Critical points of normal functions. II.

Critical points of simple functions

Generic critical points of normal matrix ensembles

The evolution of the degenerate complex curve associated with the ensemble at a generic critical point is related to the finite time singularities of Laplacian Growth. It is shown that the scaling behavior at a critical point of singular geometry x ∼ y is described by the first Painlevé transcendent. The regularization of the curve resulting from discretization is discussed. PACS numbers: 05.30...

On the Critical Points of Harmonic Functions

Critical Points of Functions on Singular Spaces

We compare and contrast various notions of the “critical locus” of a complex analytic function on a singular space. After choosing a topological variant as our primary notion of the critical locus, we justify our choice by generalizing Lê and Saito’s result that constant Milnor number implies that Thom’s af condition is satisfied.

Fixed Points of Univalent Functions Ii

For a closed nowhere dense subset C of ∂D a bounded univalent holomorphic function f on D is found such that C equals the cluster set of its fixed points.