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Matthew J. Colbrook

Publications and source records attributed to Matthew J. Colbrook.

At least 19 recordsLinked to original sources
arXiv2026

Spectral isometries of unital $C^*$-algebras

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 central unitary multiple of a Jordan isomorphism. The main step is to show that an entire one-parameter group of unital spectral isometries consists of algebra automorphisms. To prove this, we extend the group to the bidual, which splits into finite and properly infinite summands. On the finite part, we use quadratic and cubic trace identities and on the properly infinite part, we apply a structure theorem for maps that preserve square-zero elements. To show that both summands are invariant, we use holomorphic dependence on the parameter together with a uniform resolvent bound, which ensures that self-adjoint elements have images with real spectrum. We then apply the group theorem to inner automorphisms conjugated by the original spectral isometry. This reduces the problem to the commutative case, where Nagasawa's theorem completes the proof.

math.OA↗
arXiv2026

Homogeneous polynomial approximation on symmetric convex bodies: A proof of Kroó's conjecture

We resolve Kroó's 2004 conjecture on homogeneous polynomial approximation in every dimension, without smoothness or curvature assumptions. For every origin-symmetric convex body $K\subset\mathbb R^d$ with nonempty interior, every continuous real function on $\partial K$ is uniformly approximated by sums of homogeneous polynomials of degrees $n$ and $n+1$ as $n\to\infty$. Moreover, approximation of the constant function by homogeneous polynomials of degree $2n$ is uniform over all such bodies in each fixed dimension. Via homogenization, we also resolve Kroó's 2019 weighted approximation conjecture for every convex admissible weight on Euclidean space. The key construction produces homogeneous polynomials with a common Gaussian boundary profile and localization errors controlled by arbitrary polynomial weights. Its analytic core is a weighted Gaussian approximation theorem: for \(d\ge2\), sufficiently broad Gaussians on $\mathbb R^{d-1}$ can be approximated by $e^{-f}F$, where $f$ is nonnegative, convex and Lipschitz with $f(0)=0$, and $F$ is a Schwartz function with Fourier support in an arbitrarily small ball about the origin. Small Fourier support allows Taylor truncation to a lower degree, while the remainder enforces decay through homogenization. We then use positive spherical quadrature and degree raising to yield the full approximation theorem.

math.NA↗
arXiv2026

From boundary geometry to counterexamples in Smale's mean value problem

This paper provides a human digestion of the recent counterexample to Smale's mean value problem found by GPT-6 Astra. We give a geometric mechanism for constructing counterexamples. An outward normal that re-enters an analytic Jordan domain, subject to a curvature condition, yields a holomorphic model whose normalized divided differences exceed one in modulus on the boundary. Polynomial perturbations then concentrate every critical point near the boundary whilst preserving the strict inequality. This turns boundary geometry into counterexamples in every sufficiently large degree. We also locate these critical points to first order and show that their limiting distribution is the logarithmic equilibrium measure on the boundary. We end with an explicit example of degree $29\,984$, verified by interval arithmetic.

math.CV↗
arXiv2026

Convex Bernstein limits and sharp recovery on expanding intervals

We determine when approximation by convex polynomials on expanding intervals converges to best approximation on the real line by convex entire functions of exponential type at most $σ>0$. For intervals of radius $(n-γ_n)/σ$, where $0\leγ_n<n$ and $γ_n=o(n)$, one margin sequence gives this limit for every convex function of polynomial growth if and only if $γ_n/[n^{1/3}(\log n)^{2/3}]\to\infty$. When this condition fails, a fixed uniformly strictly convex entire function has approximation errors with unbounded limsup. We apply the limit theorem to prove existence of the convex Bernstein constant for $|x|^λ$ for every $λ\ge1$, and identify it with an attained entire-function minimum. We also give a sufficient margin for growth bounded by $\exp(a|x|^α)$, $0<α<1$. For arbitrary subexponential growth on the real line, each target with finite entire-function error admits a suitable margin, whereas no single margin is sufficient for the whole class. Further results concern attainment, right continuity with respect to the exponential type, weighted growth conditions, and the endpoints of fixed-fraction approximation intervals.

math.NA↗
arXiv2026

On the planar Neumann--Poincaré inverse spectral problem

Does the Neumann--Poincaré spectrum determine a planar domain up to an extended Möbius 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 of normalised exterior conformal maps. For every $m\ge1$, the first $m$ positive eigenvalues determine $m$ real Laurent coefficients on a small truncated convex cone, provided the higher coefficients are a prescribed analytic remainder of quadratic order. Recovery is Lipschitz stable, uniformly down to the disk. The proof combines the Grunsky representation with a nondegeneracy argument for weighted Hankel matrices and differentiated perturbation estimates. Explicit two- and three-parameter examples, single-mode families with repeated leading eigenvalues, and Möbius equivalence classes illustrate the result. Direct proofs for complete polarization data show what spectral weights add to the eigenvalues.

