Bentancur, L., Henrion, D., & Velasco, M. (2026). Mollified Christoffel–Darboux Kernels and Density Recovery on Varieties. arXiv preprint arXiv:2603.09462. Submitted for publication.
Armentano, D., Bentancur, L., Carrasco, F., Fiori, M., Valdés, M., & Velasco, M. (2026). Characterization of logarithmic Fekete critical configurations of at most six points in all dimensions. Journal of Symbolic Computation, 137, 102570. https://doi.org/10.1016/j.jsc.2026.102570
My interst areas are the following:
Christoffel–Darboux Kernels, Mollification, and Density Recovery
The Christoffel–Darboux (CD) function, derived from the CD kernel, is a central tool in approximation theory, orthogonal polynomials, and moment-based methods. Its importance lies in its ability to efficiently encode geometric and statistical information associated with a measure, with applications to support inference, anomaly detection, density approximation, optimal sampling, and machine learning. One of the main goals of my doctoral research is to study how classical CD kernels can be regularized in order to obtain more stable procedures with improved analytical properties.
Recently, together with Mauricio Velasco and Didier Henrion, we introduced mollified CD kernels on algebraic varieties. These objects improve support recovery and yield convergence rates for density recovery without requiring prior knowledge of the equilibrium measure of the support, which is a limitation of the classical approach.
We are currently exploring several extensions of this line of research. These include applications of moment methods and CD functions to optimal stopping problems and anomaly detection; univariate versions of the CD function with lower computational cost; and the use of the mollified version in optimal sampling problems.
The Fekete problem consists of finding, for a given positive integer N, a configuration of N points on the sphere that maximizes the product of their pairwise distances. In other words, the goal is to distribute points on the sphere so that they are as far apart from each other as possible.
This problem can be equivalently reformulated as maximizing the logarithmic energy of a configuration, defined as the sum of the logarithms of the distances between all pairs of points. Stephen Smale included a computational version of this question as Problem 7 in his list of mathematical problems for the twenty-first century: construct point configurations whose logarithmic energy differs from the optimal value by at most a quantity of order log(N).
Exact solutions are currently known only for N = 2, 3, 4, 5, 6, and 12. The problem remains widely open and is connected to many active research directions, including finding exact solutions for small values of N, constructing low-energy configurations, improving asymptotic estimates for the energy, and understanding the abundance and structure of critical points.
Together with Diego Armentano, Federico Carrasco, Marcelo Fiori, Pedro Raigorodsky, Matías Valdés, and Mauricio Velasco, we are currently working on several aspects of this problem. By writing the equations that characterize the critical points of the logarithmic energy—namely, the vanishing of the Lagrangian together with the constraints that the points lie on the sphere and remain distinct—we obtain an algebraic variety describing the set of critical configurations. This allows us to study the ideal generated by these equations using tools from computational algebraic geometry, such as Gröbner bases, in order to count and classify critical points for small values of N.