## Research

### Research groups

### Papers and preprints

**Pablo Angulo, Daniel Faraco, Luis Guijarro, Mikko Salo:**

*Limiting Carleman weights and conformally transversally anisotropic manifolds*

arXiv
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.

**Pablo Angulo, Daniel Faraco, Carlos García-Gutiérrez**

*Exact computation of the 2+1 convex hull of a finite set*

arXiv
We present an algorithm to exactly calculate the
\(\mathbb{R}^2\oplus\mathbb{R}\)-separately convex hull of a finite set of points
in \(\mathbb{R}^3\), as introduced in \cite{KirchheimMullerSverak2003}. When
\(\mathbb{R}^3\) is considered as certain subset of \(3\times 2\) matrices, this
algorithm calculates the rank-one convex hull. If \(\mathbb{R}^3\) is identified
instead with a subset of \(2\times 3\) matrices, the algorithm is actually
calculating the quasiconvex hull, due to a recent result by
\cite{HarrisKirchheimLin18}.

The algorithm combines outer approximations based in the locality theorem \cite[4.7]{Kirchheim2003} with inner approximations to $2+1$ convexity based on "$(2+1)$-complexes". The departing point is an outer approximation and by iteratively chopping off "$D$-prisms", we prove that an inner approximation to the rank-one convex hull is reached.

The algorithm combines outer approximations based in the locality theorem \cite[4.7]{Kirchheim2003} with inner approximations to $2+1$ convexity based on "$(2+1)$-complexes". The departing point is an outer approximation and by iteratively chopping off "$D$-prisms", we prove that an inner approximation to the rank-one convex hull is reached.

**Pablo Angulo, Víctor Gallego, David Gómez-Ullate, Pablo Suárez-García**

*Bayesian Factorization Machines for Risk Management and Robust Decision Making*

Contributed talk at MAF2018
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.

**Víctor Gallego, Pablo Suárez-García, Pablo Angulo, David Gómez-Ullate**

*Assessing the effect of advertising expenditures upon sales: a Bayesian structural time series model*

arXiv
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.

**Pablo Angulo, Daniel Faraco, Luis Guijarro:**

*Sufficient conditions for the existence of limiting Carleman weights*

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
$LCW$.

**Pablo Angulo:**

*Linking curves, sutured manifolds and the Ambrose conjecture for generic 3-manifolds*

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 \(3\).**Pablo Angulo:**

*On the set of metrics without local limiting Carleman weights*

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.

**Pablo Angulo, Daniel Faraco, Luis Guijarro and Alberto Ruiz:**

*Obstructions to the existence of limiting Carleman weights*

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.

**Pablo Angulo:**

*Cut and conjugate points of the exponential map, with applications*

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**.**Pablo Angulo, Luis Guijarro, Gerard Walschap**

*Twisted submersions in nonnegative sectional curvature*

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.**Pablo Angulo, Luis Guijarro**

*Balanced split sets and Hamilton-Jacobi equations*

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.

**Pablo Angulo, Luis Guijarro**

*Cut and singular loci up to codimension 3*

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 \(n-3\).

### Talks

- Bayesian Regression with Factorization Machines for Risk Management and Robust Decision Making at MAF2018 (pdf)
- "Necessary, and often sufficient, conditions for the existence of Conformal Factorizations" at the 7th ECM at Berlin (pdf)
- The Ambrose conjecture at the 7th ECM at Berlin (pdf)
- Optimal budget allocation in media campaigns (pdf)
- Presentación de mi tesis doctoral (pdf)
- The singular sets of the solutions to the Hamilton-Jacobi equations ( pdf)