# Research

## Analysis

### Tuesday, August 3, 2010

Title: A solution to the Gromov-Vaserstein Problem
Speaker: Frank Kutzschebauch, University of Berne
Switzerland
Time: 1:00pm–2:00pm
Place: PHY 108

#### Abstract

Any matrix in $$\operatorname{Sl}_n\,(\mathbb{C})$$ can (due to the Gauss elimination process) be written as a product of elementary matrices. If instead of the complex numbers (a field) the entries in the matrix are elements of a ring, this becomes a delicate question. In particular the rings of maps from a space $$X\to\mathbb{C}$$ are interesting cases. A deep result of Suslin gives an affirmative answer for the polynomial ring in $$m$$ variables in case the size of the matrix ($$n$$) is greater than 2. In the topological category the problem was solved by Thurston and Vaserstein. For holomorphic functions on $$\mathbb{C}^m$$ the problem was posed by Gromov in the 1980's. We report on a complete solution to Gromov's problem. A main tool is the Oka-Grauert-Gromov-$$h$$-principle in Complex Analysis. This is joint work with Björn Ivarsson.