Abstract We establish a patchworking theorem à la Viro for the Log-critical locus of algebraic curves in ( ℂ ∗ stretchy="false">) 2 {(\mathbb{C}^{\ast})^{2}} . As an application, we prove existence projective arbitrary ...

Eszter Klein’s theorem claims that among any 5 points in the plane, no three collinear, there is the vertex set of a convex quadrilateral. An application of Ramsey’s theorem then yields the classical Erdős–Szekeres theorem [19]: For every integer n ≥ 3 there is an N0 such that, among any set of N ≥ N0 points in general position in the plane, there is the vertex set of a convex n-gon. Let f(n) d...

Contents Introduction. 1 Chapter 1. Four-dimensional Theory. 4 1. Setup. 4 2. A slice theorem for singular connections. 9 3. Laplacians on forms. 17 Chapter 2. Three-dimensional Theory. 22 1. Setup. 22 2. The Chern-Simons functional. 24 3. The deformation complex. 27 Chapter 3. Singular connections on manifolds with cylindrical ends. 33 1. Setup. 33 2. The translation-invariant case. 35 3. Glob...

1. Introduction and statement of results. "Let three points be specified on a line. Then one of the points is in the interior of the line segment joining the other two, and one of the points is exterior to the line segment joining the other two." This elementary statement concerning the ordering of three points on a line is capable of various extensions. Thus, e.g., it is easy to prove the foll...