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.
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'AnellsMesh-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 NombresGeneralized 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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