Programs of the courses

Functions of One complex variable:

Preliminaries, power series, elementary trascendental functions, integration along path, Cauchy theorem for star regions, Cauchy integral formula, Taylor series, properties of holomorphic functions, isolated singularities, meromorphic functions, Casorati-Weierstrass theorem, Laurent series in the annulus, the general Cauchy theorem, residus and residus theorem, calculus of definite integrals.

Geometry V:

Elliptic functions: Liouville's theorems, Weierstrass pi function, the field E(Gamma) of elliptic functions, the addition theorem for the Weierstrass pi function, analitic prolongation and the concept of analitic function, Algebraic functions.

Real and Complex varieties I, II:

Riemann surfaces I:

Definitions and examples, topology of compact Riemann surfaces, functions on a CRS, morphism of CRS, Riemann-Hurwitz's formula, differential forms and Integration on CRS, the residus theorem, divisors and linear equivalence of divisors, the Riemann-Roch theorem (as much as possible).

Riemann surfaces II:

Weierstrass points: weight and number, hyperelliptic RS, Hurwitz's theorem on Aut(X), Abel's theorem and Jacobi's inversion, the Jacobian, embeddings into projective spaces and algebraic curves, the canonical model, the geometry of curves of low genus. Seminar.