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Complex Lagrangians in the moduli space of Higgs bundles

Speaker: Lucas Branco - PUC-Rio
Tue 19 Mar 2024, 15:30 - SALA 236Differential Geometry

Abstract: Motivated by mirror symmetry, we discuss two classes of complex Lagrangian subvarieties inside the moduli space of G-Higgs bundles on a curve, where G is a complex reductive Lie group. One emerges from real forms of G, while the other stems from symplectic representations.

We then focus on two examples: (a) the real form SL(n,H) of SL(2n,C), and (b) the standard representation of the symplectic group.


Neural Implicit Representations

Speaker: Tiago Novello - IMPA
Wed 20 Mar 2024, 10:30 - AUDITORIO 3Computer Graphics

Abstract: The revolution in the media industry caused by neural networks has motivated the development of new representations adapted to machine learning methods. In this context, this talk presents the ongoing research in the area of implicit neural representations (INRs) of media objects at the Visgraf Laboratory.

The parameters of an INR are implicitly determined by its loss function, which, in addition to the data constraints, considers possible differential properties that the function must satisfy. We present architectures, loss functions, training, and sampling strategies for INRs. In particular, we explore the use of this model to parameterize media objects, 3D scenes, flows for use in animation and morphing, and multiscale neural networks to represent objects at different levels of detail.


Monodromia de estruturas projetivas em superfícies de tipo finito

Speaker: Genyle Nascimento - UERJ
Thu 21 Mar 2024, 15:30 - SALA 224Holomorphic Foliations

Abstract: Nesta palestra, falaremos sobre a monodromia de estruturas projetivas com singularidades do tipo fuchsiana, i.e., mostraremos que toda representação do grupo fundamental de uma superfície de Riemann de tipo finito em $PSL_2(\mathbb{C})$ pode ser representada como a holonomia de estrutura projetiva ramificada com singularidades de tipo fuchsiana sobre as cúspides. Por fim, vamos explorar o problema de minimizar ângulos nessas estruturas.


An overdetermined eigenvalue problem and the Critical Catenoid conjecture

Speaker: José Espinar - Universidad de Granada
Tue 26 Mar 2024, 15:30 - SALA 236Differential Geometry

Abstract: We consider the eigenvalue problem $\Delta_{\mathbb{S}^2}\xi + 2\xi=0 $ in $\Omega$ and $\xi = 0$ along $\partial \Omega$, being $\Omega$ the complement of a disjoint and finite union of smooth and bounded simply connected regions in the two-sphere $\mathbb{S}^2$. Imposing that $|\nabla \xi|$ is locally constant along $\partial \Omega$ and that $\xi$ has infinitely many maximum points, we are able to classify positive solutions as the rotationally symmetric ones. As a consequence, we obtain a characterization of the critical catenoid as the only embedded free boundary minimal annulus in the unit ball whose support function has infinitely many critical points.