AnàlisiCarleson's $\epsilon^2$-conjecture in higher dimensions, the Alt-Caffareli-Friedman formula, and Faber-Krahn inequalities
Resum: In this talk I will report on a recent work with Ian Fleschler and Michele Villa where we extend the $\epsilon^2$-conjecture of Carleson about the characterization of tangent points for Jordan domains in the plane to the higher dimensional setting. The generalization that we obtain does not require any connectivity condition for the domains under consideration, and so it is new even in dimension 2. I will also explain interesting connections with the Alt-Caffarelli-Friedman monotonicity formula and with the so-called Faber-Krahn inequalities that quantify the size of the first eigenvalue of the Dirichlet Laplacian on a given domain. In fact, one of the ingredients of our work is the obtention of new Faber-Krahn inequalities which measure the deviation of a given domain from a ball with the same measure in terms of Newtonian or logarithmic capacities.
Geometria TropicalCossos de Newton-Okounkov II
Resum: Veurem els principals resultats que s'obtenen de la teoria de cossos de Newton-Okounkov sobre el creixement asimptòtic de semigrups: el teorema del volum i els teoremes tipus Fujita.
Models Estocàstics i Deterministes A model of competition through growth reduction
Resum: In this talk I'll present a determinsitic model and a stochastic model of a hierarchically size-structured population. Both models consider that the amount of resources an individual has access to is affected by those individuals that are larger than it, and that the intake of resources by an individual only affects directly the growth rate of the individual (the fertility and the mortality rates of the individual are determined by its size, and hence are affected by the intake of resources indirectly). The talk will focus on the relation between the two models.
Teoria d'AnellsWebbing transformations and C*-algebras
Resum: In the recent light of the emergence of new invariants for non-simple C*-algebras, we expose a categorical construction that we refer to as the webbing transformation, allowing to generically merge distinct C*-invariants together. E.g. the Cuntz semigroup together with K-theoretical data. One of the benefits is to naturally incorporate the data encoded within any (closed two-sided) ideals. In this talk, we will first define our categorical framework and study properties of these webbed objects, including an ideal-quotient theory, to then venture into their possible impact on the classification of non-simple C*-algebras.
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