Seminários

Threshold Behavior in Schrödinger-type Problems

Resumo: In the context of nonlinear, focusing dispersive and wave equations, the associated ground state often gives a quantitative threshold, under which the long-time behavior is well-understood. Such thresholds arise from sharp Gagliardo-Nirenberg-Sobolev inequalities, and initial data below these quantities are subject to the so-called energy trapping, which guarantees coercivity for quantities related to the associated potential and kinetic energies. Combined with the concentration-compactness approach, the energy trapping can be used to prove a dichotomy for the asymptotic behavior of $H^1$ solutions. In this talk, we discuss recent results for Schrödinger-type problems with initial data exactly at the mass-energy threshold in the $L^2$-supercritical setting.

Ahlfors current associated with an entire curve

Resumo: An entire curve on a complex manifold is a curve obtained as a non-trivial image of a holomorphic map from C. It is known that existence of an entire curve on a complex manifold puts great restrictions on its geometry. I would explain how one can use the entire curve to obtain a positive, closed current, known as Ahlfors current. Informally, this current is an integration current of this curve. This argument is used in Nevanlinna theory to count the zeros of an entire function.

Recent developments in Fourier interpolation theory

Resumo: In this talk we will discuss some problems related to the theory of Fourier interpolation. The goal is to talk about the general problem of how to obtain new interpolation formulas from a previously known one. This talk is based on joint work with Iker Gardeazabal (BCAM).

On isometric rigidity

Resumo: Nash's Theorem states that any Riemannian manifold $M^n$ can be isometrically immersed into some Euclidean space. The isometric deformation problem is the unicity-related question. Namely, to describe the moduli space of isometric immersions $f:M^n\rightarrow\mathbb{R}^{n+q}$ that $M^n$ can have for certain $q$.

This talk is about isometric rigidity, and it is divided into three parts. First, we discuss the isometric rigidity of hypersurfaces in higher codimensions. Then, we will provide some examples of isometrically deformable hypersurfaces. Finally, we will talk about the Chern-Kuiper's inequalities and how they are related to the isometric rigidity theory.

The results discussed here are a summary of my Ph.D. thesis.