Ludovick Bouthat

I am Ludovick Bouthat, currently an NSERC postdoc in mathematics at the University of Delaware and the University of Regina. Prior to that, I did my PhD at Université Laval under the supervision of Pr. Javad Mashreghi and Pr. Frédéric Morneau-Guérin.

My research lies at the intersection of matrix theory, operator theory, and functional analysis, with a focus on norms and eigenvalues of matrices. This naturally led me to explore a variety of mathematical areas, including probability, group and semigroup theory, geometry, and both classical and complex analysis, and to connect them through results that highlight deep interactions between structure and function. Much of my work centers on understanding how geometric phenomena manifest in various ways in linear and multilinear algebraic settings.

I am also actively engaged in the mathematical community, notably having served as Chair of the Student Committee of the Canadian Mathematical Society and Student Director on its Board of Directors. This notably helped me be the recipient of the Vanier scholarship in 2022. For more details about my academic involvement, outreach activities, awards, and teaching experience, please refer to my CV.

Some Publications (Images are PDF links)

On the Geometry of the Birkhoff Polytope. I. The operator ℓp-norms AND On the Geometry of the Birkhoff Polytope. II. The Schatten p-norms

This series of two papers authored by Javad Mashreghi, Frédéric Morneau-Guérin, and myself, was published in 2024 in Acta Sci. Math. (Szeged). Therein, we study the geometry of the Birkhoff polytope, i.e., the compact convex set of all n×n doubly stochastic matrices. While its faces, edges and facets as well as its volume have been intensely studied, other geometric characteristics such as the center and radius were left off, despite their natural uses in some areas of mathematics. In these papers, we completely characterize the Chebyshev center and the Chebyshev radius of the Birkhoff polytope associated with the metrics induced by the operator ℓp-norms and the Schatten p-norms for the range 1≤p≤∞. While studying these properties, the intrinsic connection to the minimal trace, which naturally appears in the assignment problem, is also established.

On the monotonicity of left and right Riemann sums

This single-authored 42 pages paper was published in Sampling Theory, Signal Processing, and Data Analysis in 2024. It is dedicated to proving general theorems about the monotonicity of left and right Riemann sums, a problem first raised by Fejér in 1950. We provide a complete much-needed review of the literature on the problem and offer several new sufficient and necessary conditions for the monotonicity of Riemann sums. Additionally, we present a new insightful proof of a fundamental theorem related to these sums using tools from the theory of majorization. Lastly, we delve deeper into a question posed by Borwein, almost resolving it completely.

On a question of Erdős on doubly stochastic matrices

This paper was co-authored with my collaborators, Javad Mashreghi and Frédéric Morneau-Guérin, in the journal Linear and Multilinear Algebra in 2024. In a celebrated 1959 paper of Marcus and Ree, it is shown that if a matrix A=[ai,j] is an n×n doubly stochastic matrix, then there is a permutation σ∈Sn such that Σai,j² ≤ Σai,σ(i). Erdős asked for which doubly stochastic matrices the inequality is saturated. Although Marcus and Ree provided some insight for the set of solutions, the question fell into oblivion for several years. In this paper, we provide a complete answer in the particular, yet non-trivial, case of n=3. These partial results generated a new axis of research which has already produced several new results.

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