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.