IMPA - O Instituto de Matemática Pura e Aplicada

An oriented surface in $\mathbb{R}^3$ is a Weingarten surface if its principal curvatures $\kappa_1 \geq \kappa_2$ satisfy a nontrivial relation $W(\kappa_1,\kappa_2)=0$. It is elliptic when the PDE governing its local graph is elliptic, in which case we can locally write $\kappa_2 = f(\kappa_1)$ with $f$ strictly decreasing. Minimal surfaces and CMC surfaces, which give rise to quasilinear PDEs, are classical examples. Therefore, studying elliptic Weingarten surfaces gives a natural generalization to the fully nonlinear setting.

In this talk, we review key results on complete elliptic Weingarten surfaces in $\mathbb{R}^3$ and present recent joint work with José A. Gálvez, Yuanyuan Lian, and Kai Zhang, on the asymptotic behavior at infinity of uniformly elliptic Weingarten surfaces of minimal type and finite total curvature. We classify this behavior depending strongly on the value $f'(0)$ and establish a maximum principle at infinity.

Essa palestra apresenta os resultados da primeira fase do projeto: VQ-VAE, World Model (GPT causal) treinado offline com teacher forcing e política treinada via imaginação sobre esse World Model congelado e apresenta uma comparação com PPO direto no ambiente real.

Live @ https://www.youtube.com/live/c7mhtJe0Gfs

We study the Cauchy problem for the isentropic hypo-viscous compressible Navier-Stokes equations under general pressure laws. By using convex integration method we provides the first non-uniqueness result of weak solutions to viscous compressible fluid. Due to the difficulties caused by the relative rigidity of the pressure and by the viscosity, our proof features new constructions of building blocks for both the density and momentum, which are, particularly, designed to respect the compressible structure. We provide first applications of a temporally intermittent but spatially homogeneous building block scheme and a mild density perturbation. It also applies to the compressible Euler equations and the hypo-viscous incompressible Navier-Stokes equations. This is a joint work with Peng Qu, Zirong Zeng, and Deng Zhang.

For a compact symplectic manifold, the quantum volume element is the summation of quantum products of elements in dual bases of the cohomology space. This is a deformation of the Euler class of the manifold. In this talk I will describe application of this element to Virasoro conjecture for Gromov-Witten invariants and complexity of quantum cohomology.

We show that for the standard map family, for all parameter values except one, the principal fixed point of the mapping has a transverse homoclinic point.