Título: HOLOMORPHIC APPROXIMATION THEORY FOR NULL CURVES IN THE SPECIAL LINEAR GROUP.
Tutor: Antonio Alarcón López.

Einstein metrics with non-positive scalar curvature are conjectured to be linearly stable. On spin manifolds, the existence of parallel spinors provides a spinorial criterion for stability. In this talk I explain how this mechanism extends to non-spin manifolds using twisted spinors. In particular, we show that Einstein manifolds carrying parallel twisted pure spinors are linearly semistable. This yields stability results for a large class of spaces, including quaternionic Kähler manifolds of dimension greater than 8.

The virial theorem originates in classical mechanics and expresses a balance between kinetic energy and the radial component of the force acting on a system. Remarkably, this principle persists in quantum mechanics. For the Schrödinger equation, virial identities arise from simple dynamical considerations and have become fundamental tools in the analysis of partial differential equations, with applications ranging from dispersive estimates to nonlinear dynamics. In this talk I will discuss how the virial principle extends to relativistic quantum mechanics, governed by the Dirac operator. In this setting the classical dynamical derivation breaks down due to the peculiar structure of relativistic dynamics. Instead, virial-type identities emerge from algebraic relations involving commutators and anticommutators of the Dirac Hamiltonian. This viewpoint reveals an unexpected connection between classical mechanics, quantum dynamics, and operator theory. As an application, we obtain constraints on the spectral properties of relativistic quantum Hamiltonians, leading in particular to conditions that exclude the existence of bound states.

I will prove Gromov's conjecture that every 3-manifold of positive scalar curvature contains a short closed geodesic. The proof uses Min-Max theory of minimal surfaces and a combinatorial version of mean curvature flow. This is a joint work with Davi Maximo and Regina Rotman.

El establecimiento de resultados tipo Talenti para la simetrización de la solución de una ecuación de Poisson, planteada sobre un dominio regular $D$ en una variedad Riemanniana $(M,g)$, se encuentra estrechamente vinculado a la existencia de una desigualdad isoperimétrica satisfecha por los dominios de la variedad. Esta relación ha sido explorada previamente, tanto en el contexto compacto como en el no-compacto. Hasta la fecha, el conocimiento sobre comparaciones en espacios de curvatura negativa es limitado, más allá de lo establecido por McDonald (donde se prueba una comparación tipo Talenti en espacios hiperbólicos). La dificultad radica en que, en espacios de curvatura negativa, no se ha establecido una desigualdad isoperimétrica de forma general. Presentaremos un esquema de la demostración de una serie de comparaciones tipo Talenti para variedades Riemannianas completas y no compactas que cumplen una de las siguientes condiciones: • Poseen una constante isoperimétrica positiva. • Su perfil isoperimétrico está controlado en cierto sentido. Estas hipótesis son integradoras en el sentido de que, en el caso no compacto, engloban los contextos ya estudiados e incluyen además a las variedades de Cartan-Hadamard. Trabajo en colaboración con V. Gimeno.

Let $M$ be a properly embedded, connected, complete surface in $\mathbb{R}^3$ with boundary a convex planar curve $C$, satisfying an elliptic equation $H=f(H^2-K)$, where $H$ and $K$ are the mean and the Gauss curvature respectively – which we will refer to as Weingarten equation. In this talk, we discuss how the symmetries of $C$ may induce symmetries of the whole $M$. When $M$ is contained in one of the two halfspaces determined by $C$, we give sufficient conditions for $M$ to inherit the symmetries of $C$. In particular, when $M$ is vertically cylindrically bounded, we get that $M$ is rotational if $C$ is a circle. In the case in which the Weingarten equation is linear, we give a sufficient condition for such a surface to be contained in a halfspace. Both results are generalizations of results of Rosenberg and Sa Earp, for constant mean curvature surfaces, to the Weingarten setting. In particular, our results also recover and generalize the constant mean curvature case.

Zoll manifolds are Riemannian manifolds all of whose geodesics are closed and have the same length. Beyond the round sphere, nontrivial examples were constructed by Funk and Guillemin, initiating a rich line of research. In this talk, I introduce a free-boundary analogue of this notion. A compact Riemannian manifold with boundary is said to be Zoll with boundary if every geodesic issuing orthogonally from the boundary returns orthogonally and is nowhere tangent to it. I will show that such manifolds exhibit strong rigidity: all free-boundary geodesics have the same length and share the same Morse index. Using Morse index theory and algebraic topology, we obtain a complete geometric and topological classification. In particular, when the boundary is connected, the manifold is a tubular neighborhood of a closed embedded submanifold (the “soul”), and the boundary fibers over the soul either as a sphere bundle or as a nontrivial two-fold covering. This is joint work with Paolo Piccione and Roney Santos.

In this talk, I will describe a classification result for translating solitons of the mean curvature flow under two natural quantitative/topological assumptions: finite genus and entropy equal to 2. Roughly speaking, these hypotheses place the translator at the borderline between the simplest nontrivial singularity models and genuinely higher–complexity behavior: finite genus controls the global topology, while the entropy bound rigidifies the possible asymptotic geometry and rules out many exotic configurations. I will explain how one combines geometric measure theory and PDE tools for the translator equation to extract strong structural information from the entropy constraint, leading to a precise description of all such examples. Along the way, I will discuss the role of blow-down and compactness arguments, how the entropy pinching interacts with curvature and topology, and what the result says about the landscape of low-entropy translators. This is joint work with E. S. Gama and N. M. Møller.

We prove the existence of complete minimal surfaces, in Euclidean 3-space, of arbitrary positive genus and least total absolute curvature that have precisely two ends: a catenoidal end and an Enneper-type end. This talk is based on a joint work with Rivu Bardhan, Indranil Biswas, and Pradip Kumar. https://arxiv.org/abs/2509.03925