Seminários

Moser's lemma for holomorphically symplectic manifolds

Resumo: Moser's lemma states that any smooth family of symplectic structures with constant cohomology class is trivialized by a diffeomorphism.

I will give a holomorphically symplectic version of this statement. Let $(M, I_t, \Omega_t)$ be a smooth family of holomorphically symplectic manifolds, with the cohomology class $[\Omega_t]\in H^2(M,{\mathbb C})$ constant. Then this family defines a smooth family of cohomology classes $\eta_t \in H^{0,1}(M, I_t)$ such that whenever $\eta_t\equiv 0$, the family $(M, I_t, \Omega_t)$ is trivialized by a diffeomorphism. This gives a simple and elementary proof of injectivity of period map for deformations of holomorphically symplectic manifolds. I will use Moser lemma to prove several holomorphically symplectic versions of Weinstein tubular neighbourhood theorem.

Asymptotic Plateau problem for prescribed mean curvature hypersurfaces

Resumo: I will talk on a recent joint paper with Jean-Baptiste Casteras and Jaime Ripoll.

Let $N$ be an $n$-dimensional Cartan--Hadamard manifold that satisfies the so-called strict convexity condition and has strictly negative upper bound for sectional curvatures, $K\le-\alpha^2<0$. Given a suitable subset $L\subset\partial_\infty N$ of the asymptotic boundary of $N$ and a continuous function $H\colon N\to [-H_0,H_0],\ H_0<(n-1)\alpha$, we prove the existence of an open subset $Q\subset N$ of locally finite perimeter whose boundary $M$ has generalized mean curvature $H$ towards $N\setminus Q$ and $\partial_\infty M=L$. By regularity theory, $M$ is a $C^2$-smooth $(n-1)$-dimensional submanifold up to a closed singular set of Hausdorff dimension at most $n-8$. In particular, $M$ is $C^2$-smooth if $n\le 7$. Moreover, if $H\in [-H_0,H_0]$ is constant and $n\le 7$, there are at least two disjoint hypersurfaces $M_1$, $M_2$ with constant mean curvature $H$ and $\partial_\infty M_i=L,\ i=1,2$.

Our results generalize those of Alencar and Rosenberg, Tonegawa, and others.

Visual SLAM in Human Populated Environments: Exploring the Trade-off between Accuracy and Speed of YOLO and Mask R-CNN

Resumo: Simultaneous Localization and Mapping (SLAM) is a fundamental problem in mobile robotics. However, the majority of Visual SLAM algorithms assume a static scenario, limiting their applicability in real-world environments. Dealing with dynamic content in Visual SLAM is still an open problem, with solutions usually relying on direct or feature-based methods. Deep learning techniques can improve the SLAM solution in environments with a priori dynamic objects, providing high-level information of the scene. This paper presents a new approach to SLAM in human populated environments using deep learning-based techniques. The system is built on ORB-SLAM2, a state-of-the-art SLAM system. The proposed methodology is evaluated using a benchmark dataset, outperforming other Visual SLAM methods in highly dynamic scenarios.