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Renormalized potential energies and their asymptotics

Wednesday 5th December 2012 - 11:00 to 11:40
INI Seminar Room 1
Session Title: 
Knots in mathematics: Knot energies
Session Chair: 
Jason Cantarella
Energy of a knot was originally defined as the integration of the renormalized potential of a certain kind. Here, the renormalization can be done as follows: Suppose we are interested in a singular integral $\int_\Omega\omega$, which blows up on a subset $X\subset\Omega$. Remove an $\delta$-tubular neighbourhood of $X$ from $\Omega$, consider the integral over the complement, expand it in a Laurent series of $\delta$, and take the constant term. This idea gave rise to a M\"obius invariant surface energy in the sense of Auckly and Sadun, and recently, to generalization of Riesz potential of compact domains. If we integrate this generalized Riesz potential over the domain, we may need another renormalization around the boundary, according to the order of the generalized Riesz potential.

In this talk I will give "baby cases" of the application of the above story to the study of knots or surfaces.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons