**Content:**

1.The electromagnetic feld

Facts from Physics: a) Electrostatics, b) Magnetostatics, c) Electromagnetism.

Differential forms on R^n. Potentials, gauge invariance and wave equation.

4-dimensional form of Maxwell equations.

2.Matter and gauge fields

Space-time.Spinors. Action functionals for matter fields.

Noether's theorem and conservation laws.

Multiplets of matter fields.

From global to local symmetries.

Yang-Mills gauge theories.

3.Background from Geometry

Vector bundles, metrics, connections and curvature.

Principal fible bundles and associated bundles. Hopf bundles.

Chern classes. Grassmannian connections.

4.Instantons

Absolute minima of the Y.M. action functional. The basic instanton.

Sl(4,C) and the conformal group SO(1,5).

Multiinstantons and their moduli spaces.

5.Global analysis of gauge theories

Results from Sobolev space theory.

The space of connections and the Y.M. functional.

The group of gauge transformations: action on the space of connections and the gauge orbit space.

Gauge fixing and Gribov ambiguity. Gribov region.

Hints to functional integral quantization: the Faddeev-Popov method.

Family of meadure spaces parametrized by instanton moduli spaces.

**References**

J.D. Jackson ``Classical electrodynamics'' Wiley (1962)

Eguchi, Gilkey, Hanson Phys. Rep. 66 (1980) 213-393

M.F. Athiyah "Geometry of Yang-Mills Fields" Lezioni Fermiane SNM Pisa (1979),