Seminários

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.

Recent works by M. Gromov, A. Petrunin, O. Chodosh, and C. Li deals with the question: Given a closed smooth manifold M, minimize the maximum normal curvatures among all immersions of M into (high-dimensional) Euclidean spaces, under the constraint that the image lies in a closed ball of radius one.

Replacing this constraint with the condition that the (extrinsic) diameter is at most two, one arrives at a different, but still natural, question.

The result I will present is a first step in the exploration of this new question, namely: For any immersed submanifold M with diameter at most two, the maximum normal curvature is at least one, and equality holds if and only if M is diffeomorphic to a sphere or (real/complex/quaternionic/octonionic) projective space, embedded either as an affine sphere, or as a "Veronese" variety.

Time allowing, I will discuss: (1) a generalization to submanifolds of spheres and hyperbolic spaces, (2) the proof, which uses A. Schur's "Bow" Lemma and K. Sakamoto's classification of submanifolds with planar geodesics, and (3) natural open questions.

Singular Riemannian foliations are certain partitions of Riemannian manifolds, and the traditional sources of examples are isometric group actions and isoparametric hypersurfaces. When the ambient manifold is a sphere, it has long been known that such examples are, in the appropriate sense, algebraic. In 2018, Lytchak and Radeschi have shown algebraicity for a general singular Riemannian foliation in a sphere. In joint work with Samuel Lin and Marco Radeschi, we extend the Lytchak-Radeschi theorem from spheres to any compact normal homogeneous space, a class that includes all compact symmetric spaces. Time-permitting, I'll comment on the ingredients of the proof(s).