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Diffeomorphisms of unit circle, shape analysis and some non-linear PDEs

Presented by: 
Irina Markina Universitetet i Bergen
Date: 
Thursday 10th October 2019 - 16:00 to 17:00
Venue: 
INI Seminar Room 2
Abstract: 
In the talk, we explain how univalent functions can be used to 
analyze plain shapes. In its turn, the univalent functions defined on 
the unit disc are closely related to the group of oriented preserving 
diffeomorphisms of the unit circle. A moving plain shape gives rise to a 
curve on the group of diffeomorphisms. The requirement to describe a 
shape modulo its rotation and/or scaling leads to a curve subordinated 
to some constraints. A geodesic curve of the motion of a shape is a 
solution to some non-linear partial differential equation. The choice of 
metric leads to different PDEs, that are generalizations of equations 
originated in fluid dynamics, such us inviscid Burgers' equation, 
Camassa-Holm, Hunter-Saxton, and KdV.





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