BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//132.216.177.160//NONSGML kigkonsult.se iCalcreator 2.18//
BEGIN:VEVENT
UID:20220516T110848EDT-2809cHr7Ve@132.216.177.160
DTSTAMP:20220516T150848Z
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 RESUME / ABSTRACT :\n What physical systems c
an be non-computational? (Roger Penrose\, 1989). Is hydrodynamics capable
of calculations? (Cris Moore\, 1991). Can a mechanical system (including t
he trajectory of a fluid) simulate a universal Turing machine? (Terence Ta
o\, 2017).\n\nThe movement of an incompressible fluid without viscosity is
governed by Euler equations. Its viscid analogue is given by the Navier-S
tokes equations whose regularity is one of the open problems in the list o
f problems for the Millenium by\n\nthe Clay Foundation. The trajectories o
f a fluid are complex. Can we measure its levels of complexity (computatio
nal\, logical and dynamical)?\n\nIn this talk\, we will address these ques
tions. In particular\, we will show how to construct a 3-dimensional Euler
flow which is Turing complete. Undecidability of fluid paths is then a co
nsequence of the classical undecidability of the halting\n\nproblem proved
by Alan Turing back in 1936. This is another manifestation of complexity
in hydrodynamics which is very different from the theory of chaos.\n\nOur
solution of Euler equations corresponds to a stationary solution or Beltra
mi field. To address this problem\, we will use a mirror [5] reflecting Be
ltrami fields as Reeb vector fields of a contact\n\nstructure. Thus\, our
solutions import techniques from geometry to solve a problem in fluid dyna
mics. 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 th
at the associated metric is not the euclidean metric. We will announce an
euclidean construction and its implications to complexity and undecidabili
ty.\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 E
uler flows and flexibility of Reeb\n\nembeddings. https://arxiv.org/abs/19
11.01963.\n\n[2] R. Cardona\, E. Miranda\, D. Peralta-Salas\, F. Presas. C
onstructing Turing complete Euler flows in\n\ndimension 3. Proc. Natl. Aca
d. Sci. 118 (2021) e2026818118.\n\n[3] R. Cardona\, E. Miranda\, D. Peralt
a-Salas. Turing universality of the incompressible Euler equations\n\nand
a conjecture of Moore. Int. Math. Res. Notices\, \, 2021\;\, rnab233\,\n\n
https://doi.org/10.1093/imrn/rnab233\n\n[4] R. Cardona\, E. Miranda\, D. P
eralta-Salas. Computability and Beltrami fields in Euclidean space.\n\nhtt
ps://arxiv.org/abs/2111.03559\n\n[5] J. Etnyre\, R. Ghrist. Contact topolo
gy and hydrodynamics I. Beltrami fields and the Seifert conjecture.\n\nNon
linearity 13 (2000) 441–458.\n\n[6] C. Moore. Generalized shifts: unpredic
tability and undecidability in dynamical systems. Nonlinearity\n\n4 (1991)
199–230.\n\n[7] T. Tao. On the universality of potential well dynamics. D
yn. PDE 14 (2017) 219–238.\n\n[8] T. Tao. On the universality of the incom
pressible Euler equation on compact manifolds. Discrete\n\nCont. Dyn. Sys.
A 38 (2018) 1553–1565.\n\n[9] T. Tao. Searching for singularities in the
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