AnàlisiQuantitative Unique Continuation for the Neumann Problem in Planar $C^{1,\alpha}$ Domains
Resum: In this paper, we study the quantitative unique continuation property of second-order elliptic operators under vanishing Neumann boundary conditions over $C^{1,\alpha}$ domains in two dimensions. Combining the monotonicity formula with the tool of quasiconformal mapping in two dimensions, we establish an almost sharp upper bound for the doubling index, which further implies an explicit bound on the size of level sets. As byproducts, vanishing orders, critical points, Cauchy uniqueness, and related topics are also discussed.
GeometriaHolomorphic geometric structures on Hopf manifolds.
Resum: Complex Hopf manifolds were the first known examples of compact non-Kähler complex manifolds. Their study is closely related to a classical result in holomorphic dynamics: the Poincaré–Dulac theorem. This theorem provides geometric information on Hopf manifolds, in particular the existence of holomorphic geometric structures. The aim of this talk is to present a kind of converse. I will explain how to construct holomorphic geometric structures on Hopf manifolds, such as G-structures or Cartan connections, without using the Poincaré–Dulac theorem. I will also show how the existence of these structures yields a new proof of the latter. These results extend recent work of Ornea and Verbitsky, obtained in the non-resonant case.
On rigidity of Poisson homeomorphisms
Resum: Symplectic topology/geometry is well known for being a huge source of interesting interactions between flexible, i.e. topological, and rigid, i.e. geometric, phenomena. Diffeomorphisms preserving symplectic structures, also called symplectomorphisms, are in particular an interesting class of diffeomorphisms as they exist in abundance, containing for instance all the time-1 Hamiltonian flows, but they are also more rigid than volume preserving diffeomorphisms, due e.g. to Gromov's non-squeezing theorem. Recently there has been a rising interest in the study of symplectic homeomorphisms, i.e. of homeomorphisms that are uniform C^0-limits of symplectomorphisms: indeed, they are in certain aspects more flexible than their smooth counterparts, while preserving some of their rigid features. In this talk I will report on a joint work in progress with Robert Cardona, where we study analogous C^0 questions in the more general context of Poisson geometry. Namely, I will discuss to what extent known results about symplectic homeomorphisms generalize to Poisson homeomorphisms, i.e. homeomorphisms that are uniform C^0-limits of Poisson diffeomorphisms.
Geometria TropicalLocalització i feixos II
Resum: Seguirem explorant la relació entre la localització i la propietat de feix.
Localització 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 Inclusions of operator algebras from tensor categories
Resum: The works of V. Jones and A. Ocneanu suggested the interpretation of subfactors as quantum symmetries. There is indeed a group-like object one can extract from a given subfactor, called the standard invariant, nowadays described as a W*-tensor category. In this talk, I will summarize this story, and report on my joint work with R. Hernández Palomares, where we study C*-algebraic inclusions through the subfactor philosophy, motivated by a topological quantum field theoretic construction called factorization homology.
Singularity categories of path algebras over quasi-Frobenius rings
Resum: The singularity category of a ring, introduced by Buchweitz in 86, is an important invariant that describes stable homological properties of R. In this talk we will look at the singularity category of the path algebra RQ of a finite acyclic quiver Q over a quasi-Frobenius ring R. We will first recall how it is related to the monomorphism category of RQ. Then we will present some new tools aimed at determining its indecomposables and its Auslander—Reiten quiver. In particular, we will give applications to cyclic abelian groups and truncated polynomial rings. This is joint work with Nan Gao, Julian Külshammer and Chrysostomos Psaroudakis.
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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