Briot-Bouquet’s Theorem in high dimension

Infinitesimal Hartman-Grobman Theorem in Dimension Three.

In this paper we give the main ideas to show that a real analytic vector field in R3 with a singular point at the origin is locally topologically equivalent to its principal part defined through Newton polyhedra under non-degeneracy conditions.

Noncommutative Tsen’s Theorem in Dimension One

Let k be a field. In this paper, we find necessary and sufficient conditions for a noncommutative curve of genus zero over k to be a noncommutative P1-bundle. This result can be considered a noncommutative, onedimensional version of Tsen’s theorem. By specializing this theorem, we show that every arithmetic noncommutative projective line is a noncommutative curve, and conversely we characterize...

A Fiber Dimension Theorem for Essential and Canonical Dimension

The well-known fiber dimension theorem in algebraic geometry says that for every morphism f : X → Y of integral schemes of finite type, the dimension of every fiber of f is at least dimX−dimY . This has recently been generalized by P. Brosnan, Z. Reichstein and A. Vistoli to certain morphisms of algebraic stacks f : X → Y , where the usual dimension is replaced by essential dimension. We will p...

Bouquets of Genes

Entropy and a Convergence Theorem for Gauss Curvature Flow in High Dimension

In this paper we prove uniform regularity estimates for the normalized Gauss curvature flow in higher dimensions. The convergence of solutions in C∞-topology to a smooth strictly convex soliton as t approaches to infinity is obtained as a consequence of these estimates together with an earlier result of Andrews. The estimates are established via the study of an entropy functional for convex bod...