Speaker
Description
Karpelevich’s theorem describes the single-eigenvalue region
$
\Theta_n=\{\lambda\in{\bf C}:\lambda\in\sigma(A)\ {\rm for\ some\ }A\in{\bf R}^{n\times n}
\ {\rm row\mbox{-}stochastic}\}.
$
The set of row-stochastic matrices is the polytope $\mathrm{conv}(V_n)$, where $V_n$ consists of the $n^n$
deterministic Markov kernels ($0$-$1$ matrices with exactly one $1$ in each row).
For a stochastic class $C=\mathrm{conv}(V)$, call $W\subseteq V$ spectrally complete if
$
\Theta(\mathrm{conv}(W))=\Theta(C),
$ and minimal if it is inclusion-minimal.
Since these eigenvalue regions are typically star-shaped about $0$, spectral completeness is essentially radial, writing $
\rho_C(\theta)=\sup\{r\ge0:\, r e^{i\theta}\in\Theta(C)\}, $ equality $\rho_{\mathrm{conv}(W)}(\theta)=\rho_C(\theta)$ for all $\theta$ implies $\Theta(\mathrm{conv}(W))=\Theta(C)$.
I develop this subset-selection viewpoint in three settings.
(1) Doubly stochastic matrices: vertices are permutation matrices; simulations up to $n=25$ support the Harlev-Johnson-Lim Boundary Conjecture, with no new phenomena beyond the exceptional $n=5$ behavior, and indicate that a very small family of permutation pairs is already spectrally complete.
(2) Monotone stochastic matrices: building on work of Vagenende et al., I focus on the
explicit boundary description for $n=3$ and explain how AI-assisted was used to
identify extremal families and supporting inequalities that were then verified rigorously.
(3) Prescribed zero patterns: in the nonnegative inverse eigenvalue problem with fixed support (Ran-Teng, dimension $3$), restricting the zero pattern yields another spectral subset-selection problem.
I conclude by outlining how AlphaEvolve-style workflows could systematize the discovery ofbspectrally complete subsets and hopefully guide us towards alternative Karpelevich-type proofs.