GEMS — Geometry Meetings Seminar 

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.

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.

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.

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.