**Target skills and knowledge**: This course will review some mathematical concepts that have interesting non-trivial applications in modern physics. The main objective is to show when and how mathematical rigour becomes effective in giving a better description of physical phenomena.

**Planned learning activities and teaching methods:** Lectures. Assignments.

**Exercises**

First part, Second part.

**Lectures**: Lectures will be delivered at the Physics and Astronomy department:

Day |
Time |
Room |

MON |
17.00 - 19.30 |
Aula B |

**1. Sigma Models**.

Symmetry breaking: explicit and spontaneous breakings. Sigma models.

**2. Homogeneous manifolds**.

Group manifolds. Cosets, coset representatives, coset manifolds. Pions.

**3. Homotopy**.

Skyrmions. Fundamental group. Homotopy groups.

**4. Fibre Bundles**.

Fibre bundles. Connection, local connection, Curvature, local curvature.

**5. Simple topological solutions - Solitons **.

The kink. Solitons with more scalars - skyrmions. Solitons in gauge theories. The vortex. Strings. Non-topological solitons.

**6. Monopoles**.

SU(2) monopoles. Fibre bundle interpretation. Other models. Dyons.

**7. Tunnelling**

Decay of metastable vacua in QM. Instantons. Determinant zero modes. Decay of the false vacuum in scalar field theory. Thin-wall approximation.

**8. Instantons and sphalerons in gauge theories.**

Euclidean Yang-Mills theories. Instantons in Yang-Mills theories. Classical vacua and theta-vacua. Sphalerons.

**9. Complex geometry.**

Almost complex manifolds. Complex Manifolds. Symplectic manifolds. Kaehler manifolds. Group structures and differential properties.