BEGIN:VCALENDAR VERSION:2.0 PRODID:-//132.216.98.100//NONSGML kigkonsult.se iCalcreator 2.20.4// BEGIN:VEVENT UID:20260911T165448EDT-6982PUsp8I@132.216.98.100 DTSTAMP:20260911T205448Z DESCRIPTION:TITLE : Looking at hydrodynamics through a contact mirror: From Euler to Turing and beyond\n\nPLACE : ZOOM\n\nhttps://umontreal.zoom.us/j /93983313215?pwd=clB6cUNsSjAvRmFMME1PblhkTUtsQT09\n ID de réunion : 939 833 1 3215\n Code secret : 096952\n \n RESUME / ABSTRACT :\n What physical systems can be non-computational? (Roger Penrose\, 1989). Is hydrodynamics capabl e of calculations? (Cris Moore\, 1991). Can a mechanical system (including the trajectory of a fluid) simulate a universal Turing machine? (Terence Tao\, 2017).\n\nThe 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\n\nthe Clay Foundation. The trajectories of a fluid are complex. Can we measure its levels of complexity (computat ional\, logical and dynamical)?\n\nIn this talk\, we will address these qu estions. In particular\, we will show how to construct a 3-dimensional Eul er flow which is Turing complete. Undecidability of fluid paths is then a consequence of the classical undecidability of the halting\n\nproblem prov ed by Alan Turing back in 1936. This is another manifestation of complexit y in hydrodynamics which is very different from the theory of chaos.\n\nOu r solution of Euler equations corresponds to a stationary solution or Belt rami field. To address this problem\, we will use a mirror [5] reflecting Beltrami fields as Reeb vector fields of a contact\n\nstructure. Thus\, ou r solutions import techniques from geometry to solve a problem in fluid dy namics. But how general are Euler flows? Can we represent any dynamics as an Euler flow? We will address this universality problem using the Beltram i/Reeb mirror again and Gromov's h-principle. We will also consider the no n-stationary case. These universality features illustrate the complexity o f Euler flows. However\, this construction is not 'physical' in the sense that the associated metric is not the euclidean metric. We will announce a n euclidean construction and its implications to complexity and undecidabi lity.\n\nThese 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.\n\n[ 1] R. Cardona\, E. Miranda\, D. Peralta-Salas\, F. Presas. Universality of Euler flows and flexibility of Reeb\n\nembeddings. https://arxiv.org/abs/ 1911.01963.\n\n[2] R. Cardona\, E. Miranda\, D. Peralta-Salas\, F. Presas. Constructing Turing complete Euler flows in\n\ndimension 3. Proc. Natl. A cad. Sci. 118 (2021) e2026818118.\n\n[3] R. Cardona\, E. Miranda\, D. Pera lta-Salas. Turing universality of the incompressible Euler equations\n\nan d a conjecture of Moore. Int. Math. Res. Notices\, \, 2021\;\, rnab233\,\n \nhttps://doi.org/10.1093/imrn/rnab233\n\n[4] R. Cardona\, E. Miranda\, D. Peralta-Salas. Computability and Beltrami fields in Euclidean space.\n\nh ttps://arxiv.org/abs/2111.03559\n\n[5] J. Etnyre\, R. Ghrist. Contact topo logy and hydrodynamics I. Beltrami fields and the Seifert conjecture.\n\nN onlinearity 13 (2000) 441–458.\n\n[6] C. Moore. Generalized shifts: unpred ictability and undecidability in dynamical systems. Nonlinearity\n\n4 (199 1) 199–230.\n\n[7] T. Tao. On the universality of potential well dynamics. Dyn. PDE 14 (2017) 219–238.\n\n[8] T. Tao. On the universality of the inc ompressible Euler equation on compact manifolds. Discrete\n\nCont. Dyn. Sy s. A 38 (2018) 1553–1565.\n\n[9] T. Tao. Searching for singularities in th e Navier-Stokes equations. Nature Rev. Phys. 1 (2019) 418–419.\n DTSTART:20220114T160000Z DTEND:20220114T170000Z SUMMARY:Eva Miranda (Polytechnic University of Catalonia\, Spain) URL:https://www.mcgill.ca/mathstat/channels/event/eva-miranda-polytechnic-u niversity-catalonia-spain-336082 END:VEVENT END:VCALENDAR