Geometria TropicalLocalització i feixos
Resum: Analitzaré per quines localitzacions es compleix la propietat de feix, per a varies categories de semianells.
Models Estocàstics i DeterministesMathematical Models for Understanding and Managing Biological Invasions
Resum: Invasive species pose major challenges for biodiversity and ecosystem management. In this talk, I will present recent mathematical approaches that help explain how invasions emerge, spread, and can be controlled.
Diffusion limit for Markovian models of evolution in structured populations with migration
Resum: The evolution of microbial subpopulations that migrate within spatial structures has gained interest in recent years. Questions of relevance include, for instance, the ability of a migrant mutant to take over the population (fixate). Estimating fixation probabilities is, however, usually hindered by the lack of analytical formulas and by computational complexity of simulation-based strategies when considering large populations. In this work, we study several population genetics models where the population is divided into $D$ subpopulations (called demes) consisting of two types of individuals, mutants and wild-types, that evolve through discrete Markovian updates. We prove that under certain assumptions all the considered models converge to the same diffusion approximation, which we call \textit{universal}. This diffusion approximation is amenable to simulation strategies that underly methods of statistical inference while significantly reducing computational costs. In all models, each Markovian update follows two phases: First, a local growth phase in each subpopulation, where the growth of each type of individual depends on its fitness, and then a sampling phase that implements migration between subpopulations. Our proof relies on existing diffusion approximation results for degenerate diffusions, see [1], but requires further technicalities due to fact that sample sizes in each deem are not necessarily fixed but change randomly with each update.
Teoria d'Anells
Pure C*-algebras and extensions
Resum: In this talk I will review how the concept of pureness (that is, almost unperforation and almost divisibility) is equivalent to the combination of a weak comparison property and what has been termed functional divisibility. I will discuss how said comparison property suffices to show pureness for simple nonelementary, unital C*-algebras with a unique quasitracial state. I will also consider permanence properties of pureness, namely its behaviour under extensions.
Pure C*-algebras and extensions
Resum: In this talk I will review how the concept of pureness (that is, almost unperforation and almost divisibility) is equivalent to the combination of a weak comparison property and what has been termed functional divisibility. I will discuss how said comparison property suffices to show pureness for simple nonelementary, unital C*-algebras with a unique quasitracial state. I will also consider permanence properties of pureness, namely its behaviour under extensions.
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