Faculty Research Perspectives
Winding numbers and fourier series: an interesting but difficult marriage
Haim Brezis February 18, 3:30 PM in Hill 705
Abstract. A few years ago -- following a suggestion by I. M. Gelfand-- I discovered an intriguing formula connecting the winding number (also called index or 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 also discuss new estimates for the degree. There are still many intriguing problems. The initial motivation for this direction of research came from the analysis of the Ginzburg-Landau model in Physics.