IMPA - O Instituto de Matemática Pura e Aplicada

The splitting field $\mathcal{K}$ of an elliptic surface $\mathcal{E}$ defined over $\mathbb{Q}(t)$ is defined as the smallest subfield of $\mathbb{C}$ such that all $\mathbb{C}(t)$-rational points are actually defined over $\mathcal{K}(t)$, i.e., $\mathcal{E}(\mathbb{C}(t)) \cong \mathcal{E}(\mathcal{K}(t))$.

In this talk, I will present the results of our recent study (in a joint work with Arman Shamsi Zargar) regarding the arithmetic of Shioda's elliptic surfaces $\mathcal{E}_m$ defined by the equation: \[ \mathcal{E}_m: y^2 = x^3 + t^m + 1.\] While the structure and fundamental quantities of the Mordell-Weil lattices for these surfaces over $\mathbb{C}(t)$ were previously studied by Shioda and Usui, in our work, we explicitly determine the splitting fields $\mathcal{K}_m$ and provid a set of linearly independent generators for the Mordell-Weil lattices for positive integers $1 < m \le 12$.

Gromov conjectured that the integral of the mean curvature of the boundary of a compact Riemannian manifold is bounded above by a constant that depends only on the induced metric on the boundary and a lower bound for the scalar curvature in the interior. We prove Gromov's conjecture in various cases, including spin and non-spin manifolds, with an upper bound that also depends on a lower bound for the mean curvature. To achieve this, we use surgery to reduce the statement to the case when the boundary is a sphere. This case can be addressed using instances of the positive mass theorem.

This is joint work with Georg Frenck (University of Augsburg) and Sven Hirsch (Columbia University).

It is well known that the collection of linear frames of a smooth n-manifold $M$ defines a principal $GL(n, R)$-bundle over $M$ (called the frame bundle); more generally, this construction makes sense for any vector bundle over $M$. Conversely, any principal bundle together with a representation induces an associated vector bundle; these processes establish therefore a correspondence between vector bundles on one side, and principal bundles with representations on the other side.

 In differential geometry there are several natural instances where diagrams of Lie groupoids and vector bundles, together with suitable compatibilities, appear. They are known as vector bundle groupoids (VB-groupoids) and their theory has been fairly developed in the past decades, with applications e.g. to Poisson geometry, representations up to homotopy, deformation theory, and non-commutative geometry. On the other hand, little is known about the principal bundle counterpart of these objects.

 In this talk, I will recall all the notions mentioned above, and then introduce a special class of frames of VB-groupoids which interact nicely with the groupoid structure. I will then use them to associate to any given VB-groupoid a diagram of Lie groupoids and principal bundles, together with the action of a (strict) Lie 2-groupoid $GL(l, k)$; this will lead to the general notion of a principal bundle groupoid (PB-groupoid). Moreover, I will sketch how to generalise the standard correspondence between vector bundles and principal bundles to a correspondence between VB-groupoids and PB-groupoids. I will conclude discussing a few examples, current works in progress, and future applications.

 This is joint work with Alfonso Garmendia.

Let $X$ be a Fano manifold of dimension five and let $\sigma\in {H^0}(X,\wedge^2 T_{X})$ be a holomorphic Poisson structure. We prove that the rank zero locus $D_0(\sigma)$  has an irreducible component of dimension at least $1$ and that the rank at most two locus $D_2(\sigma)$  has an irreducible component of dimension at least $3$. This confirms Bondal's conjecture in dimension five. The proof uses: (i) an algebraic integrability criterion for codimension one foliations on weak Fano manifolds; (ii) a lemma of Gualtieri-Pym on degeneracy loci of ample Poisson modules; and (iii) a generalization of a result by Esteves-Kleiman on the zero loci of Pfaff fields along invariant subvarieties. 

We study long-waves propagation in two-dimensional waveguide channels with irregular geometries, focusing on curved and expanded junctions. These domains are mapped to simpler canonical channels through the Schwarz–Christoffel conformal transformation. Using modal decomposition in the conformal domain, we show that the averaged solution is governed by the fundamental mode, enabling an effective one-dimensional reduction. In this reduced model, junctions appear as delta-type perturbations, establishing a natural connection with quantum graph theory. We developed analytical and numerical approaches to quantify the effects of channel width, angle, smoothness, and wavelength on the wave dynamics.

Propomos uma nova formulação da equação de Korteweg-de Vries (KdV) na linha real, por meio de uma transformação de calibre. Embora a KdV e a equação calibrada sejam equivalentes para soluções suaves, a última se comporta melhor com baixa regularidade em espaços de Fourier-Lebesgue. Em particular, as regularidades admissíveis vão além da escala $H^{-1}$, que é um limite bem conhecido para a KdV. Como subproduto, ao inverter a transformação de calibre, conseguimos aprimorar a teoria conhecida para KdV e derivar um bem-posto local nítido em espaços de Fourier-Lebesgue com grande expoente de integrabilidade. Nossa estratégia se baseia em uma redução de forma normal infinita e estimativas de restrição de Fourier, juntamente com uma exploração completa de cancelamentos algébricos. Além disso, nosso método é totalmente independente da estrutura KdV completamente integrável e se estende a outros modelos não integráveis com não linearidades quadráticas.