Hölder continuity of the Lyapunov Exponents of random GL(d) cocycles
Graduate Students’ ColloquiumAbstract: Consider a probability measure $\nu$ in $GL(d)$ and a linear cocycle $\mathcal{F}:GL^\mathbb{N}(d)\times \mathbb{R}^d\rightarrow GL^\mathbb{N} (d)\times \mathbb{R}^d$ defined as $\mathcal{F}((g_j)_j,v)=((g_{j+1})_j,g_0 v)$. The cocycle associated with a random product of $2\times 2$ invertible matrices (i.e., $d=2$) under a probability distribution $\nu$ have two (possibly equal) Lyapunov exponents $\lambda _1\ge\lambda\ _2$. When $\lambda\ _1>\lambda\ _2$ we can prove that those two exponents are pointwise Hölder continuous with respect to the probability measure $\nu$. A natural question arises: is it true in any dimension? In this talk, we'll discuss what a stochastic dynamical system is, explain how we define a random $GL(d)$-cocycle and look into calculating Lyapunov exponents based on a given probability measure in this context. Additionally, we'll discuss about the modulus of continuity of those exponents considering any dimension $d$ and address the challenges posed by assuming $d > 2$. This is a work in progress with El Hadji Yaya Tall, Adriana Sánchez and Marcelo Viana.
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.
Grafos de isogenias de curvas elíticas
AlgebraAbstract: 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
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.