Pablo Angulo en la Universidad Politécnica de Madrid

arXiv TAMS We analyze the structure of the set of limiting Carleman weights in all conformally flat manifolds, 3-manifolds, and 4-manifolds. In particular we give a new proof of the classification of Euclidean limiting Carleman weights, and show that there are only three basic such weights up to the action of the conformal group. In dimension three we show that if the manifold is not conformally flat, there could be one or two limiting Carleman weights. We also characterize the metrics that have more than one limiting Carleman weight. In dimension four we obtain a complete spectrum of examples according to the structure of the Weyl tensor. In particular, we construct unimodular Lie groups whose Weyl or Cotton-York tensors have the symmetries of conformally transversally anisotropic manifolds, but which do not admit limiting Carleman weights.

arXiv Rank-one convexity is a weak form of convexity related to convex integration and the elusive notion of quasiconvexity, but more amenable both in theory and practice. However, exact algorithms for computing the rank one convex hull of a finite set are only known for some special cases of separate convexity with a finite number of directions. Both inner approximations either with laminates or \(T_4\)'s and outer approximations through polyconvexity are known to be insufficient in general. We study \(\mathbb{R}^2 \oplus \mathbb{R}\)-separately convex hulls of finite sets, which is a special case of rank-one convexity with infinitely many directions in which \(T_4\)'s are known not to capture the rank one convex hull. When \(\mathbb{R}^3\) is identified with a subset of \(2\times 3\) matrices, it is known to correspond also to quasiconvexity. We propose new inner and outer approximations built upon systematic use of known results, and prove that they agree. The inner approximation allows to understand better the structure of the rank one convex hull. The outer approximation gives rise to a computational algorithm which, in some cases, computes the hull exactly, and in general builds a sequence that converges to the hull. We use and systematize all previous attempts at computing D-convex hulls, and bring new ideas that may help compute general D-convex hulls.

Contributed talk at MAF2018 Book chapter at MAF2018 proceedings MAF2018 short paper Contributed talk at EURO2018 When considering different allocations of the marketing budget of a firm, some predictions, that correspond to scenarios similar to others observed in the past, can be made with more confidence than others, that correspond to more innovative strategies. Selecting a few relevant features of the predicted probability distribution leads to a multi-objective optimization problem, and the Pareto front contains the most interesting media plans. Using expected return and standard deviation we get the familiar two moment decision model, but other problem specific additional ob- jectives can be incorporated. The Factorization Machine kernel, initially introduced for recommendation systems, but later used also for regression, is a good choice for incorporating interaction terms into the model, since they can effectively exploit the sparse nature of typical datasets found in econometrics.

arXiv ASMB We propose a robust implementation of the Nerlove--Arrow model using a Bayesian structural time series model to explain the relationship between advertising expenditures of a country-wide fast-food franchise network with its weekly sales. Thanks to the flexibility and modularity of the model, it is well suited to generalization to other markets or situations. Its Bayesian nature facilitates incorporating \emph{a priori} information (the manager's views), which can be updated with relevant data. This aspect of the model will be used to present a strategy of budget scheduling across time and channels.

arXiv Forum of mathematics - Sigma In 1411.4887, we found some necessary conditions for a Riemannian manifold to admit a local limiting Carleman weight (LCW), based upon the Cotton-York tensor in dimension \(3\) and the Weyl tensor in dimension \(4\). In this paper, we find further necessary conditions for the existence of local LCWs that are often sufficient. For a manifold of dimension \(3\) or \(4\), we classify the possible Cotton-York, or Weyl tensors, and provide a mechanism to find out whether the manifold admits local LCW for each type of tensor. In particular, we show that a product of two surfaces admits a LCW if and only if at least one of the two surfaces is of revolution. This provides an example of a manifold satisfying the eigenflag condition of \cite{AFGR} but not admitting \(LCW\).

arXiv DCDS We present a new strategy for proving the Ambrose conjecture, a global version of the Cartan local lemma. We introduce the concepts of linking curves, unequivocal sets and sutured manifolds, and use them to show that any sutured manifold satisfies the Ambrose conjecture. We then prove that the set of sutured Riemmanian manifolds contains a residual set of the metrics on a given smooth manifold of dimension \(3\).

arXiv Inverse Problems and Imaging In the paper AFGR it is shown that the set of Riemannian metrics which do not admit global limiting Carleman weights is open and dense, by studying the conformally invariant Weyl and Cotton tensors. In the paper LS it is shown that the set of Riemannian metrics which do not admit local limiting Carleman weights at any point is residual, showing that it contains the set of metrics for which there are no local conformal diffeomorphisms between any distinct open subsets. This note is a continuation of AFGR, in order to prove that the set of Riemannian metrics which do not admit local limiting Carleman weights \emph{at any point} is open and dense.

arXiv APDE We give a necessary condition for a Riemannian manifold to admit limiting Carleman weights in terms of its Weyl tensor (in dimensions 4 and higher), or its Cotton-York tensor in dimension 3. As an application we provide explicit examples of manifolds without limiting Carleman weights and show that the set of such metrics on a given manifold contains an open and dense set.

arXiv Entrada en Teseo The goal of this thesis is to study the singularities of the exponential map of \emph{Riemannian and Finsler manifolds} (a concept related to caustics and catastrophes), and the object known as the cut locus (aka ridge, medial axis or skeleton, with applications to differential geometry, control theory, statistics, image processing...), to improve existing results about its structure, to look at it in new ways, and to derive applications to the Ambrose conjecture and the Hamilton-Jacobi equations.

arXiv Arch. Mat. In Wil, B. Wilking introduced the dual foliation associated to a metric foliation in a Riemannian manifold with nonnegative sectional curvature, and proved that when the curvature is strictly positive, the dual foliation contains a single leaf, so that any two points in the ambient space can be joined by a horizontal curve. We show that the same phenomenon often occurs for Riemannian submersions from nonnegatively curved spaces even without the strict positive curvature assumption, and irrespective of the particular metric.

arXiv Calc. Var. PDE We study the singular set of solutions to Hamilton-Jacobi equations with a Hamiltonian independent of \(u\). In a previous paper, we proved that the singular set is what we called a balanced split locus. In this paper, we find and classify all balanced split loci, identifying the cases where the only balanced split locus is the singular locus, and the cases where this does not hold. This clarifies the relationship between viscosity solutions and the classical approach of characteristics, providing equations for the singular set. Along the way, we prove more structure results about the singular sets.

arXiv Ann. Inst We give a new and detailed description of the structure of cut loci, with direct applications to the singular sets of some Hamilton-Jacobi equations. These sets may be non-triangulable, but a local description at all points except for a set of Hausdorff dimension \(n-2\) is well known. We go further in this direction by giving a classification of all points up to a set of Hausdorff dimension \(n-3\).