Critical points of normal functions. I.

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.

An Application of Higher Order Fixed Points of Normal Functions

We define higher order fixed points of normal functions, describe them and apply to obtain a constructive proof that, if κ is the least ordinal such that the ultrapower κ/F is non-trivial, then that ultrapower has at least κ elements. AMS Mathematics Subject Classification (2000): 03E04, 03E20, 03C20.