AnàlisiCaloric capacities of Cantor sets
Resum: The study of removable singularities focuses on characterizing the sets that allow extensions of solutions to partial differential equations (PDEs). In this way, it builds a bridge between this area and the metric–geometric properties of sets that, in some sense, do not pose an obstacle when solving such equations, the so-called removable sets. At this point, capacities play a key role. These functions assign a non-negative value to subsets of the ambient space and satisfy the property that those with zero value are precisely the ones that are removable for the PDE associated with the capacity.
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.
Sistemes DinàmicsClassification of quadratic differential systems with two invariant conics: parabola and ellipse
Resum: We consider the family QS of quadratic polynomial differential systems. According to Darboux's theory of integrability, the existence of invariant algebraic curves for a polynomial differential system aids in determining a first integral of the system. Systematic studies of quadratic differential systems with invariant conics began toward the end of the 20th century and the beginning of the 21st century. However, it is only in the past 15 years, with the application of the theory of algebraic invariants for polynomial differential equations (developed by Sibirschi), that some global classifications of systems in QS possessing invariant conics have been established. Each of these classifications addresses a subfamily of QS systems, which each possess either at least one invariant hyperbola, parabola, or ellipse.
Teoria d'AnellsThe Cuntz semigroup of rings of continuous functions on one-dimensional spaces
Resum: A well known theorem of Dolezal and Sibuya states that given a square matrix A(t) that is a continuous function of the scalar parameter $t \in R$ and whose rank is constant as the parameter varies, one can span the range and kernel of A(t) by linearly independent vectors that are also continuous functions on t. In the talk we will explore further generalizations of this result in which the parameter space R is replaced by a one-dimensional topological space. This will allow us to explore the relation between the Cuntz semigroup of C([0, 1]) viewed as a C*-algebra and as a ring, and also for other rings of continuous functions on one-dimensional spaces. I will also characterize the class of a countably generated projective ideal by its trace ideal. We will see that the situation is different when considering real-valued or complex-valued functions.
Mesh-comparable components of the AR-quiver
Resum: Given a $k$-algebra $A$ (where $k$ is a field), one way of organizing the category mod$A$ of the finitely generated right $A$-modules is through the so-called Auslander-Reiten quiver $\Gamma$(mod$A)$. The vertices of such quiver correspond to the isoclasses of the indecomposable objects in mod$A$ and the arrows indicate the existence of irreducible morphisms between them. Recall that, for $A$-modules $X,Y$, $\mathrm{rad}_A(X,Y)$ denotes the set of non-isomorphisms $X \rightarrow Y$. Clearly, one can extend it to general modules as follows: $\mathrm{rad}_A(\oplus_{i=1}^n X_i,\oplus_{j=1}^m Y_j) = \oplus_{i=1}^n\oplus_{j=1}^m \mathrm{rad}_A(X_i,Y_j)$. Using the fact that $\mathrm{rad}_A$ is an ideal of the category $\textrm{mod} A$, one can consider its powers, defined recursively by: $\mathrm{rad}_A^0 = \mathrm{Hom}_A, \mathrm{rad}_A^1= \mathrm{rad}_A, \mathrm{rad}_A^n = \mathrm{rad}_A^{n-1} \cdot \mathrm{rad}_A$, where the product $\cdot$ stands for composition of morphisms. We also define $\mathrm{rad}_A^{\infty} = \cap_{n \geq 0} \mathrm{rad}_A^n$. A morphism is called irreducible if it belongs to $\mathrm{rad}_A(X,Y) \setminus \mathrm{rad}_A^2(X,Y)$. Irreducible morphisms are of key importance, since, as shown by Auslander-Reiten theory, these morphisms generate any other morphism modulo $\mathrm{rad}^{\infty}$.
Teoria de NombresAutomorphism of quotient modular curves
Resum: The automorphism group of a modular curve plays an important role in studying the points on the curve. In this talk, I will briefly discuss the methods for computing the full automorphism groups of modular curves and quotients of modular curves.
Generalized class groups acting on oriented elliptic curves with level structure
Resum: Certain isogeny-based cryptographic protocols rely on a class group action. We study a large family of generalized class groups of imaginary quadratic orders . We prove that they act freely and (essentially) transitively on the set of primitively O-oriented elliptic curves over a field k equipped with appropriate level structure. This extends, in several ways, a recent observation due to Galbraith, Perrin, and Voloch for the ray class group. We show that this leads to a reinterpretation of the action of the class group of a suborder O’ of O on the set of O’-oriented elliptic curves and discuss several other examples. This is joint work with W. Castryck, J. Komada Eriksen, G. Lorenzon, and F. Vercauteren, with ongoing follow-up work joint with J. Macula and E. Orvis.
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