Seminário GEMS — Encontros de Geometria 

A Lorentzian spacetime $(M,g)$ can be generalized via the inclusion of a positive density function $h$ which modifies the Riemannian volume element, giving rise to a smooth metric measure spacetime $(M,g,h \operatorname{dvol}_g)$. In the study of these manifolds with density, we use weighted invariants which retain geometric significance while incorporating the density function in natural ways.

Starting with the weighted Einstein-Hilbert functional, through a variational approach, we define a weighted analogue of the associated Euler-Lagrange equations (the weighted Einstein field equations). We organize the resulting vacuum solutions according to the causal character of the gradient of $h$, which strongly influences the geometry of the underlying manifold: isotropic, when the gradient of $h$ is lightlike; and non-isotropic, when it is timelike or spacelike. In order to illustrate the properties of different solutions, in this talk I will go over some local rigidity results for both families under conditions on curvature-related tensors. I will also give some examples realized on geometrically relevant Kundt-type spacetimes that often arise in the study of the weighted Einstein field equations.

This is joint work with Miguel Brozos-Vázquez.

References

• M. Brozos-Vázquez, D. Mojón-Álvarez. The vacuum weighted Einstein field equations, Math. Z. 310 (2025), no. 3, Paper No. 44, 38 pp.
• M. Brozos-Vázquez, D. Mojón-Álvarez. The vacuum weighted Einstein field equations on pr-waves. Springer Proc. Math. Stat., 512 Springer, Cham, 2025, 67-80.
• M. Brozos-Vázquez, D. Mojón-Álvarez. Vacuum Einstein field equations in smooth metric measure spaces: the isotropic case. Class. Quantum Grav. 39 (13) (2022) 135013, 20 pp.

We study finite modular radicial extensions in characteristic $p > 0$. We introduce trace maps attached to monomial bases and use them to study dual modules in purely inseparable settings. This clarifies the interaction between trace maps, Frobenius, and differential structures in modular radicial extensions. We also discuss exact sequences associated with modular radicial extensions and their relation with Frobenius and differential structures.

Joint work with Qing Liu.

I will discuss a notion of tensor product of (twisted) quiver representations with relations in the category of $O_X$-modules. As a first application of our notion, we see that tensor products of polystable quiver bundles are polystable, and later we use this to both deduce a quiver version of the classic Segre embedding and to identify distinguished closed subschemes of $\operatorname{GL}(n,\mathbb{C})$-character varieties of free abelian groups.

I will introduce the problem of finding the Euler stratification of an emdedded scaled projective toric variety. The strata where the Euler characteristic is one is of special interest in Algebraic Statistics because that means a certain associated statistical model has a maximum likelihood estimator with a closed form expression as a rational function of the data. I will explain what is known for this case in 2D and present baby step progress in the 3D case.

Lie groupoids can encode geometric objects such as smooth actions, and foliations; deformations of Lie groupoids also relate to deformations of these objects. In this talk we’ll first see the deformation cohomology of a (real) Lie groupoid and mention relations to deformations the mentioned examples.

We will then see the cohomology controlling deformations of a complex Lie groupoid: it combines deformation cohomology of the groupoid structure and Kodaira-Spencer cohomology of the underlying complex manifold, via a double complex.

The talk is based on ongoing work with Luca Vitagliano.

A striking difference between Riemannian and pseudo-Riemannian metrics is that pseudo-Riemannian ones often fail to be geodesically complete even in the compact case. We will present some developments in the classification of Lie groups with all their left-invariant pseudo-Riemannian metrics complete. More concretely, we will discuss the specifics of geodesic completeness when the manifold in question is a Lie group and recall the Euler-Arnold theorem as well as the seminal work of Marsden for the compact (homogeneous) case. We will see how an interpretation in Riemannian terms of his techniques provided us with tools for characterising completeness even for general manifolds. As for Lie groups, we will show how a certain notion of “linear growth” allowed us to establish large classes of Lie groups whose left-invariant metrics are all complete. Time permitting, we will also discuss the generalisation of the Euler-Arnold formalism to the holomorphic-Riemann setting and discuss the classification of geodesic completeness for 3-dimensional (non-unimodular) Lie groups.

This is a series of joint works with S. Chaib, A. Elshafei, H. Reis, M. Sánchez and A. Zeghib.

The ODE/IM correspondence is a conjectural and surprising link between nonlocal observables of integrable quantum field theories and monodromy data of linear analytic ODEs.

Let $g$ be a simple Lie algebra over the complex field, $(g,1)$ the corresponding untwisted Kac-Moody algebra and $\operatorname{Lan}(g,1)$ the Langlands dual of $(g,1)$.

In 2011, extending previous results in the physics literature, B. Feigin and E. Frenkel conjectured that observables of the Quantum $g$-KdV model can be expressed in terms of monodromy data of a class of $\operatorname{Lan}(g,1)$-affine opers known as Feigin-Frenkel-Hernandez opers.

In this talk, we introduce the relevant objects (such as affine opers and Bethe equations) and describe the state-of-.the-art of the Feigin-Frenkel conjecture.

Out of a given holomorphic Lie algebroid $L$ on a compact Riemann surface $X$, one can consider a corresponding $L$-connection on a vector bundle over $X$. This naturally degenerates onto a (twisted) Higgs bundle on $X$. Via a generalization of the classical construction by Simpson of $\lambda$-connections, such degeneration induces an associated one at the level of moduli spaces, using the so-called $L$-Hodge moduli space. We use this to study geometric and topological properties of the moduli spaces of $L$-connections on $X$, as simple as their dimension or more complicated like their motivic class. This joint work with David Alfaya.