Abstract:
Recently analytic and geometric methods have been been found useful in knot theory. The idea is that for any smooth imbedded curve $c$ in $R^3$ to define an energy functional and look at the curve in the knot class that minimizes the energy. However the natural question of what is the curve that minimizes the energy over all curves has been open. Freedman, He, and Wang have made the natural conjecture that for energies on unit speed curve of length $L$ of the form
E(c) = \int_0^L\int_0^L (|c(s)-c(t)|^{-a} - D(c(s),c(t))^{-a}) ds dt
(where $D(c(s),c(t))$ is the distance on the curve between $c(s)$ and $c(t)$) that the planar circles of radius $L(2\pi)$ are the minimizers and they prove this for $a=2$. In joint work with Mohammad Ghomi we use the elementary theory of Fourier series to show that the Freedman-He-Wang conjecture is true for an even larger class of energies. This talk should be accessible to graduate students that know what a Fourier series is.