Complex Analysis and Topology

This course develops Cauchy's theory for functions of a complex variable and introduces the concepts of conformal mapping, the fundamental group, and coverings.

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

A bachelor's degree program in mathematics.

Recommended prerequisites: the material covered in the third-year (“L3”) “Complex Analysis” course in the Bachelor’s program in Mathematics at the University of Montpellier.

1. Bachelor's Degree Review: holomorphic functions, entire series expansions, Cauchy's formula and theorem, Morera's theorem, the maximum principle.
2. Complex analysis: singularities, meromorphic functions, the residue theorem, the open map theorem, biholomorphisms, Riemann's conformal mapping theorem.
3. Fundamental group and coverings: homotopy of paths and mappings, retraction by deformation; definition of the fundamental group and coverings; the fundamental group of the circle, the degree of a mapping from 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 roots functions.

Hourly volumes:

            CM: 27 hours

            TD: 24 hours

            TP: 0

            Land: 0