Armentano, D., Bentancur, L., Carrasco, F., Fiori, M., Valdés, M., & Velasco, M. (2025). Characterization of logarithmic Fekete critical configurations of at most six points in all dimensions. arXiv. https://doi.org/10.48550/arXiv.2502.10152
My interst areas are the following:
The Christoffel funcion on the sphere
The Christoffel function and the Christoffel-Darboux (CD) kernel are fundamental tools in approximation theory and orthogonal polynomials. They are useful because they efficiently represent certain properties of measures, and they have recently been applied to data analysis problems such as support inference, anomaly detection, and density approximation. These methods are used in areas like probability, control theory, mathematical physics, and machine learning.
One important property of the Christoffel function is that it has an explicit rational form based on the inverse of the moment matrix of the measure under study. The elements of this matrix can be estimated from data, allowing for efficient computations in moderate dimensions.
Recently, Lasserre proposed a modification of the Christoffel function that makes it possible to recover the density of the measure without knowing the equilibrium measure of its support, which is only known in simple cases. Inspired by this idea, together with my advisor Mauricio Velasco, we are exploring other regularized versions of the Christoffel function on the sphere and studying convergence rates for density recovery. Additionally, in collaboration with Ernesto García, we are analyzing applications of this method to obtain densities of quasi-stationary distributions for certain stochastic processes, a problem of interest in reinforcement learning.
Polyhedral hierarchies for polynomial optimization in homogeneous varieties
The study of polynomial non-negativity and its relation to the sum of squares is a classical theme in algebraic geometry. This area underwent significant changes in the early 2000s, primarily due to the contributions of Lasserre and Parrilo, who introduced the use of semidefinite programming to search for certificates of non-negativity. This approach offered an effective and efficient solution to polynomial optimization problems. Hierarchies are constructed using sums of squares of polynomials of increasing degrees to approximate the set of non-negative polynomials, with representation theory used to describe the invariants of actions on these spaces.
In a recent paper, Cristancho and Velasco constructed new polyhedral hierarchies for approximating the cone of non-negative polynomials on the unit sphere of arbitrary dimension. Additionally, computable bounds for the convergence rate of this method were proved.
The objective of my PhD thesis is to construct convergent polyhedral hierarchies for the cone of positive polynomials on the Grassmannian variety in the first case, and then on general homogeneous algebraic varieties. To achieve this, quadrature rules will be constructed on the variety, and the decomposition into invariants of the polynomial ring given by the action of the orthogonal group will be studied.
Fekete problem
The Fekete problem consists of finding x1,…,xNx_1, \dots, x_Nx1,…,xN points on the 2-sphere that maximize the product of their distances, given a natural number NNN. In other words, it seeks configurations of points on the sphere that are as far apart from each other as possible. This can be easily rewritten as the problem of maximizing the logarithmic energy of a configuration, which is the sum of the logarithms of the distances between all pairs of points. Smale's 7th problem in his list of problems for the 21st century is to construct point configurations whose logarithmic energy is within a quantity of order log(N)\log(N)log(N) of the minimum logarithmic energy value, that is, the value in the point configurations that solve the problem. Specifically, solutions to the problem are only known for N=2,3,4,5,6,N = 2, 3, 4, 5, 6,N=2,3,4,5,6, and 121212. This problem remains genuinely open, presenting various related questions, such as finding exact solutions for small numbers of points, constructing low-energy point configurations, improving the asymptotic estimates of the energy value, proving the abundance of critical points, among others.
Together with Diego Armentano, Federico Carrasco, Marcelo Fiori, Pedro Raigorodsky, Matías Valdés, and Mauricio Velasco, we are working on certain aspects of this problem. When writing the equations that the critical points of the problem must satisfy, which involve setting the Lagrangian to zero, along with the constraints that the points lie on the sphere and are distinct, we found that the critical points form an algebraic variety. From this, we can work on the ideal generated by these equations, using computational algebraic geometry tools such as Gröbner bases, to count the number of critical points and classify them for some small values of NNN.
Representation theory of affine extensions of abelian varieties
My master thesis objective was to explore the fundamental properties of representation theory concerning affine extensions of abelian varieties. This theory is presented as a generalization of the representation theory of schemes in affine groups. An affine extension S of an abelian variety A by an affine group scheme H is a short exact sequence of group schemes 1 \to H \to G \to A \to 0. A representation of S is an action of G on a homogeneous vector bundle E over A such that if q(g)=a, then the action by g takes the fiber over b to the fiber over a+b, so that the corresponding morphism is a linear transformation. We present the construction of this representation theory of S and the proof of a theorem of the "Tannaka duality'' type recently developed by Rittatore, del Ángel, and Ferrer. We study basic properties of this theory, such as the characterization of semisimplicity and the unipotent case, obtaining results that link these cases with classical representation theory for the affine case.