报告题目：Undecidability of certain fluid paths, the Navier-Stokes problem and 29000 rubber ducks lost in the ocean
主 讲 人：Prof. Dr. Eva Miranda
单 位：Universitat Politècnica de Catalunya and Paris Sciences et Lettres
ZOOM ID：567 306 5241
The movement of an incompressible fluid without viscosity is governed by Euler equations. Its viscid analogue is given by the Navier-Stokes equations.In 1992, 29000 rubber ducks were lost in the pacific ocean in a storm. They were travelling from Hong-Kong to Tacoma on a cargo ship. These rubber ducks have appeared in several parts of the planet many years after. The erratic trajectories of the rubber ducks (also known as friendly floatees) have been the object of several studies of currents in Oceanography (works of Curtis Ebbesmeyer and James Ingraham).In this talk, I will prove that certain Fluid paths are undecidable. This would explain why it is undecidable to know where the friendly floatees would wash up.In other words, a love message in a bottle on a 3-dimensional sea could maybe not reach its recipient.I will do this by constructing 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 Turing in 1936. Our solution of Euler equations corresponds to a stationary solution or Beltrami field. In order to solve this problem, we will take advantage of the existence of a mirror reflecting Beltrami fields as Reeb vector fields of a contact geometry. Thus, our solutions import techniques from Geometry to solve a problem in Fluid Dynamics. Our work is motivated by Tao's approach to the problem of Navier Stokes (one of the unsolved problems in the millenium list of the Clay institute) which I will also explain.
Eva Miranda is a Full professor at UPC-IMTech, member of CRM and chercheur afflilié at Observatoire de Paris. She is director of the Lab of Geometry and Dynamical Systems. Since 2018 she is member of the Governing Board of BGSMath and since 2020 she is member of the Board of Trustees at Institut Henri Poincaré (Paris).Her research is at the crossroads of Differential Geometry, Mathematical Physics and Dynamical Systems. She works with objects appearing on the interface of Geometry and Physics such as integrable systems and group actions acquainting for symmetries of the systems. She is particularly interested in building bridges between different areas such as Geometry, Dynamical Systems, Mathematical Physics and, more recently, Fluid Dynamics. She has published over 50 articles including Ann. Sci. Éc. Norm. Supér. (4), Adv. Math., PNAS, J. Math. Pures Appl. (9), and Comm. Math. Phys. She has supervised a total of 6 Ph.D. theses and is currently supervising 3 more. Eva Miranda has been awarded the ICREA Academia Prize in 2016. In 2017 she was awarded a Chaire d'Excellence of the Fondation Sciences Mathématiques de Paris. Miranda has been plenary speaker in the top workshops in her field and invited speaker at the 8th European Congress of Mathematicians.