Let M be a spacetime and consider the Lorentzian distance function from a point p in M. Under suitable conditions this function is differentiable, at least in a ''sufficiently near chronological future'' of the point p. In this talk we will study the Lorentzian distance function restricted on certain spacelike hypersurfaces and spacelike submanifolds of M, and derive sharp estimates for the mean curvature of such submanifolds under appropriate hypotheses on the curvature of the ambient spacetime. Our results are obtained as an application of the corresponding comparison results of the (Lorentzian) Hessian and Laplacian operators of the Lorentzian distance, as well as the generalized Omori-Yau maximum principle for the (Riemannian) Hessian and Laplacian operators.
This was joint work with Ana Hurtado and Vicente Palmer (2010) and, more recently, with G. Pacelli Bessa and Jorge H.S. de Lira (2016). Our study is strongly based on a previous work by F. Erkekoglu, E. García-Río and D.N. Kupeli (2003), where they established the basis for the comparison analysis of the Lorentzian distance funtion
Las visiones lagrangiana y hamiltoniana de las teorías de campos clásicos son bien entendidas, aunque algunas construcciones geométricas como el triple de Tulczyjew son recientes. Sin embargo, hay un problema que sigue estando abierto y para el que sólo se han obtenido resultados parciales: ¿cuál es una definición canónica de un formalismo hamiltoniano para sistemas de orden superior a 2? Presentaremos los elementos básicos de la teoría de jets y daremos indicaciones geométricas sobre cómo resolver este problema.
The generalized Ricci flow is a geometric evolution equation which couples Riemannian metrics on a manifold with a family of evolving closed three forms. The equation has its origins in the renormalization group flow of two-dimensional sigma models in mathematical physics, and extends the fundamental Hamilton/Perelman theory of Ricci flow. A key feature of the flow is that it preserves the cohomology class of the three-form, and hence it naturally controls three-volumes for the evolving metric. In this talk I will give an introduction to this topic, with a special emphasis on examples. If time allows, I will explain the expected relation between the flow and Kodaira's classification compact complex surfaces, and present some global existence results. Based on joint work with Joshua Jordan (UC Irvine) and Jeffrey Streets (UC Irvine), arXiv:2008.07004 and arXiv:2106.13716.
In this talk, we consider a bounded domain in the Euclidean plane and examine the Laplacian eigenvalue problem supplemented with specific boundary conditions. A famous conjecture by Berry proposes that in chaotic dynamical systems, eigenfunctions resemble random monochromatic waves; however, this behavior is generally not expected in integrable dynamical systems. In this talk, we explore the behavior of high-energy eigenfunctions and their connection to Berry’s random wave model. We do so by studying a related property called Inverse Localization, which describes how eigenfunctions can approximate monochromatic waves in small regions of the domain.
The aim of this talk is to explain to the audience the path we followed from a question asked to us by some researchers in 2015: Why do we insist on using the notion of diameter in generalized Minkowski spaces given by Danzer, Grünbaum, and Klee (1963) and not the standard one.
This question led us to study relations between different means of convex bodies. To do so, we made use of the Asymmetry measure of Minkowski, i.e. the smallest rescaling factor of K such that it contains a translation of -K, allowing us to do a meaningful comparison between those means.
Finally, we will also connect this topic to some open questions in convex geometry, some of them now partly solved due to those techniques.
Joint work with René Brandenberg, Katherina von Dichter, and Florian Grundbacher.
La conjetura del catenoide crítico (Nitsche, 1985) se ha convertido en la última década en uno de los problemas de unicidad fundamentales en la teoría de superficies mínimas. Dicha conjetura afirma que todo anillo mínimo embebido con borde libre en la bola unidad tridimensional ha de ser una porción de catenoide. En esta charla expondremos el origen y la motivación de dicha conjetura, su relación con otras ramas del análisis geométrico (sistemas integrables, autovalores de Steklov), y los avances más recientes sobre este problema, que aún sigue abierto.