GDMSeminar: Abstracts of the Talks (2021/11)
Jair Koiller ( Universidade Federal de Juiz de Fora, Brazil)
Vortex pairs on surfaces: can it be a tool for topology in the large?
Tuesday, May 25, 2021 , 15:15 GMT
Abstract:
In 1999 Yoshifumi Kimura mentioned in his paper 'Vortex motion on surfaces with constant curvature'(//doi.org/10.1098/rspa.1999.0311) that a vortex dipole (two infinitesimally close opposite vortices) on a curved surface should move along a geodesic: "curvature checker", as he interestingly defined.
A proof outline was given in 2008 by Stefanella Boatto and JK (arXiv:0802.4313, //link.springer.com/book/10.1007/978-1-4939-2441-7). In this talk I present some results of ongoing work with Umberto Hryniewicz, Alejandro Cabrera and Anilatmaja Aryasomayajula. Regarding vortex pairs at a small finite distance, we show that close-by pairs can actually be called "topology checkers".
In fact we suggest the idea that, very much like geodesics (perhaps only more so), the study of vortex pair dynamics could be a good way to probe the topology in the large. This is because the Hamiltonian for vortex dynamics on surfaces involves the Laplace Beltrami operator Green's function and its regularizations (Robin's function and its partner, Batman).
Time permitting I will briefly review joint work with Adriano R. Rodrigues and Cesar Castilho (//doi.org/10.1063/1.3146241, doi: 10.3934/jgm.2018007, http://mi.mathnet.ru/eng/rcd389) on far-away vortices on a surface with antipodal symmetry. I will also advertise work by Clodoaldo Ragazzo and Humberto Viglioni (//doi.org/10.1098/rspa.2017.0447, //link.springer.com/article/10.1007/s00332-017-9380-7) on a the motion of a single vortex.
During the talk we will make some queries for the audience, and research suggestions will be presented in the end.