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Geometric function theory and vortex motion: the role of connections

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CATW01 - The complex analysis toolbox: new techniques and perspectives

We discuss point vortex dynamics on a closed two-dimensional Riemann manifolds from the point of view of affine and other connections. The speed of a vortex then comes out as the difference between two affine connections, one derived from the coordinate Robin function and the other being the Levi-Civita connection associated to the Riemannian metric.  

In a  Hamiltonian formulation of the vortex dynamics, the Hamiltonian function consists of two main terms. One of them is a quadratic form based on a matrix whose entries are Green and Robin functions, while the other describes the energy contribution from those circulating flows besides those which are implicit in the Green functions. These two terms are not independent of each other, and one major issue is trying to understand the exchange of energy between them.

This talk is part of the Isaac Newton Institute Seminar Series series.

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