Seminários

Stability of extremal domains for the first Dirichlet eigenvalue of the Laplacian operator

Resumo: In this talk, we will discuss the concept of stable extremal domains for the first Dirichlet eigenvalue of the Laplacian operator. We will classify the stable extremal domains in the 2-sphere and higher-dimensional spheres when the boundary is minimal. Additionally, we will establish topological bounds for stable domains in general compact Riemannian surfaces, assuming either nonnegative total Gaussian curvature or small volume. This is a joint work with Marcos P. Cavalcante (UFAL).

Differentiable Unsigned Distance Fields for Implicit Neural Representations

Resumo: In recent years, there has been a growing interest in training Neural Networks to approximate Unsigned Distance Fields (UDFs) for representing open surfaces in the context of 3D reconstruction. However, UDFs are non differentiable at the zero level set which leads to significant errors in distances and gradients, generally resulting in fragmented and discontinuous surfaces. We propose to learn a differentiable pseudo-distance distance field, which defines a new Eikonal problem with distinct boundary conditions. This allows our formulation to integrate seamlessly with state-of-the-art periodic neural networks, largely applied in the literature to represent signed distance fields. Moreover, the unlocked field’s differentiability allows the accurate computation of essential topological properties such as normal directions and curvatures, pervasive in downstream tasks like rendering and simulation among others.

Conjuntos de rotação em gênero maior ou igual a $2$ e ação no grafo fino de curvas

Resumo: Apos a definição do conjunto de rotação ergódico para homeomorfismos de superfícies fechadas, vou expor um teorema de decomposição para esses conjuntos no caso de superfícies de gênero maior ou igual a $2$. Esse enunciado sera acompanhado com um álbum panini de exemplos. Trabalho comum com A. Garcia e P. Lessa.

Twistor space of a compact hypercomplex manifold is never Moishezon

Resumo: Moishezon manifolds are compact, complex manifolds that admit many curves and divisors which can be used to study the geometry of the ambient manifold. Twistor spaces of compact hyperkahler manifolds are very far from being Moishezon. I am going to explain why the twistor space of a compact hypercomplex manifold is never Moishezon and neither Fujiki class C (in particular, never Kahler and projective). It is the work in progress.

On the construction of compact rotationally invariant Ricci surfaces

Resumo: The starting point of this talk is the intrinsic study of local minimal isometric immersions of a Riemannian surface ($\Sigma,d \sigma^2$) into a $3$-dimensional space form of curvature $c$. Ricci’s theorem, generalized by Lawson, states that such immersions exist if the Gaussian curvature $K$ of $d\sigma^2$ satisfies $K<c$ and the equation: $$(c-K)\Delta K + |\nabla K|^2 + 4K(c-K)^2 = 0$$

A Riemannian metric $d\sigma^2$ that satisfies this condition is called a Ricci metric of type $c$, making ($\Sigma, d\sigma^2$) a Ricci surface of type $c$.

Minimal surfaces in a $3$-dimensional space form of curvature c are among the first examples of such surfaces. In this talk, we explore Ricci metrics of type c with rotational invariance. We begin by presenting a classification for the case $c = 0$ and construct new examples of immersed Ricci surfaces in $\mathbb{R}^3$. Next, we develop a two-parameter family of non-isometric Ricci metrics of type $c$ for $c >0$, which can be realized on a torus, and we show how some of these surfaces can be immersed in the $3$-sphere.

Cascading upper bounds for triangle soup Pompeiu-Hausdorff distance

Resumo: We propose a new method to accurately approximate the Pompeiu-Hausdorff distance from a triangle soup A to another triangle soup B up to a given tolerance. Based on lower and upper bound computations, we discard triangles from A that do not contain the maximizer of the distance to B and subdivide the others for further processing. In contrast to previous methods, we use four upper bounds instead of only one, three of which newly proposed by us. Many triangles are discarded using the simpler bounds, while the most difficult cases are dealt with by the other bounds. Exhaustive testing determines the best ordering of the four upper bounds. A collection of experiments shows that our method is faster than all previous accurate methods in the literature.

Isometric immersions of bordered surfaces and geometry of infinite Grassmannian

Resumo: We will discuss Kapovich-Millson space of polygons, its smooth analog -- Millson-Zombro space, and their Grassmannian nature. The first half of the talk will be devoted to a theorem of the speaker (joint with S. Anan'in) which roughly says that the space of isometric immersions of a polyhedral disk in R^3 has the natural structure of a Lagrangian subset in the Kapovich-Millson space. In the second half we'll talk about a conjectural picture relating the space of smooth isometric immersions of a disk, and some Lagrangians in the Millson-Zombro space. Supporting evidence, partial results and reductions will be given.