January 14, 2022

### Looking at hydrodynamics through a contact mirror: From Euler to Turing and beyond

Eva Miranda

January 14, 2022 from 11:00 to 12:00 (Montreal/EST time) Zoom meeting

Colloquium presented by **Eva Miranda (Polytechnic University of Catalonia, Spain)**

What physical systems can be non-computational? (Roger Penrose, 1989). Is hydrodynamics capable of calculations? (Cris Moore, 1991). Can a mechanical system (including the trajectory of a fluid) simulate a universal Turing machine? (Terence Tao, 2017).

The movement of an incompressible fluid without viscosity is governed by Euler equations. Its viscid analogue is given by the Navier-Stokes equations whose regularity is one of the open problems in the list of problems for the Millenium by

the Clay Foundation. The trajectories of a fluid are complex. Can we measure its levels of complexity (computational, logical and dynamical)?

In this talk, we will address these questions. In particular, we will show how to construct a 3-dimensional Euler flow which is Turing complete. Undecidability of fluid paths is then a consequence of the classical undecidability of the halting

problem proved by Alan Turing back in 1936. This is another manifestation of complexity in hydrodynamics which is very different from the theory of chaos.

Our solution of Euler equations corresponds to a stationary solution or Beltrami field. To address this problem, we will use a mirror [5] reflecting Beltrami fields as Reeb vector fields of a contact

structure. Thus, our solutions import techniques from geometry to solve a problem in fluid dynamics. But how general are Euler flows? Can we represent any dynamics as an Euler flow? We will address this universality problem using the Beltrami/Reeb mirror again and Gromov's h-principle. We will also consider the non-stationary case. These universality features illustrate the complexity of Euler flows. However, this construction is not "physical" in the sense that the associated metric is not the euclidean metric. We will announce an euclidean construction and its implications to complexity and undecidability.

These constructions [1,2,3,4] are motivated by Tao's approach to the problem of Navier-Stokes [7,8,9] which we will also explain.

[1] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. Universality of Euler flows and flexibility of Reeb

embeddings. https://arxiv.org/abs/1911.01963.

[2] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. Constructing Turing complete Euler flows in

dimension 3. Proc. Natl. Acad. Sci. 118 (2021) e2026818118.

[3] R. Cardona, E. Miranda, D. Peralta-Salas. Turing universality of the incompressible Euler equations

and a conjecture of Moore. Int. Math. Res. Notices, , 2021;, rnab233,

https://doi.org/10.1093/imrn/rnab233

[4] R. Cardona, E. Miranda, D. Peralta-Salas. Computability and Beltrami fields in Euclidean space.

https://arxiv.org/abs/2111.03559

[5] J. Etnyre, R. Ghrist. Contact topology and hydrodynamics I. Beltrami fields and the Seifert conjecture.

Nonlinearity 13 (2000) 441–458.

[6] C. Moore. Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity

4 (1991) 199–230.

[7] T. Tao. On the universality of potential well dynamics. Dyn. PDE 14 (2017) 219–238.

[8] T. Tao. On the universality of the incompressible Euler equation on compact manifolds. Discrete

Cont. Dyn. Sys. A 38 (2018) 1553–1565.

[9] T. Tao. Searching for singularities in the Navier-Stokes equations. Nature Rev. Phys. 1 (2019) 418–419.

**Address**