The project

Research topic: The entry door of our project is motivated by the work of Tao to find a counterexample to the Navier-Stokes conjecture, one of the open problems in the millennium list of the Clay foundation. In the recent exploration of some of the facets of Tao’s program, we have unveiled deep connections to unexplored pathways in hydrodynamics. In particular, we showed the existence of undecidable fluid paths in hydrodynamics (Cardona, Miranda, Peralta-Salas, Presas, PNAS, 2021; Cardona, Miranda, Peralta-Salas, Journal de Mathématiques Pures et Appliquées, 2022) and chaos in the incompressible Euler equation on manifolds of high dimension (Torres de Lizaur, Inventiones Mathematicae, 2022). This investigation took us to cumbersome labyrinths connected to the mathematical foundations of computer science and symbolic dynamics. Our research project focuses on the interplay of pure mathematics and theoretical aspects of computer science. The investigation of the several levels of complexity (topological, dynamical, computational or logical) will play a central role in the project. The project unfolds in three working packages: on embeddings, on computational and logical complexity and on dynamical complexity of fluids. The resolution of the Navier-Stokes conjecture in the Clay list is a breakthrough in mathematics. Even if the initial motivation for this project is to disentangle the Navier-Stokes enigma a priori it is not clear that Tao’s approach can succeed; however understanding the rich connection to computational mathematics and diverse areas of pure mathematics is an extraordinary accomplishment in mathematics and at the frontier of science.