This paper provides a human digestion of a proof of the spectral isometry conjecture for $C^*$-algebras found by GPT-6 Astra. We prove that every unital surjective complex-linear spectral isometry between unital $C^*$-algebras is a Jordan isomorphism. Without the assumption that the map preserves the identity, it is a...
Does the Neumann--Poincar\'e spectrum determine a planar domain up to an extended M\"obius transformation? We construct a pair showing that the answer is no, even for analytic, strictly convex, algebraic boundaries. Their operators are similar. We then prove a positive result for prescribed finite-dimensional families...
We disprove Kenig's 1994 spectral-radius conjecture for the double-layer operator in every dimension $d\ge2$. For every $t>1/2$ sufficiently close to $1/2$, we construct a bounded simply connected Lipschitz domain with connected boundary such that the double-layer operator on $L^2$ with respect to the surface measure h...
The Dirichlet endpoint considered here, the nonzero Neumann datum forces the harmonic source modes to remain, producing a coupled interior--boundary system involving the full zero-Dirichlet inverse and its Neumann trace, together with a separate sign-recovery problem.
Matthew J. Colbrook, Siavash Sadeghi, George Stepaniants· 1 citation
The representation number of a graph is the least positive integer $k$ for which its vertices can be arranged in a word, each occurring $k$ times, so that two distinct letters alternate precisely when the corresponding vertices are adjacent. We prove that every bipartite graph on $N\ge9$ vertices has representation num...
Matthew J. Colbrook, Catherine Drysdale· 0 citations
A bounded simply connected noncircular domain of $\Omega$ with real-analytic Jordan boundary and a nonconstant function $u$ such that $u$ is a Neumann eigenfunction which is constant on the boundary, and $\Omega$ is a counterexample to Schiffer's conjecture.
Matthew J. Colbrook, George Stepaniants· 5 citations
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
Matthew J. Colbrook, Siavash Sadeghi· 0 citations
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