Universitat Politècnica de Catalunya and Centre de Recerca Matemàtica
Title: From Alan Turing to Contact geometry: towards a "Fluid computer”
Abstract: Abstract: 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 in the positive the questions raised by Moore and Tao. This is a recent result that mixes up techniques from Alan Turing with modern Geometry (contact geometry) to construct a "Fluid computer" in dimension 3. This construction shows, in particular, the existence of undecidable fluid paths. I will also explain applications of this mirror to the detection of escape trajectories in Celestial
mechanics (for which I'll need to extend the mirror to a singular set-up). This mirror allows us to construct a tunnel connecting problems in Celestial mechanics and Fluid Dynamics.
Robert Cardona, Eva Miranda, Daniel Peralta-Salas, and Francisco Presas, Constructing Turing complete Euler flows in dimension 3. Proc. Natl. Acad. Sci. USA 118 (2021), no. 19, Paper No. e2026818118, 9 pp.
- Etnyre, R. Ghrist, Contact topology and hydrodynamics: I. Beltrami fields and the Seifert conjecture. Nonlinearity 13, 441 (2000).
Eva Miranda, Cédric Oms and Daniel Peralta-Salas, On the singular Weinstein conjecture and the existence of escape orbits for b-Beltrami fields. Commun. Contemp. Math. 24 (2022), no. 7, Paper No. 2150076, 25 pp.
- Tao, Finite time blowup for an averaged three-dimensional Navier–Stokes equation. J. Am. Math. Soc. 29, 601–674 (2016).
Alan Turing, On Computable Numbers, with an Application to the Entscheidungsproblem. Proceedings of the London Mathematical Society. Wiley. s2-42 (1): 230–265. doi:10.1112/plms/s2-42.1.230. ISSN 0024-6115., (1937).