Speaker: Mohammad Ghomi (University of South Carolina)
Title: Knot energies, unfoldings, and distortion
Abstract: Given a simple closed curve (a knot) C in Euclidean space, one
may measure the distance between pairs of points p and q of C in two
ways: the intrinsic distance d(p,q) measured along the curve, and the
extrinsic distance |p-q| measured in the ambient space. Comparing
these two notions of distance provides a way to study the geometry and
topology of C.
In particular, there is Gromov's distortion, which is the supremum of
the ratio d(p,q)/|p-q|, and the family of the knot energies defined by
O'Hara which may be regarded as weak forms of distortion. In recent
joint work with A. Abrams, J. Cantarella, J. Fu, and Ralph Howard, the
speaker has proved that these energies are uniquely minimized by the
round circle, as had been conjectured by Freedman, He, and Wang. We
discuss an outline of this proof, which is in part based on Fourier
series. Further, we mention ongoing work with Ralph Howard on
"unfoldings", an operation which preserves the intrinsic distances of a
space curve, while increasing the extrinsic distances.
Finally we mention some open problems and related questions. It is not
yet known for instance whether there exists in each knot class a
representative whose distortion is bounded above by some universal
constant.