Einstein-like geometric structures on surfaces

An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenerate immersed hypersurface in flat affine space. The author has defined for AH structures Einstein equations, which specialize on the one hand to the usual Einstein Weyl equations and, on the other hand, to the equations for affine hyperspheres. Here these equations are solved for Riemannian signature AH structures on compact orientable surfaces, the deformation spaces of solutions are described, and some aspects of the geometry of these structures are related. Every such structure is either Einstein Weyl (in the sense defined for surfaces by Calderbank) or is determined by a pair comprising a conformal structure and a cubic holomorphic differential, and so by a convex flat real projective structure. In the latter case it can be identified with a solution of the Abelian vortex equations on an appropriate power of the canonical bundle. On the cone over a surface of genus at least two carrying an Einstein AH structure there are Monge-Amp`ere metrics of Lorentzian and Riemannian signature and a Riemannian Einstein K"ahler affine metric. A mean curvature zero spacelike immersed Lagrangian submanifold of a para-K"ahler four-manifold with constant para-holomorphic sectional curvature inherits an Einstein AH structure, and this is used to deduce some restrictions on such immersions.

Reduction of critical mass in a chemotaxis system by external application of a chemoattractant

In this paper we study non-negative radially symmetric solutions of a parabolic-elliptic Keller-Segel system. The system describes the chemotactic movement of cells under the additional circumstance that an external application of a chemo attractant at a distinguished point is introduced.

A Schwarz lemma for Kahler affine metrics and the canonical potential of a proper convex cone

This is an account of some aspects of the geometry of Kahler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a version for Kahler affine metrics of Yau s Schwarz lemma for volume forms. By a theorem of Cheng and Yau, there is a canonical Kahler affine Einstein metric on a proper convex domain, and the Schwarz lemma gives a direct proof of its uniqueness up to homothety. The potential for this metric is a function canonically associated to the cone, characterized by the property that its level sets are hyperbolic affine spheres foliating the cone. It is shown that for an n -dimensional cone, a rescaling of the canonical potential is an n -normal barrier function in the sense of interior point methods for conic programming. It is explained also how to construct from the canonical potential Monge-Ampère metrics of both Riemannian and Lorentzian signatures, and a mean curvature zero conical Lagrangian submanifold of the flat para-Kahler space.

Percepción dispersa: Arquitectura y tactilidad en la sociedad de la comunicación

El tema de la tesis se centra en el pensamiento de Walter Benjamin y en el concepto de percepción distraída, desarrollado en el ensayo La obra de arte en la época de su reproducibilidad técnica (1936). El título percepción dispersa deriva de una profundización de los conceptos benjaminianos en términos lingüísticos, adoptando otros significados de la palabra distracción que relacionan este concepto a cuestiones de ámbito espacial y atmosférico. Benjamin sostiene que la arquitectura, en una época en la que se desarrollan y se difunden nuevas técnicas de reproducción/ comunicación, no se puede percibir solo de manera visual sino también táctil, interrelacionando la difusión de nuevas técnicas de reproducción, la tensión entre diferentes modalidades perceptivas y la arquitectura: según él, la dispersión de la vista a favor de la percepción táctil/háptica es un rasgo característico de una sociedad en la que se desarrollan y se difunden nuevas técnicas de reproducción. La tesis profundiza en estos temas, actualizando el concepto de distracción/dispersión para valorar las repercusiones de esta intuición en el ámbito de una parte de la producción arquitectónica contemporánea: se propone una contextualización histórica y una conceptual, relativas a los proyectos recientes de algunos de los autores que profundizan en el impacto de las nuevas tecnologías en el ámbito del proyecto de arquitectura. La hipótesis que se quiere demostrar es que se está desarrollando una tendencia que lleva proyectos o edificios de alto nivel tecnológico y informático desde un ámbito de (supuesto) predominio de la percepción visual, hasta una arquitectura que proporciona experiencias multisensoriales: espacios que se pueden medir con el movimiento, los músculos y el tacto, que desarrollan un conjunto de sensaciones táctiles/hápticas y no solo visuales. Finalmente se quiere verificar si, revertiendo siglos de evolución visual, la visión táctil y cercana siga siendo uno de los polos dialécticos que estructuran la experiencia de la arquitectura, como previsto por Benjamin hace 75 años. ABSTRACT The thesis focuses on the thought of Walter Benjamin and the concept of distracted perception, developed in the essay The Work of Art in the Age of Mechanical Reproduction (1936). The title derives from a linguistic approach to this concept which further explores the meaning of the word distraction, finally relating this kind of perception to spatial and atmospheric issues. Benjamin argues that architecture, thanks to the development of new reproduction/communication technologies, cannot be perceived exclusively in a visual way but also in a tactile way, interrelating new technologies of reproduction, perceptual modalities and architecture: according to him, the dispersion of the view in favor of tactile/ haptic perception is a fundamental characteristic of a technologically developed society. The thesis explores these issues, updating the concept of distraction/dispersion to assess its implications in the works of some architects that are exploring with their buildings the impact of new communication technologies in the field of architectural design. The hypothesis at the base of this work is that we are facing a new trend in this particular field: many of the most advanced buildings provide multisensory experience, and define spaces that can be better perceived with the sense of touch (haptic sense), going beyond a mere visual perception. Finally the thesis wants to verify if, reversing centuries of visual evolution, tactile and close vision remains one of the dialectical poles that stay at the base of the experience of architecture, as foreseen by Benjamin 75 years ago. ABSTRACT La tesi si basa sul pensiero di Walter Benjamin e sul concetto di percezione distratta, sviluppato nel saggio La opera d’arte nell’epoca della sua riproducibilità tecnica (1936). Il titolo, percezione dispersa, deriva da un approfondimento dei concetti benjaminiani in termini linguistici, adottando ulteriori significati della parola che il filosofo utilizza per indicare la distrazione e che relazionano questo termine a questioni di ambito spaziale ed atmosferico. Benjamin sostiene che l’architettura, in un’epoca nella quale si sviluppano e si diffondono nuove tecniche di riproduzione (o meglio: nuove tecnologie di comunicazione), non possa essere percepita solamente in maniera visuale, ma anche tattile, mettendo in relazione la diffusione di nuove tecnologie, la tensione tra differenti modalità percettive e l’architettura: secondo Benjamin, la dispersione della vista a favore di una ricezione tattile/aptica della realtà è una caratteristica tipica di quei momenti storici nei quali si manifestano grandi trasformazioni di ambito sociale e culturale dovute allo sviluppo di nuove tecnologie di comunicazione. La tesi approfondisce questi temi, aggiornando il concetto di distrazione/dispersione per valutarne le ripercussioni su una certa parte della produzione architettonica contemporanea: si propone una contestualizzazione storica ed una concettuale, relative ai progetti di alcuni architetti che lavorano da tempo sull’impatto delle nuove tecnologie nell’ambito del progetto di architettura. La ipotesi che si dimostra con questa tesi è che si sta sviluppando una contro-tendenza tattile, a seguito della quale molti degli edifici tecnologicamente ed informaticamente più avanzati strutturano la relazione con i propri utenti sulla base di esperienze multisensoriali, definendo spazi che possono essere percepiti attraverso una percezione aptica, piuttosto che visuale. In definitiva la tesi verifica che, superando secoli di evoluzione visuale, la visione tattile e ravvicinata e - più in generale - una percezione di tipo aptico continuano ad essere uno dei poli dialettici che strutturano l’esperienza dell’architettura, come previsto da Benjamin più di 75 anni fa.