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An overdetermined eigenvalue problem and the Critical Catenoid conjecture

Expositor: José Espinar - Universidad de Granada
Ter 02 Apr 2024, 15:30 - SALA 236Seminário de Geometria Diferencial

Resumo: 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.


Homological reduction of Poisson and Courant structures.

Expositor: Pedro H. Carvalho - IMPA
Qua 03 Apr 2024, 15:30 - SALA 236Seminário de Geometria Simplética

Resumo: We show how to obtain homological models for some instances of the reduction of Poisson manifolds and Courant algebroids:

1) For Poisson structures, we obtain a homotopy Poisson algebra extending the classical Kostant-Sternberg BRST algebra, which is a differential graded Poisson algebra constructed from the usual hamiltonian data on a Poisson manifold, to a more general hamiltonian setting. In this context, particular cases of interest are quasi-Poisson and hamiltonian quasi-Poisson manifolds;

2) For Courant algebroids, we describe a homological model for the (Bursztyn-Cavalcanti-Gualtieri) reduction of exact Courant algebroids. In particular, we conclude that the underlying algebraic structure of such a model is that of a differential graded Courant-Dorfman algebra.

Our approach to these problems is based on a graded symplectic perspective for the reduction of Poisson and Courant structures and our results follow from an appropriate interpretation of the BFV models for hamiltonian reduction of graded symplectic manifolds of degrees one and two.


A vertex-centric representation for adaptive diamond-kite meshes

Expositor: Luiz Henrique de Figueiredo - IMPA
Qua 17 Apr 2024, 10:30 - AUDITORIO 3Seminário de Computação Gráfica

Resumo: We describe a concise representation for adaptive diamond-kite meshes based solely on the vertices and their stars. The representation is exact because it uses only integers, is much smaller than standard topological data structures, and is highly compressible. All topological elements are reconstructed in expected constant time per element.