Eva Miranda will speak in the geometry seminar at 2pm on 15/12/2022. Eva’s title is “Two sides of a mirror” and her abstract is below.
Is hydrodynamics capable of performing computations? (Moore 1991). Can a mechanical system (including a fluid flow) simulate a universal Turing machine? (Tao, 2016).
Etnyre and Ghrist unveiled a mirror between contact geometry and fluid dynamics reflecting Reeb vector fields as Beltrami vector fields. With the aid of this mirror, we can answer affirmatively the questions raised by Moore and Tao in a geometrical manner by combining techniques from Alan Turing with contact geometry to construct a “Fluid computer” in dimension 3. This construction shows, in particular, the existence of undecidable fluid paths.
Furthermore, the application of the h-principle to the contact side of the mirror discloses the universal nature of Euler flows on the other side: Any dynamical system can be presented as an Euler flow (indeed as a Beltrami field) in higher dimensions. As contact geometry provides a natural language for many problems in Celestial mechanics, this mirror allows us to construct a “tunnel” connecting problems in Celestial mechanics to Fluid Dynamics.
Time permitting, I will also briefly explain applications of this mirror to the detection of escape trajectories in Celestial mechanics.