Complex Analysis and Topology

This course develops Cauchy's theory for functions of a complex variable and introduces the concepts 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 course "Complex Analysis" from the Bachelor's degree in Mathematics at the University of Montpellier.

1. Bachelor's degree revisions: holomorphic functions, entire series expansion, Cauchy's formula and theorem, Morera's theorem, maximum principle.
2. Complex analysis: singularities, meromorphic functions, residue theorem, open mapping theorem, biholomorphisms, Riemann's conformal representation theorem.
3. Fundamental group and coverings: homotopy of paths, maps, deformation retraction; definition of the fundamental group and coverings; the fundamental group of the circle, the degree of a map from the circle to itself; statement of the Seifert-Van Kampen theorem, applications (e.g., fundamental groups of graphs); Riemann surfaces of complex logarithm and complex root functions.

Hourly volumes:

            CM: 27 hours

            TD: 24 hours

            TP: 0

            Land: 0