## Hilbert's 13th Problem

**From** 2012-07-15 **to** 2014-07-14, Grant Agreement terminated

## Project details

### Total cost:

EUR 209 033,4

### EU contribution:

EUR 209 033,4

### Coordinated in:

United Kingdom
## Objective

"The aim of this fellowship is to enable Dr Christopher Good, as Scientist in Charge, and Dr Ziqin Feng, as Researcher, to carry out some innovative and mutually beneficial research utilizing their complementary skill sets.

The 13th Problem from Hilbert's famous list asks whether every continuous (respectively smooth) function of three variables can be written as a superposition (or, in modern parlance, composition) of continuous (respectively smooth) functions of two variables. Hilbert conjectured that the answer to this problem was `no.' However, in 1957, Kolmogorov together with his student Arnold gave a positive solution in the continuous case: every continuous function of n variables taken from the closed unit interval can be represented as a linear superposition of one-variable functions and the two-variable function addition. One might expect this result to have applications (for example to data analysis), since it allows for multi-dimensional functions to be expressed as `simpler' functions of one variable and addition. However, whilst being of great theoretical interest, Kolmogorov's result is highly non-constructive and does not obviously allow for this. Together with Professor Paul Gartside, Feng has made highly non-trivial extensions to Kolmogorov's theorem that suggest ways around these restrictions. This project aims to realize potential applications by providing improved algorithms, implementing the extensions in high-level computer code.

Vitushkin gave a negative answer to the smooth (differentiable) version of Hilbert's 13th problem in 1954, proving, in particular, that there are continuously differentiable functions of three variables which can not be written as a superposition of continuously differentiable functions of two variables. The project also aims to investigate just how smooth one can take the functions arising in Kolmogorov's theorem to be. Questions along these lines will be addressed through combinatorical analysis of Vitushkin's work, the topology of critical points, and approximation theory in function spaces."

## Related information

## Coordinator

THE UNIVERSITY OF BIRMINGHAM

United Kingdom

**EU contribution: **EUR 209 033,4

Edgbaston

B15 2TT BIRMINGHAM

United Kingdom

**Record Number**: 102944 / **Last updated on**: 2015-03-11
**Last updated on** 2015-03-11