Coherent Structures and Lattice-Boltzmann Hydrodynamics in Turbulent Pipe Flows
Applied and Computational MathematicsAbstract: 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.
Seminario de alunos (Riemann-Hilbert correspondence)
AlgebraAbstract: Durante esse semestre teremos um seminário de alunos tentando cobrir a correspondência de Riemann-Hilbert. As palestras acontecerão segundas às 15:00 e quintas às 14:30 na sala 232. Nessa quinta 25/04, estaremos fazendo um review de D-modulos seguindo o livro de Hotta et. al. "D-modules, perverse sheaves, and representation theory"
The Mandelbrot set and its satellite copies
Ergodic TheoryAbstract: For a polynomial on the Riemann sphere, infinity is a (super) attracting fixed point, and the filled Julia set is the set of points with bounded orbit. Consider the quadratic family P_c(z)=z^2+c. The Mandelbrot set M is the set of parameters c such that the filled Julia set of P_c is connected. Douady and Hubbard proved the existence of homeomorphic copies of M inside of M, which can be primitive (roughly speaking the ones with a cusp) or a satellite (without a cusp). Lyubich proved that the primitive copies of M are quasiconformally homeomorphic to M, and that the satellite ones are quasiconformally homeomorphic to M outside any small neighbourhood of the root. The satellite copies are not quasiconformally homeomorphic to M (as we cannot straighten a cusp quasiconformally), but are they mutually quasiconformally homeomorphic? In a joint work with C. Petersen we prove that this question has in general a negative answer, but positive in the case the satellite copies have rotation number with same denominator.
On linear convergence of iterative methods for the variational inequality problem
OptimizationAbstract: In this review talk, we will discuss Paul Tseng's classical paper on linear convergence of some algorithms for variational inequalities (VIs). Namely, we will introduce extragradient, proximal point, and matrix splitting methods, and we intend to reiterate some results and proofs from the original paper. This theme is of great importance due to the wide applicability of VIs in optimization and equilibrium problems.
Degenerate twistor deformations of holomorphically symplectic manifolds
Geometric structures on manifoldsAbstract: Let $\pi:\; M\to X$ be a Lagrangian fibration on a holomorphically symplectic manifold. Adding a closed (1,1)-form to the holomorphic symplectic form $\Omega$, we obtain a holomorphic symplectic form on a complex deformation of $(M, \Omega)$. This deformation is called "degenerate twistor deformation". In the hyperkahler case the degenerate twistor deformation can be obtained as the limit of twistor deformations. A limit of a Kahler manifolds is not necessarily Kahler, and for many years it was unknown whether the degenerate twistor deformation of a compact hyperkahler manifold is Kahler. I will explain the construction of the degenerate twistor deformation and prove that it is Kahler, using the Harvey-Lawson duality criterion of Kahlerness. This is a joint work with Andrey Soldatenkov.
Geometric planted matchings beyond the Gaussian model
Graduate Students’ ColloquiumAbstract: We consider the problem of recovering an unknown matching between a set of $n$ randomly placed points in $\mathbb{R}^d$ and random perturbations of these points. This can be seen as a model for particle tracking and more generally, entity resolution. We use matchings in random geometric graphs to derive minimax lower bounds for this problem that hold under great generality. Using these results we show that for a broad class of distributions, the order of the number of mistakes made by an estimator that minimizes the sum of squared Euclidean distances is minimax optimal when $d$ is fixed and is optimal up to $n^{o(1)}$ factors when $d=o(log(n))$. Joint work with Roberto I. Oliveira.
The coarse distance from dynamically convex to convex
Differential GeometryAbstract: 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.
Distribuições holomorfas com feixes tangentes totalmente decomponíveis
Holomorphic FoliationsAbstract: 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.
Neural Conjugate Flows: Physics-Informed Architectures with Differentiable Flow Structure
Computer GraphicsAbstract: We present Neural Conjugate Flows, a novel design for physics-informed neural networks with flow structure. We prove that this architecture is an universal approximator for solutions of differential equations and demonstrate how its group and topological lead to computational and theoretical gains when simulating dynamical systems with neural networks.
Data Science and Innovation
Special TalkAbstract: Data science and innovation have become overloaded terms, leading to some confusion. To be successful, the innovation process involves not only inventions (e.g., new methods) but also context, e.g., user behavior and timing, e.g., market readiness. In this talk, I discuss the impact of data science on innovation, using selected success stories (some of which I was involved in). I also give hints to promote innovation within companies, in particular, using open innovation. Finally, I describe some innovations in the context of the Inria-Brasil partnership.
Category Theory applied to Data Visualization
Computer GraphicsAbstract: Category Theory (CT) is a branch of mathematics that studies general abstract structures through their relationships, and it is unmatched in its ability to organize and relate abstractions. In recent years, CT has found applications in a wide range of disciplines, such as chemistry, biology, natural language processing, and database theory. We present a novel application by formalizing Data Visualization within Category Theory. This formalization creates a bridge between Mathematics and Data Visualizations. Moreover, it provides a framework to express complex visualizations, which can be implemented computationally by leveraging the well-established connection between CT and Functional Programming.