Upcoming: With my ICREA Academia 2021 award I plan to open several INIREC grants (initiation to research) for undergraduate and master students financed with my award. The research will be done related to one of the three pillars of my ICREA project.

Eligible research topics:

My ICREA Academia project has 3 axes related to 3 long-standing conjectures in mathematics: the (singular) Weinstein conjecture, the [Q,R]=0 conjecture for Poisson manifolds, and the Navier-Stokes problem (on Clay's Millennium list).

1- The Singular Weinstein conjecture: The existence of periodic orbits on a given energy level set of a Hamiltonian system is a central question in symplectic geometry motivated by its applications to Celestial Mechanics (trajectories that return to the initial point). A special class of Hamiltonian flows are the Reeb flows in contact geometry. The Weinstein conjecture asserts that on a compact manifold any such flow has a periodic orbit. Although proved in many cases, the general question remains widely open. We have formulated the singular version of the Weinstein conjecture (MirandaOms, Adv. Math 21) to include singular orbits motivated by the escape trajectories in astrodynamics. This conjecture is also related to the existence of singularities of the n-body problem (first problem in the list by Barry Simon for the new century).

Poincaré already envisaged the existence of infinite periodic orbits of the restricted 3-body problem accumulating to infinity. The trajectories of some Euler flows (Beltrami fields) on manifolds with boundary show a similar behavior. Both situations have an underlying singular structure (b-symplectic or b-contact). This project opens the door to b-Floer theory and b-contact homology which I plan to investigate further.

2- The quantization of Poisson manifolds and [Q,R]=0: Finding a model for quantization of Poisson manifolds is pending since Weinstein and Konsevich (Fields) and the conjecture that quantization commutes with reduction (Guillemin-Sternberg) remains terra incognita in this scenario. We have given models of quantization for b-Poisson manifolds that satisfy [Q,R]=0 (Guillemin-Miranda-Weitsman, Adv. Math 18). We plan to extend the theory to more general Poisson manifolds.

3- The Navier-Stokes conjecture: aims at proving regularity of the solutions of NS equations. In 2017 Tao proposed a new approach to find potential counterexamples to the NS conjecture based on the idea that sufficiently complicated initial solutions might lead to blow-up. Motivated by this, we have constructed Turing complete Euler flows and proved universality. In our constructions (Cardona-Miranda-Peralta Salas-Presas PNAS, and Cardona-Miranda-Peralta Salas, IMRN) the metric is not prescribed so we can import tecniques from contact geometry as the h-principle. In out recent article (Cardona-Miranda-Peralta Salas, JMPA) we have constructed a Turing complete Euler flow on the Euclidean space and investigate the relation between computational complexity and dynamical complexity. Such examples should serve as toy model for Tao's program.

About the project: The ICREA Acadèmia programme started in 2008 and offers research intensification grants to outstanding university professors who already hold permanent positions in the Catalan research system and are in an expanding phase of their careers. The grant is for five years and is intended to promote the awardees’ research by relieving them from teaching duties.

More information about ICREA academia in this link.

More information about this ICREA grant on the media: https://www.crm.cat/eva-miranda-full-professor-at-upc-and-crm-researcher-has-been-awarded-an-icrea-academia-prize-2021/

Principal Researcher: Eva Miranda at UPC.

How to apply? Send me an email with your cv and a motivation letter.