Abstract: To define a complex holomorphic function, we take the definition of a real differentiable function and allow all the variables involved to be complex. As the miraculous theorems of complex analysis show, this quiet move from R to C introduces all sorts of rigidity. When we further require our functions to be injective, powerful connections between geometry and analysis arise.
Focusing on the Koebe function $k(z) = z + 2z^2 + 3z^3 + …,$ this talk provides an introduction to functions that are both holomorphic and injective (univalent). We will prove the Second Coefficient and Koebe One-Quarter theorems and discuss (without proof) de Branges’ Theorem and related statements.
If you have seen calculus and are willing to take some statements on faith, I hope this talk will excite you to study this magical area of math; for students who have seen complex analysis, I hope this talk will serve as an invitation to further study.