AnàlisiQuadratic spectral concentration of characteristic functions
Resum: A theorem of Donoho and Stark states that decreasing rearrangement increases the quadratic spectral concentration of a square integrable function supported on a sufficiently small set. Importantly, their condition on the smallness of the support turns out to be necessary. In this talk, we restrict ourselves to considering only characteristic functions and, in this setting, we are able to relax the condition of Donoho and Stark. We also discuss various properties of the sets of fixed measure maximizing the quadratic spectral concentration of their characteristic functions. As a corollary, we obtain a sharp (up to a constant) estimate for the $L^2$-norms of non-harmonic trigonometric polynomials with alternating coefficients $\pm 1$.
Gradients of single layer potentials for elliptic operators with coefficients of DMO-type and applications to elliptic measure
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Geometria TropicalUna proposta de feix estructural per l'espectre de congruències primeres d'un semianell commutatiu.
Resum: Explicaré com es podria dotar d'un feix "natural" a l'espectre de congruències primeres (com a espai topològic) d'un semianell commutatiu, introduïdes per Joó i Mincheva (en el cas idempotent) el 2017. Primer repassaré la seva definició i alguna de les seves propietats, per introduir seguidament la meva proposta de feix, que és vàlida per a qualsevol semianell commutatiu i coincideix amb el feix estructural per l'espectre d'un anell commutatiu sempre 2 sigui invertible.
Models Estocàstics i DeterministesInference for a SPDE related to an ecological niche
Resum: In this talk, we use a stochastic partial differential equation (SPDE) as a model for the density of a population. Indeed, we are interested in modeling animal density under the influence of random external forces/stimuli given by the environment. We want to study statistical properties for two crucial parameters of the SPDE that describe the dynamic of the system. To do that we use the Galerkin projection to transform the problem, passing from the SPDE to a system of independent SDEs; in this manner, we are able to find the Maximum likelihood estimator of the parameters. We validate the method by using simulations of the SPDE. We show consistency and asymptotic normality of the estimators; the latter it could be proved using the Malliavin-Stein method. These would allow us to fit the model to actual data.
Sistemes DinàmicsA sufficient condition in order that the real Jacobian conjecture in the plane holds.
Resum: Let $F$ be a polynomial map from the real plane to the real plane with a non-zero Jacobian determinant at any point of the real plane. I will review some sufficient conditions for the injectivity of the map $F$ and, therefore, the validity of the (strong) real Jacobian conjecture. Using tools of the qualitative theory of the differential equations in the plane, I will present a new and simple sufficient condition for the injectivity of $F$. This study was carried out in collaboration with M. Domingues and J. Llibre.
The $\lambda$-Lemma for Homoclinic Tangles of Piecewise Smooth Vector Fields
Resum: In this talk we will provide extensions of the $\lambda$-Lemma (also known as Inclination Lemma) for piecewise smooth vector fields and maps. In order to achieve our main result, we investigate the regularity of time-$T$-maps of piecewise smooth vector fields defined at crossing orbits. We prove that these maps are homeomorphisms and also piecewise smooth diffeomorphisms.
Teoria d'AnellsOn the space of Sylvester rank functions
Resum: Given a ring R, let P(R) be the space of Sylvester rank functions on the category of finitely presented modules mod-R. Recall that P(R) can also be described as the space of normalized additive functions on the Abelianization Ab(R), which is the category of finitely presented objects in the additive functor category [R-mod,Ab]. Compared to mod-R, the category Ab(R) has a much richer structure, which can be used for the study of P(R). In this talk we will discuss two such applications: for the first one we will describe the space $P_{ex}(R)$ of exact rank functions in terms of effaceable functors, while for the second we will describe the irreducible discrete rank functions using the topology of the Ziegler spectrum. As a concrete application, we will give a complete description of the space P(R) when R is a Dedekind domain with a polynomial identity
A Glimpse into Quivers and Identities
Resum: A polynomial identity for an algebra $A$ is a polynomial in non-commutative variables that vanishes under all evaluations in $A$. Algebras satisfying at least one such non-trivial identity are known as PI-algebras.
C*-àlgebres sobre el cercle: classificació i limitacions
Resum: Primer, repassem el programa tradicional de classificació d’Elliott per a les AT-àlgebres, és a dir, C*-àlgebres obtingudes com a límits inductius de 'building blocks' sobre C(T). Després de recordar els primers resultats satisfactoris tant en el cas simple com en el de rang real zero, passem a desenvolupaments més recents que involucren el semigrup de Cuntz i les seves versions refinades. En particular, discutim la classificació dels *-homomorfismes de C(T) a varios codominis, i mostrem las limitacions actuals que trobem per obtenir la classificacio completa d'aquestes C*-àlgebres.
Projective modules over locally semiperfect algebras with semisimple artinian ring of quotients
Resum: Let $\Lambda$ be a locally semiperfect module-finite algebra with semisimple artinian ring of quotients over a noetherian domain of Krull dimension 1. Such algebras arise naturally in representation theory of finite groups and, more generally, in the theory of maximal orders. In this context, every countably generated projective module can be realized as the lifting of a finitely generated projective module modulo an idempotent ideal $I$. We study the structure of $\Lambda/I$, which is itself locally semiperfect, and provide an equational description of the finitely generated projective modules over $\Lambda/I$, as well as the countably generated projective modules over $\Lambda$, in the case when $\Lambda$ is the group algebra of a finite group.
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