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### 2010/2011 Seminar Series

 18 octubre 2010Marta PellicerModelització de la fase inflamatòria en la cicatrització de feridesModelling the inflamatory phase in wound healingWound healing is an extremely complicated process and still not fully understood, moreover when diabetis mellitus is present. The first phase of this process, the inflammatory phase, is where there exists a major difference between diabetic and nondiabetic wound healing. This work (in progress) is related with the modeling and analysis of the dynamics of some of the main agents involved in this first phase. We propose a reaction-difusion system as a model, that aims at generalizing the previous existing approach of J.Sherratt and H.Waugh, where an ODE system was proposed as a first approach for this situation. Joint work with N. Cónsul (U. Politècnica de Catalunya) and S.M. Oliva (U. de Sao Paulo, Brazil).15 de novembre de 2010Maria AguarelesEstructura de solucions amb defectes topològics al plaStructure of topological defect solutions in the planeUna de les tècniques més utilitzades per a l'anàlisi de procesos en xarxa són les funcions generatrius de probabilitat (FGP). Una de les aplicacions més recents d'aquesta tècnica és l'estudi d'epidèmies en xarxes on neixen i moren nodes (individus). En la xerrada es vol fer una presentació de com s'utilitzen les FGP en uns models senzills d'epidèmies en xarxes i discutir la seva possible aplicació per resoldre problemes oberts en xarxes dinàmiques. In this talk we will deal with some non-linear partial differential equations of the form -Laplace(u) = F(u) that have solutions with a non-vanishing degree or winding number. These type of solutions arise in many physical contexts like for instance in superconductivity or nematic liquid crystals where they are usually known as vortices or topological defects. In particular we will focus on solutions in the whole plane with a single zero, in which case the partial differential equation reduces to an ordinary differential equation by means of a radial symmetry of the form u=f(r)e^{in\phi}, being n the degree of the solution. We will henceforth work with this ordinary differential equation on f(r) and we will prove existence of these solutions using a rather tricky fixed point theorem while uniqueness will be proved using a so-called sliding method that we will briefly explain. We will also show that with our derivation one also gets monotonicity and some other properties at the origin and at infinity, such as for instance the fact that the derivatives of the solution at the origin become exponentially small as the degree increases. Joint work with I. Baldomà (U. Politècnica de Catalunya).20 de desembre de 2010Esther BarrabésUn cas límit pel problema de l'anell planetari de MaxwellA limit cas of the "Ring Problem"We study the dynamics of an extremely idealized model of a planetary ring. In particular, we study the motion of an infinitesimal particle moving under the gravitational influence of a large central body and a regular n-gon of smaller bodies as n tends to infinity. Our goal is to gain insight into the structure of thin, isolated rings. Joint work with J.M. Cors (U. Politècnica de Catalunya) and G.R.Hall (Boston University)22 de febrer de 2011Albert AvinyóObtenció de funcions que equalitzen una densitat donada mitjançant el mètode del transport de la calorHeat transport method to construct density-equalizing maps  The problem of producing a transformation of a domain into itself such that its jacobian determinant is given at each point is an interesting geometrical problem with some applications in others areas of science and technology. When this transformation is applied to geographical maps in order to make each country or region to have an area proportional to a given quantity, like the number of inhabitants, the gross domestic product or others, the output is usually called a Geographical Cartogram. In this talk we analyze mathematically the heat transport method to produce these transformations, but in some particular cases where the initial data has line or angle discontinuities in the plane. In this situation, the conclusion reinforces the conjecture that the algorithm is always well-posed, in accordance with the experience of the extensive numerical uses of it in the last years. Joint work with J. Solà-Morales and M. València of Universitat Politècnica de Catalunya22 de març de 2011David JuherMaximitzant entropia de cicles en arbres Maximizing entropy of cycles on trees In this talk we give a partial characterization of the periodic tree patterns of maximum entropy. More precisely, let n be a natural number and let K be the maximum of the topological entropies of all n-periodic tree patterns. We prove that each n-periodic pattern with entropy K is irreducible and simplicial, and that all such patterns are maximodal -in the sense that its monotone representative has no local homeomorphisms at the points of the invariant set. Joint work with Ll. Alsedà i F. Mañosas (U. Autònoma de Barcelona)12 d'abril de 2011Laura GarciaEvitant collisions entre satèl.lits al voltant de la Terra Avoiding collisions between satellites around the Earth Whether talking about formations of spacecraft around Earth, or one spacecraft which can collide with an obstacle (such as space junk), we must find a rapid methodology to identify collisions and find a new trajectory which spend as little fuel as possible and avoids collisions between spacecraft. The methodology presented is a combination of a methodology based on finite elements (which gives better results in terms of fuel consumption) and an analytical methodology (which is much faster in terms of computing time). Joint work with Luke Sauter and Phil Palmer from Surrey Space Centre17 de maig de 2011Sara CostaCoexistència d'un conjunt innombrable de conjunts atractors en el cilindreCoexistence of uncountable many attracting sets on the cylinderWe work with two-dimensional skew-products defined on the cylinder from two functions f and p sucht that f is a continuous degree-one circle map, and p satisfies some properties of concavity and monotonicity. We prove that  when f has no periodic points (rotation interval is a singleton), there exists finitely many attracting sets. In contrast, when the rotation interval of any of the lifting of f is non-degenerate, we prove the existence of uncountably many attracting sets, each of them related with an irrational rotation number. Joint work wiht Ll. Alsedà from UAB.