**1st module**

**Prof. Edoardo Ballico**

**A.A. 1998/99**

**PROGRAMME**

(a) Introduction to analytic functions of one complex variable:

1) complex numbers (a reminder);

2) complex-valued functions;

3) the Cauchy - Riemann conditions;

4) holomorphic functions;

5) Abel's theorem;

6) the complex exponential;

7) curvilinear integrals;

8) Cauchy formula and its applications;

9) Goursat's theorem;

10) analytic continuation and the maximum principle;

11) singularities of holomorphic functions;

12) residues

13) (optional) definitions and basic properties of holomorphic functions
of several complex variables; Hartogs Theorem.

(b) Introduction to singular homology theory following the first 3 chapters
of W. S. Massey, Singular Homology Theory, Springer-Verlag:

1) Definitions and fundamental theorems (homotopy, exact sequence of
a triple, excission and Mayer - Vietoris exact sequence);

2) Computations of the homology groups of important examples (spheres,
graphs and compact topological surfaces);

3) Applications: fixed point theorems, invariance of the domain and
Jordan-Brower.