IMPA - O Instituto de Matemática Pura e Aplicada

We propose a new formulation of the Korteweg-de Vries equation (KdV) on the real line, via a gauge transform. While KdV and the gauged equation are equivalent for smooth solutions, the latter is better behaved at low regularity in Fourier-Lebesgue spaces. In particular, the admissible regularities go beyond the  Hˆ{-1}-scale, which is a well-known threshold for KdV. As a byproduct, by reversing the gauge transform, we are able to improve on the known theory for KdV and derive sharp local well-posedness in Fourier-Lebesgue spaces with large integrability exponent. Our strategy is based on an infinite normal form reduction and Fourier restriction estimates, together with a thorough exploitation of algebraic cancellations. Additionally, our method is totally independent of the KdV completely integrable structure, and extends to other non-integrable models with quadratic nonlinearities.

Generative AI is transforming storytelling into living story worlds where humans and artificial agents co-create narrative events in real time. This talk presents a framework spanning Story World, Narrative Universe, and Story, and introduces the MMSW Runtime, a distributed platform for multi-agent, multimodal narrative interaction with persistent memory, enabling new workflows for interactive media, education, and collaborative storytelling.

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

O problema de classificar pares de Calabi-Yau logarítmicos até a equivalência de preservação de volume (ou crepante) é muito desafiador. Para essa tarefa, você pode usar a coregularidade, o invariante de preservação de volume mais importante de um par log Calabi-Yau.   Recentemente, Ducat fez alguns progressos em relação a pares da forma (P^3,D) e de coregularidade ≤ 1. Embora invariantes e refinamentos na classificação tenham sido estudados especialmente no caso de coregularidade 0, o caso de alta coregularidade é o mais difícil de analisar. Nesta palestra, ilustrarei essa afirmação abordando o caso ausente de coregularidade 2 para pares da forma (P^3,D) e compartilhando algumas descobertas interessantes. Este é um trabalho conjunto em andamento com Daniela Paiva, Sokratis Zikas e Felipe Zingali Meira.