Categorías (UAB-UMA)Límites y colímites
Resum: En esta (octava) sesión trataremos las nociones categóricas de límite y colímite, vía conos universales, y veremos el caso concreto en la categoría de conjuntos. Definiciones esenciales en geometría algebraica como las de producto, producto fibrado, límite, etc. serán discutidas, y como es habitual, contamos con los ejemplos de todos. Comenzamos así el capítulo 3 de nuestra referencia habitual, Category Theory in Context (Riehl, Emily).
Propiedades universales y la categoría de elementos.
Resum: En esta séptima sesión concluiremos el segundo capítulo de la referencia habitual Category Theory in Context (Emily Riehl), tratando explícitamente (1) las propiedades universales y la (2) categoría de elementos. La ubicuidad de (1) es evidente para todos, de modo que os animamos a participar proporcionando ejemplos (si el tiempo lo permite, podríamos analizar las definiciones de "producto", "producto fibrado" y "límite", y muchas otras menos básicas que propongáis). Respecto de (2), intentaremos utilizarla para dar una respuesta a la pregunta de Sergio (dada una categoría localmente pequeña $C$ y un funtor $F : C \rightarrow Set$ no representable, ¿podemos asegurar que la categoría $\mathcal{C}_F$ no tiene objetos iniciales?), en particular comentando la solución que nos ha hecho llegar Simone (gracias de nuevo por la participación).
EstadísticaFunctional Additive Models on Manifolds of Planar Shapes and Forms
Resum: In many imaging data problems, the coordinate system of recorded objects is arbitrary or explicitly not of interest. Statistical shape analysis addresses this by identifying the ultimate object of analysis as the „shape” of an observation, i.e., its equivalence class modulo translation, rotation and rescaling, or as its „form” (or „size-and-shape“) modulo translation and rotation. The shape/form space of this equivalence class is endowed with a Riemannian manifold geometry, which needs to be considered in the analysis. We introduce a flexible additive regression framework for modeling the shape or form of planar landmark configurations and/or (potentially irregularly sampled) curves in dependence on scalar covariates. Models are fit by a novel component-wise Riemannian L2-Boosting algorithm, which yields desirable means of regularization for high-dimensional scenarios and allows estimation of a large number of parameter-intense model terms with inherent model selection. The framework is illustrated in an analysis of configurations of 2D landmarks and outline segments describing the shape of astragali (ankle bones) of wild and domesticated sheep, taking also other "demographic" variables into account. Graphic illustration usually plays an essential role in practical interpretation of (potentially nonlinear) additive model effects but becomes a challenging task when the response presents an (equivalence classes of) planar curves or landmark configurations. Therefore, we also suggest a novel visualization for multidimensional functional regression models. Analogous to principal component analysis often used for visualization of functional data, a suitable tensor-product factorization identifies an optimal intrinsic coordinate system for a covariate effect. After factorization, main effect directions can be illustrated on the level of curves, while the effect into the respective direction is visualized by standard effect plots for scalar additive models. Finally, also a brief outlook to a broader context is given, addressing the problem of additional reparameterization (warping) invariance that might be required when shapes of curves rather than landmark configurations are investigated.
GeometríaComptant corbes sobre el tor punxat amb longitud de paraula acotada.
Resum: Parlarem sobre el problema de trobar una fórmula tancada pel nombre de corbes tancades sobre el tor punxat (superfície de gènere 1 sense un punt) amb longitud de paraula (nombre de lletres necessàries per representar la corba al grup fonamental) i auto-intersecció donades; presentant una caracterització de totes les paraules que representen corbes amb auto-intersecció 1 (anàleg al cas ja sabut per corbes simples) i donant un mètode per trobar la fórmula quan la caracterització és sabuda. Després discutirem com traslladar aquests resultats a la mateixa superfície amb una mètrica hiperbòlica i com es podrien derivar resultats sobre conjectures obertes. Aquesta xerrada es basa en feina conjunta amb el Mingkun Liu.
Descomposició de $3$-varietats de curvatura escalar positiva amb decreixement subquadràtic.
Resum: Una qüestió central en l'estudi de les varietats de dimensió 3 consisteix a comprendre l'estructura topològica de les 3-varietats que admeten una mètrica riemanniana completa de curvatura escalar positiva, conegudes com a varietats PSC. A les darreries dels anys setanta, els resultats obtinguts per Schoen i Yau utilitzant la teoria de superfícies minimals i, paral·lelament, els mètodes basats en la teoria de l'índex desenvolupats per Gromov i Lawson permeteren classificar les 3-varietats PSC tancades i orientables: són exactament aquelles que es descomponen en suma connexa de varietats esfèriques i de productes S2xS1.
Modelos Estocásticos y DeterministasTwo sided ergodic singular control and mean-field game for diffusions
Resum: In a probabilistic mean-field game driven by a linear diffusion an individual player aims to minimize an ergodic long-run cost by controlling the diffusion through a pair of –increasing and decreasing– cadlag processes, while he is interacting with an aggregate of players through the expectation of a similar diffusion controlled by another pair of cadlag processes. In order to find equilibrium points in this game, we first consider the control problem, in which the individual player has no interaction with the aggregate of players. In this case, we prove that the best policy is to reflect the diffusion process within two thresholds. Based on these results, we obtain criteria for the existence of equilibrium points in the mean-field game in the case when the controls of the aggregate of players are of reflection type, and give a pair of nonlinear equations to find these equilibrium points. In addition, we present an approximation result for Nash equilibria of ergodic mean-field games with a finite number of players to the mean-field game equilibria considered above when the number of players tends to infinity. These results are illustrated by several examples where the existence and uniqueness of the equilibrium points depend on the coefficients of the underlying diffusion.
Sistemas DinámicosOn a variant of Hilbert's 16th problem
Resum: We study the number of limit cycles that a planar polynomial vector field can have as a function of its number $m$ of monomials. We prove that the number of limit cycles increases at least quadratically with $m$ and we provide good lower bounds for $m\le10$. Coauthored with Armengol Gasull.
Mandelbrot set $\mathcal{M}_n$ associated to self-affine tiles and zeros of power series with integer coefficients
Resum: In this talk (joint work with D. Juher, and J. Saldaña) we introduce a connected and locally connected Mandelbrot set $\mathcal{M}_n$ characterized by the set of zeros of power series with constant term $1$ and the remaining coefficients in $\{-n+1,-n+2,\dots,n-2,n-1\}$. Our main result identifies a stability principle governing the interior of $\mathcal{M}_n$ that allows us to prove that the interior of $\mathcal{M}_n$ is dense away from $\mathcal{M}_n\cap \mathbb{R}$. The work also establishes a deep connection between $\mathcal{M}_n$ and the zero-free domain of the set of power series with coefficients in $[-n+1,n-1]$ with constant term $1$. Exotic self-similar sets found in $\mathcal{M}_n$ include the family of self-affine tiles with a collinear digit set.
Teoría de AnillosThe Cuntz Semigroup of $C([0, 1])$ viewed as a C*-algebra and as a ring
Resum: Very recently, a new invariant has been defined for any ring $R$, denoted by S$(R)$ and called the Cuntz semigroup of the ring $R$, a partially ordered abelian semigroup built from an equivalence relation on the class of countably generated projective modules. In the talk I will 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 locally compact second countable one-dimensional Hausdorff spaces. This results will allow us to study the trace ideals of countably generated ideals of these rings. We will see that the situation is different when considering real-valued or complex-valued functions.
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