Height functions on surfaces with three critical points

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.

Critical points of simple functions

Local Matching of Surfaces Using Critical Points

The local matching problem on surfaces is: Given a pair of oriented surfaces in 3-space, find subsurfaces that are identical or complementary in shape. A heuristic method is presented for local matching that is intended for use on complex curved surfaces (rather than such surfaces as as cubes and cylinders). The method proceeds as follows: (1) Find a small set of points-called "critical points"...

Points of low height on elliptic curves and surfaces I: Elliptic surfaces over P1 with small d

For each of n = 1, 2, 3 we find the minimal height ĥ(P ) of a nontorsion point P of an elliptic curve E over C(T ) of discriminant degree d = 12n (equivalently, of arithmetic genus n), and exhibit all (E, P ) attaining this minimum. The minimal ĥ(P ) was known to equal 1/30 for n = 1 (Oguiso-Shioda) and 11/420 for n = 2 (Nishiyama), but the formulas for the general (E,P ) were not known, nor wa...