IMPA - O Instituto de Matemática Pura e Aplicada

Vamos apresentar um panorama das contribuições de Bill Meeks (1947-2026) para a teoria das superfícies mínimas e de curvatura média constante, prestando assim uma homenagem a um geômetra que teve um papel de destaque na Geometria Diferencial no Brasil e no mundo. 

Despite the intrinsic complexity of turbulent flows, large-scale coherent structures are present in turbulence, and are related to quantities of interest, such as aerodynamic drag and sound radiation. This presentation shows methods to model such coherent structures, and approaches to control flows in order to reduce drag or noise radiation. Linear models will be considered first, with coherent structures modelled as dominant responses from the linearised Navier-Stokes system. Such dominant modes compare favourably with coherent structures educed from experimental or numerical data, and allow the proposition of changes to the system aiming at drag or noise reduction. Non-linear reduced-order models (ROMs) are then obtained by the Galerkin projection of the full governing equations onto modes obtained from the linearised system. Dynamical systems so obtained typically have dimensions of O(10) to O(10^3) degrees of freedom, and quantitative agreement with reference statistics is obtained for a number of canonical flow configurations. State estimation from a few sensors is explored using an extended Kálmán filter coupled with ROMs, reproducing accurately numerical results. Finally, a ROM for turbulent Couette flow is used to devise a control method aiming at turbulence suppression. The obtained strategy is seen to relaminarise turbulence in direct numerical simulations, showing that ROMs so obtained have promising applications in optimisation and control.

Given a compact Lie group, we consider its Lie-Poisson sphere foliated by coadjoint orbits. We explore equivariant symbol correspondences on leaves given by rescaling of integral orbits, called rational orbits, which are dense in the sphere. We'll show how the fuzzy structures induced by symbol correspondences on rational orbits present a suitable geometric notion of semiclassical limit with a quite simple criterion for the asymptotic emergence of the Poisson algebra of polynomials. Then, we "glue" fuzzy rational orbits to obtain a quantum version of the sphere, a Magoo sphere, and we ask what can be said about the asymptotic behavior of such structure

O toro de Clifford em uma esfera de Berger com parâmetro $\tau$ é um ponto crítico do funcional de Willmore para todo $\tau \>0$, gerando um caminho suave de superfícies de Willmore. Ao estimar o índice de Morse ao longo desse caminho, aplicamos a teoria da bifurcação para gerar novos toros de Willmore simétricos que surgem do toro de Clifford.

Scientific Machine Learning is an emerging field with significant impact on methods for solving problems in science and engineering, particularly for large-scale coupled fluid-flow and transport problems. Numerical simulations for these problems can be costly, making such approaches valuable for understanding and improving the efficiency of state quantification and prediction. This talk will review recent advances in scientific machine learning, including dynamic mode decomposition, manifold learning, graph neural networks, and neural operators (Fourier Neural Operators, DeepOnet, and hybrid schemes), as applied to coupled fluid-flow and transport problems. These problems are of interest in sustainable resource exploration, geophysics, and various industrial applications. The talk will show how data-driven information can improve the efficiency of numerical simulation software, explore parametric manifolds for unseen scenarios, and reconstruct high-dimensional simulations using lower-dimensional structures in feasible time.