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The coarse distance from dynamically convex to convex

Speaker: Vinicius Ramos - IMPA
Tue 16 Apr 2024, 15:30 - SALA 236Differential Geometry

Abstract: The Viterbo conjecture is one of the main open problems in symplectic topology. In its weakest form, it is a systolic type inequality relating the length of closed characteristics with the symplectic volume of a convex set. One of the difficulties of proving this conjecture is that convexity is not a symplectic invariant. Hofer-Wysocki-Zehnder found a symplectic condition that they named dynamical convexity that is satisfied by all convex sets and that is invariant by symplectomorphisms. It was an open question whether this notion was equivalent to convexity up to a symplectomorphism. In this talk, I will explain why these notions are not only different, but how we can find 4-dimensional dynamically convex domains that are arbitrarily far from convex domains in an appropriate distance.

Models for the Euclidean Steiner Tree Problem in n−Space n ≥ 3

Speaker: Nelson Maculan - COPPE UFRJ
Tue 16 Apr 2024, 17:00 - SALA 224Optimization

Abstract: We review the mathematical optimization models for the Euclidean Steiner Tree Problem (ESTP) in n dimensions proposed in the literature. The development of such models for the ESTP began in the late 1990s. The ESTP is a mixed integer nonlinear optimization problem with a history dating back to the 17th century. Several properties of its optimal solutions are well known, but it is still a big challenge to encode these properties in its modeling, aiming for its numerical resolution with branch-and-bound algorithms.

A vertex-centric representation for adaptive diamond-kite meshes

Speaker: Luiz Henrique de Figueiredo - IMPA
Wed 17 Apr 2024, 10:30 - AUDITORIO 3Computer Graphics

Abstract: We describe a concise representation for adaptive diamond-kite meshes based solely on the vertices and their stars. The representation is exact because it uses only integers, is much smaller than standard topological data structures, and is highly compressible. All topological elements are reconstructed in expected constant time per element.

Distribuições holomorfas com feixes tangentes totalmente decomponíveis

Speaker: Raphael Constant da Costa - UERJ
Thu 18 Apr 2024, 15:30 - SALA 224Holomorphic Foliations

Abstract: Considere uma folheação holomorfa singular em um espaço projetivo complexo de dimensão maior ou igual a 3, com grau e dimensão da folheação determinados. No contexto da obtenção de algumas componentes irredutíveis dos espaços de tais folheações, é interessante em determinadas situações que uma pequena deformação de uma folheação com feixe tangente totalmente decomponível (split) tenha não somente feixe tangente localmente livre, mas que permaneça sendo totalmente decomponível. Surge então a pergunta: dada uma folheação qualquer com feixe tangente localmente livre, é verdade que também é totalmente decomponível? Recentemente, Daniele Faenzi, Marcos Jardim e Jean Vallès mostraram que, de maneira geral, a resposta à pergunta é negativa. No nosso trabalho, exploramos condições suficientes que garantam a decomponibilidade do feixe tangente quando ele é localmente livre. Mais especificamente, dada uma distribuição singular de dimensão 2 em um espaço projetivo complexo de dimensão maior ou igual a 3, fornecemos condições sobre uma subfolheação por curvas que garantam a existência de uma outra subfolheação por curvas e uma relação de decomponibilidade entre os respectivos feixes tangentes, tanto em caráter local quanto global. Como aplicação dos resultados obtidos, mostramos que se uma folheação de codimensão 1 em um espaço projetivo complexo de dimensão 3 possuir feixe tangente localmente livre e for tangente a um campo vetorial holomorfo não-trivial, então seu feixe tangente é totalmente decomponível. Alguns resultados de divisão de formas diferenciais holomorfas por campos holomorfos tangentes são também exibidos.

Grafos de isogenias de curvas elíticas

Speaker: Felipe Voloch - University of Canterbury
Wed 24 Apr 2024, 15:30 - SALA 228Algebra

Abstract: Os grafos de isogenias de curvas elíticas são grafos cujos vértices são curvas elíticas e as arestas são isogenias de grau fixo entre as curvas. O interesse nesses grafos vem de aplicações a criptografia e de questões computacionais. Vou fazer uma exposição sobre o assunto e, se houver tempo, apresentar novos resultados para grafos de isogenias com estrutura de nível.

Coherent Structures and Lattice-Boltzmann Hydrodynamics in Turbulent Pipe Flows

Speaker: Bruno Magacho da Silva - UFRJ
Thu 25 Apr 2024, 13:30 - SALA 349Applied and Computational Mathematics

Abstract: Coherent structures (CS) are known for being part of the foundations of turbulent flow dynamics. Their appearance was believed to be chaotic and unorganized for a long time. In the last two decades, however, it has been shown through numerical simulations and experiments that a high degree of organization of the CS could be assigned to the constitution of a turbulent state. Understanding these organizational dynamics is promising to bring valuable theoretical and applied predictions, such as the average lifetime of turbulent structures, the understanding of the role of CS in scale formation, and the development of fusion reactors. A statistical analysis of an experimental turbulent pipe flow database was carried out to investigate the aforementioned mechanisms, with the transition between the identified CS studied as a stochastic process, revealing a non-Markovian memory effect for the identified structures. In parallel, simulations with the Lattice Boltzmann Method (LBM) were performed to simulate the quasi-static regime in laminar magnetohydrodynamic flows and turbulent pipe flow. The investigation on the CS dynamics was also performed for the numerical data obtained with the LBM, revealing a non-trivial memory effect with the force that was used to trigger the turbulent state and a Markovian behavior for the finely time-resolved data, indicating that the experimental behavior could be recovered for larger datasets.