A few years ago -- following a suggestion by I. M. Gelfand-- I discovered an intriguing connection between the topological degree of a map from the circle into itself and its Fourier coefficients. This relation is easily
justified when the map is smooth. However, the situation turns out to be much more delicate if one assumes only continuity, or even Holder continuity.
I will present recent developments and open problems.
I will also discuss new estimates for the degree of maps from S^n into S^n, leading to unusual characterizations of Sobolev spaces.
The initial motivation for this direction of research came from the analysis of the Ginzburg-Landau model.