Analysis seminar at the Department of Mathematical Sciences. Please post at: https://www.udel.edu/academics/colleges/cas/units/departments/mathematical-sciences/about-us/seminars-and-colloquia/

in the "Analysis" category.

TITRE : A Complete Solution to a Conjecture on Doubly Stochastic Eigenvalues

ABSTRACT : Stochastic matrices are matrices with nonnegative entries whose rows each sum to 1. When a matrix and its transpose are both stochastic, it is said to be \textit{doubly stochastic}. In 1938, Kolmogorov proposed the problem of characterizing the region of possible eigenvalues of an $n \times n$ stochastic matrix, and Karpelevich gave a complete description thirteen years later.

This talk concerns the doubly stochastic analogue: characterizing the region $\omega_n$ of eigenvalues of $n\times n$ doubly stochastic matrices, which is contained in the unit disk. Perfect and Mirsky (1965) conjectured that $\omega_n$ is the union of the regions $\Pi_k$ (the convex hulls of the $k$-th roots of unity) for $k=1,2,\dots,n$. This conjecture holds for $n=1,2,3,4$, but fails for $n=5$. The cases $n\geq 6$ remain open. In response to the scarcity of progress over the past 60 years, Levick, Pereira, and Kribs proposed a related conjecture.

In this talk, I prove a more precise version of this conjecture. The approach, based on majorization theory and geometric properties, also provides a potential general framework for characterizing $\omega_5$, as well as a numerical method for testing the conjecture for various values of $n$.