IMPA - The Institute for Pure and Applied Mathematics

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.

The founders of quantum mechanics (Born, Heisenberg, Dirac and Jordan) proposed an interpretation of the spectrum as a quantised version of normal forms in classical mechanics, hence the name quantum mechanics. The inverse of this normal form, improperly called the quantum Birkhoff normal form, defines a function which is a quantum analog of the period integrals. We show that like in the classical case, these satisfy ordinary linear differential equations whose degree is bounded by the rank of homology like for ordinary Picard-Fuchs theory. Computations for the standard cubic oscillator show that the coefficients of the quantum-period expansion are integers, suggesting a new kind of quantisation of the classical Gromov-Witten theory of $(K_{\mathbb P^2})$.

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).

Scientific Machine Learning is an emerging field with significant impact on methods for solving problems in science and engineering, particularly for large-scale coupled fluid-flow and transport problems. Numerical simulations for these problems can be costly, making such approaches valuable for understanding and improving the efficiency of state quantification and prediction. This talk will review recent advances in scientific machine learning, including dynamic mode decomposition, manifold learning, graph neural networks, and neural operators (Fourier Neural Operators, DeepOnet, and hybrid schemes), as applied to coupled fluid-flow and transport problems. These problems are of interest in sustainable resource exploration, geophysics, and various industrial applications. The talk will show how data-driven information can improve the efficiency of numerical simulation software, explore parametric manifolds for unseen scenarios, and reconstruct high-dimensional simulations using lower-dimensional structures in feasible time.