AnàlisiDesingularization of leapfrogging motion and time periodic vortices in bounded domains
Resum: In this talk I will discuss some aspects of the vortex motion for Euler equations in the planar case. In the first part, I will review some results on the desingularization of time periodic point vortices in the rigid and non rigid cases. We will in particular focus on the rigorous derivation of a leapfrogging motion with four vortices. In the second part, I will analyze how to use KAM techniques to desingularize time periodic orbits in general domains.
Entropy solutions to macroscopic IPM
Resum: The IPM system models the motion of an incompressible and viscous fluid in a porous media. The case in which the density of this fluid takes just two values separated by an interface is known as the Muskat problem. When the part of the fluid with the bigger density is always above the other part, the equation of the evolution of the interface is ill-posed. In this situation some "mixing solutions" arise in which the interface disappears and is replaced by a strip where the densities mix. We will review what is known about the existence and (lack) uniqueness of this kind of solutions and introduce the concept of subsolution. Finally, we will study the existence of some mixing solutions which came from a scheme proposed by F. Otto to select a subsolution.
Preimages of points under series of iterates of finite Blaschke products
Resum: Consider a finite Blaschke product $f,$ let $\{a_n\}$ be a sequence of complex numbers tending to zero which is not absolutely summable and formally denote
Categories (UAB-UMA)Categorías y funtores.
Resum: En esta sesión trataremos las tres primeras subsecciones (1.1, 1.2, 1.3) de Category Theory in Context (Emily Riehl). Nos centraremos en las definiciones básicas y en hacer los ejercicios 1.1.i, 1.1.ii, 1.1.iii (i), 1.2.ii, 1.2.iii, 1.2.iv, 1.2.v, 1.3.i, 1.3.ii, 1.3.iii, 1.3.viii, 1.3.ix, 1.3.x (o un subconjunto). Animamos a todos los participantes a enriquecer la charla con ejemplos concretos, tratados en el libro o no, que ilustren la presencia de las categorías en las distintas ramas de la matemáticas. También, para aquellos que tengan tiempo y ganas, hemos preparado un Reto (podéis encontrarlo en la página web, en el apartado "Calendario e índice", en el desplegable de la primera sesión), que tocaremos al final. En la web hay más detalles de todo.
GeometriaGeneralized Hilbert metrics
Resum: Hilbert metrics on convex sets of the euclidean space give a wealth of interesting metric spaces and have been used for example in the study of representations of surface groups.
Geometria TropicalDescent for modules, algebras and schemes II
Resum: I will review and reinterpret the faithful flat descent (for modules) I introduced in the previous talk, and I will consider also the case of algebras and schemes. We will relate this to Zariski descent and to Galois descent (which I will explain also).
Descent for modules, algebras and schemes
Resum: I will explain the basics for Grothendieck faithfully flat descent, which answers the following questions: given a (commutative) ring homomorphism from A to A', what natural conditions we need for the morphism to get that the tensor product is faithful? (answer: faithfully flat), what datum we need to add to get a full functor? (answer: covering datum), what it is the essential image? (answer: the ones verifying descent/cocycle conditions). I hope all this will be done for the category of modules, for the category of algebras, and for the category of schemes (maybe not in the first talk). Comparison with Galois descent, and some other special cases, can be also considered in the future.
Descent à la Milne-Weil: the case of vector spaces
Resum: In this lecture we will analyze the theory of descent for the category of vector spaces. This simplified setting will allow us to better understand some of the key features of the problem.
Descent for complex varieties (à la Milne-Weil)
Resum: Previously, we introduced the problem of determining when a complex manifold is the analytification of a variety defined over a number field. We discussed specific cases such as Belyi's theorem for Riemann surfaces and the algebraicity of complex tori with real multiplication.
Sistemes DinàmicsThe monotonicity of the period function and its application
Resum: In this talk, we will study the monotonicity of the period function of planar systems. Firstly, we will show that quadratic generalized Lotka-Volterra systems without complex invariant lines have the monotone period functions, which can be viewed as a step to solve a conjecture proposed by Chicone. Secondly, we will show that the method to prove the monotonicity of the period function can be used to deal with orbital stability of smooth solitary waves for the b-family of Camassa–Holm equations. In fact, we will prove that for $b\geq 1$, any smooth solitary wave is orbital stable, which gives a positive answer to the open problem proposed by Lafortune
Dulac's Theorem and asymptotics for return maps of polycycles
Resum: In this talk based on the preprint 'On the monograph "Finiteness Theorems for limit cycles" and a special case of alternant cycles' we will give a rudimentary introduction to the Theorem of Dulac and reduce the problem to the study of return maps of polycycles. Then we will discuss the rather subtle property of nonoscillation in this context and relate this to giving meaning to asymptotic series 'with multiple levels'.
Slow passage through PWL bifurcations: Loss of stability bifurcations.
Resum: In this talk we shed light on a problem of widespread interest in the context of slow fast dynamics, namely the phenomenon of slow passage through a bifurcation. Roughly speaking, this phenomenon consists of describing the behaviour of the flow of a one parametric family when we slowly vary the parameter and make it pass through a bifurcation value. In particular, when the bifurcation consists in the loss of stability of an equilibrium point, the orbits evolving close to a stable slow manifold are not immediately repelled when passing the bifurcation point, but continue close to a unstable slow manifold for some time, thus causing a delay in the loss of stability. In particular, in this talk we analyze the slow passage phenomenon through a PWL transcritical bifurcation with especial emphasys in the delay in the loss of stability. As a result, we describe the parameter regions at which the maximal delay behaves: (i) in a trivial way (tending to zero with the parameter); (ii) in a singular way (tending to infinity with the parameter); (iii) and in a transitional way. Furthermore, we apply this knowledge to a PWL system combining symmetrically two transcritical systems in such a way it shows an unbounded canard explosion ending at a enhanced delay phenomenon.
TopologiaCategorical Logic for the Working Mathematician
Resum: According to the Curry-Howard-Lambek correspondence, logics, type theories and structured categories are intimately related. In its cleanest form, the relation between the latter two can be realized in terms of an adjunction, given by the internal language functor and the syntactic category functor. One way of understanding this relation is that type theories provide us with a convenient language for proving things about categories by presenting free structures in a particular way. This is the narrative undertaken by Shulman in his draft 'Categorical Logic from a Categorical Point of View' of which I will give an overview. In some sense this can be taken as an answer to the question of why anyone using category theory might also want to learn type theory.
Type Theory vs Category Theory
Resum: According to the Curry-Howard-Lambek correspondence, logics, type theories and structured categories are intimately related. In its cleanest form, the relation between the latter two can be realized in terms of an adjunction, given by the internal language functor and the syntactic category functor. One way of understanding this relation is that type theories provide us with a convenient language for presenting free structured categories. This is the narrative undertaken by Shulman in his draft 'Categorical Logic from a Categorical Point of View' of which I could give an overview. In some sense this approach can be taken as answer the question of why anyone using category theory might also want to learn type theory.
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