Complex analysis and topology

This course develops Cauchy's theory for functions of one complex variable, and introduces the notions of conformal representation, fundamental group and coverings.

Master the basic tools of complex analysis and introduce those of algebraic topology.

A Bachelor's degree in Mathematics.

Recommended prerequisites: the content of the L3 "Complex Analysis" course of the Licence de Mathématiques de l'Université de Montpellier.

1. Licence revisions: holomorphic functions, integer series development, Cauchy's formula and theorem, Morera's theorem, maximum principle.
2. Complex analysis: singularities, meromorphic functions, residue theorem, open application theorem, biholomorphisms, Riemann conformal representation theorem.
3. Fundamental group and coverings: homotopy of paths, applications, shrinkage by deformation; definition of the fundamental group and coverings; the fundamental group of the circle, the degree of an application of the circle to itself; statement of the Seifert-Van-Kampen theorem, applications (e.g., fundamental groups of graphs); Riemann surfaces of the complex logarithm and complex root functions.

Hourly volumes :

            CM: 27h

            TD : 24h

            TP: 0

            Land: 0