Seminários, para a disseminação informal de resultados de investigação, trabalho exploratório de equipas de investigação, actividades de difusão, etc., constituem a forma mais simples de encontros num centro de investigação de matemática.
O CAMGSD regista e publica o calendário dos seus seminários há bastante tempo, servindo páginas como esta não só como um método de anúncio dessas actividades mas também como um registo histórico.
We consider the Cauchy problem related to the family of $k$-dispersion generalized Benjamin-Ono ($k$-DGBO) equations: \begin{equation}\label{DGBOINTRO} \begin{cases} u_t + D_x^\alpha u_x + \mu u^ku_x= 0, \quad (t,x) \in \mathbb{R} \times \mathbb{R},\\ u(0,x)=u_0(x), \end{cases} \end{equation} where $u = u(t,x)$ is real-valued, $\alpha \in [1,2]$, $\mu \in \{\pm 1\}$ and $k \in \mathbb{Z}^+$. Here, ${D^\alpha_x}$ represents the 1-dimensional fractional Laplacian operator in the spatial variable $x$. For $k \geq 4$, we establish local and global well-posedness results for \eqref{DGBOINTRO} in both the critical $\left(s= \frac{k-2 \alpha}{2k}\right)$ and subcritical $\left(s > \frac{k-2 \alpha}{2k}\right)$ regimes, addressing sharp regularity in homogeneous and inhomogeneous Sobolev spaces. Additionally, our method enables the formulation of a scattering criterion and a scattering theory for small data. We also investigate the case $k = 3$ via frequency-restricted estimates, obtaining local well-posedness results for the initial value problem associated with the $3$-DGBO equation and generalizing the existing results in the literature for the whole subcritical range. For higher dispersion, these local results can be extended globally even for rough data, particularly for initial data in Sobolev spaces with negative indices. As a byproduct, we derive new nonlinear smoothing estimates. This is a joint work with Luccas Campos (UFMG) and Felipe Linares (IMPA).
In this talk I will survey some results on the existence of periodic points of symplectomorphisms defined on closed orientable surfaces of positive genus g. Namely, I will describe some symplectic flows on such surfaces possessing finitely many periodic points and describe a non-Hamiltonian variant of the Hofer-Zehnder conjecture for symplectomorphisms defined on surfaces; this conjecture provides a quantitative threshold on the number of fixed points (possibly counted homologically) which forces the existence of infinitely many periodic points. This is joint work in progress with Marcelo Atallah and Brayan Ferreira.
I will present the super volumes of the moduli space of super Riemann surfaces. They will be defined using a family of finite measures on the moduli space of genus $g$ curves. These measures are in turn given by a construction analogous to the classical construction of the Weil–Petersson metric, using the extra data of a spin structure. The total measure gives the volume of the moduli space of super curves and can be calculated via a deep relationship with the KdV equation.
