This is an introduction to the lectures and the tutorials. We will study different topics, differential geometric topic using analysis for the lectures and topological topics with numerical experiments for the tutorials. A survey for the knot energies will also be given.
Creator:
O'Hara, Jun (Chiba University)
Created:
2019-06-17
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
We begin with the generalization of electrostatic energy of charged knots, where we come across the difficulty of divergent integrals. Two kinds of regularization will be introduced, Hadamard regularization and the regularization via analytic continuation, both from the theory of generalized functions.
Creator:
O'Hara, Jun (Chiba University)
Created:
2019-06-18
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
We will see that Hadamard regularization and the regularization via analytic continuation give essentially the same information. This part is rather technical, although it will save complicated computation afterward.
Creator:
O'Hara, Jun (Chiba University)
Created:
2019-06-20
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
We apply the regularization via analytic continuation to generalized Riesz energies of submanifolds in Euclidean spaces to obtain Brylinski's beta function, which is a meromorphic function with simple poles. We study geometric information that can be derived from Brylinski's beta function.
Creator:
O'Hara, Jun (Chiba University)
Created:
2019-06-21
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
A method to construct M\"obius invariant weighted inner products on the tangent spaces of the knot space by using M\"obius invariant knot energies will be introduced. It gives M\"obius invariant gradients of such energies."
Creator:
O'Hara, Jun (Chiba University)
Created:
2019-06-25
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
