange’s theorem. 4.2. Normal subgroups and quotient groups. 4.3. Homomorphisms and isomorphism theorems. 4.4. The dihedral group and the symmetric group. 4.5. Matrix groups. 4.6. Composition series and the Jordan–Hölder theorem. 5. Group actions and Sylow theory (4 lecture hours and 4 laboratory hours). 5.1. Action of a group on a set. 5.2. Cayley’s theorem and the orbit formula. 5.3. Sylow theorems and applications. 6. Free groups and presentations (3 lecture hours and 2 laboratory hours).