A nine-part visual and analytic guide through F. I. Karpelevič’s landmark 1951 characterization of the eigenvalues of stochastic matrices. Starting from invariant convex polygons and a discrete chip-moving game in the complex plane, the series traces how Euclidean arithmetic, geometric rigidity, boundary polynomial equations, and induction fit together before branching into the theorem’s two main conclusions: the cyclic-polygon description and the explicit Farey boundary arcs.
Research Blog & Explorations
Expository articles, visual guides, and computational experiments exploring past milestones and future research directions.
Recent work by Johnson & Paparella (2025) and Gershnik, Lewis, & Paparella (2026) showed that the character tables of finite groups act as ideal Perron similarities, turning the nonlinear symmetric eigenvalue realization problem into explicit systems of linear halfspaces. This interactive exploration tests how much of the order-five Symmetric NIEP can be carved out by finite group character tables alone—capturing nearly the entire two-dimensional Loewy–McDonald benchmark slice once reducible adjoined blocks are included, isolating the microscopic curved hyperbola gap that polyhedral cones cannot reach, and lifting the construction into the unsolved three-dimensional trace-zero frontier.