IMPA - O Instituto de Matemática Pura e Aplicada

O toro de Clifford em uma esfera de Berger com parâmetro $\tau$ é um ponto crítico do funcional de Willmore para todo $\tau \>0$, gerando um caminho suave de superfícies de Willmore. Ao estimar o índice de Morse ao longo desse caminho, aplicamos a teoria da bifurcação para gerar novos toros de Willmore simétricos que surgem do toro de Clifford.

Inspired by the progressive decoupling of linkages methodology for optimization and variational inequalities, we propose an algorithm for solving quasi-variational inequalities as a sequence of variational inequalities. Our method is shown to converge locally under some regularity conditions and globally when such conditions hold throughout the entire domain. Separately, under other type of assumptions, global convergence with linear rate is also established for the class of quasi variational inequalities with the moving set. Advantageous computational performance is shown for large-scale Walrasian equilibrium problems, formulated as generalized Nash equilibria, as well as for some other instances encountered in the related literature. Joint work with Claudia Sagastizábal and Mikhail Solodov.

In his groundbreaking work from 2002, Perelman introduced two fundamental monotonic quantities: the reduced volume and the entropy. While the reduced volume was motivated by the Bishop-Gromov volume comparison applied to a suitably constructed $N$-space, which becomes Ricci-flat as $N\rightarrow \infty$, Perelman did not provide a corresponding explanation for the origin of the entropy. In this talk, I'll show that Perelman's entropy emerges as the limit of Colding's monotonic volume for harmonic functions on Ricci-flat manifolds, when appropriately applied to Perelman's N-space. I'll also mention how the construction of these quantities actually follow a more general scheme applicable to other parabolic non-linear (and even non-local) equations.

Os fundadores da mecânica quântica (Born, Heisenberg, Dirac e Jordan) propuseram uma interpretação do espectro como uma versão quantizada das formas normais da mecânica clássica, daí o nome “mecânica quântica”. A inversa dessa forma normal, chamada incorretamente de “forma normal quântica de Birkhoff”, define uma função que é um análogo quântico das integrais de período. Mostramos que, assim como no caso clássico, essas equações satisfazem equações diferenciais lineares comuns cujo grau é limitado pelo grau da homologia, tal como na teoria de Picard-Fuchs comum. Cálculos para o oscilador cúbico padrão mostram que os coeficientes da expansão do período quântico são inteiros, sugerindo um novo tipo de quantização da teoria clássica de Gromov-Witten de $(K_{\mathbb P^2})$.

Matteo Moriconi (VFXRio/VISGRAF) and Luiz Velho (IMPA/VISGRAF) present recent advances in MMSW (Multicasting Multimodal Story Worlds) as it moves from multimodality toward transmodality and World Models. The Story World becomes a shared, updatable world state in which digital twins and robotic systems let a synthetic entity act with observable, verifiable consequences in space. In the experiments, the entity Celestia Borealis acts through a Franka Panda robotic arm. In Aurora Circuit, a bounded motor vocabulary is composed into a movement sequence, named, and then redescribed through drawing and sound. The talk presents the technical architecture and asks how meaning, perception, and action reorganize when a computational entity operates through simulated or robotic bodies.

Live @ https://www.youtube.com/live/QNvHAT-CPxo

In this talk we analyze the d-dimensional Schrödinger operator in the continuum $-\frac{1}{2}\Delta + \lambda_N^2V$ discretized on a mesh with spacing 1/N, where $\lambda_N=N^{1-\gamma}$. We first prove, that for $\gamma\in (-1,1)$, all eigenvalues of the discrete operator converge to those of the continuum as $\lambda_N\to\infty$. This recovers the semiclassical approximation of the continuum operator established by B. Simon (Ann. of Math., 1985). Subsequently, we prove a large deviation principle induced by the ground-state eigenvector. For $\gamma \in (0,1)$, the rate function is shown to coincide with the usual Agmon metric of the continuum; for $\gamma=0$, it is given by an Agmon--Finsler metric; and for $\gamma\in(-1,0)$, it is given by a flat metric. This analysis therefore reveals a finer structure within the semiclassical regime $\gamma\in(-1,1)$ at the level of ground-state asymptotics, which is not reflected in the eigenvalue asymptotics, and extends related asymptotic results in the current literature, previously known only for $\gamma=0$. This talk is based two recent joint works with  M. Keller (Uni Potsdam) and L. Pettinari (Uni Trento).