math-ph↗
arXiv2026

A counterexample to Kenig's conjecture for the Laplace double-layer operator

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 has $\pm \mathrm{i} t$ in its Fredholm essential spectrum. In particular, the essential spectral radius is greater than $1/2$. In the plane, the boundary may be chosen differentiable everywhere and smooth away from a single point. Our planar construction develops a refinement mechanism that produces unbounded growth of adjoint resolvent solutions with fixed forcing while keeping the graph slopes uniformly bounded. A closed recursion for the density and its image under an auxiliary singular integral reduces the growth estimate to a certified inequality for Hermitian $2\times2$ matrices. The certificate uses continuous estimates, exact rational arithmetic and rigorous interval enclosures. Localisation and rescaled graph insertions then produce a weakly null sequence of normalised approximate eigenvectors for the adjoint double-layer operator on one bounded boundary. A compact intertwining relation on Cartesian products of the planar domain with cubes gives the higher-dimensional examples without changing the spectral parameter.

math.AP↗
arXiv2026

Computing Spectral Properties of Differential Operators on Surfaces

Computing spectra of differential operators on surfaces is crucial in many applications, ranging from medical image analysis to nanoscale quantum structures. However, discretization of the infinite-dimensional operator and the underlying surface can lead to inaccurate and misleading results. This paper introduces the SurfSpec suite of algorithms to compute eigenvalues and spectral measures of operators on surfaces with error control and convergence guarantees. We combine methods based on complex contour integration and residuals to compute discrete spectra and high-frequency eigenmodes, including of nonlinear eigenvalue problems. We then compute convolutions of spectral measures with rational kernels to deal with continuous spectra. In both cases, the key tool is the operator resolvent applied to functions, which is computed using high-order methods that solve surface PDEs. We illustrate our algorithms on a variety of differential operators and surfaces, including a triceratops.

math.NA↗
arXiv2026

Crown graphs maximise the representation number of bipartite graphs

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 number at most $\lceil N/4\rceil$. Together with the known representation number of crown graphs, this settles the conjecture that crowns maximise the representation number among bipartite graphs of the same order. The proof develops a construction of Mozhui and Krishna by reducing the choice of a representing word to an ordering problem for neighbourhoods. We characterise the obstructions to this ordering and use probability estimates to exclude them for all sufficiently large balanced bipartitions. Two finite assertions complete the argument, each established by a checked Boolean unsatisfiability certificate. A refinement of the ordering argument treats the remaining odd part sizes directly. The main theorem and the crown-extremality corollary, including the finite certificate arguments, have also been formalised and verified in Lean 4.

math.CO↗
arXiv2026

Sharp Computational Bounds for Spectral Types of Schrödinger Operators

We prove sharp bounds for determining the spectral-type decomposition of Schrödinger operators in the spirit of Smale's program on the foundations of computation. For explicit one-dimensional self-adjoint Schrödinger operators $H=-{\mathrm d^2}/{\mathrm dx^2}+V$ on $L^2(\mathbb R)$, where $V\in C^\infty(\mathbb R;\mathbb R)$ is given by a finite description of all derivatives and derivative bounds, the pure point and absolutely continuous spectral sets cannot, in general, be recovered by any single limiting procedure. The singular continuous spectral set is strictly harder: it cannot, in general, be recovered by two nested limiting procedures. Analytic constructions of dichotomies realize the lower bounds: Gordon-type repetitions for pure point spectrum, high barriers for absolutely continuous spectrum, and an inverse spectral construction for singular continuous spectrum based on Riesz products, moment-killing perturbations, and a computational Gelfand--Levitan scheme. The finite-description framework also implies corresponding limitations on what can be certified in fixed formal systems (e.g., when used in computer-assisted proofs). Conversely, using wavelet-based certified computation, we prove matching upper bounds for broad classes of self-adjoint differential operators on $\mathbb R^d$ with coefficients of locally bounded variation and quantitative local variation control: two limits suffice for the pure point and absolutely continuous parts, and three for the singular continuous part. This provides a sharp hierarchy for spectral types.

math.NA↗
arXiv2026

A Complete Resolution of Forsythe's Conjecture for Restarted Conjugate Gradients

Forsythe's conjecture, published in 1968, asserts that for each restart length $s$, every exact-arithmetic restarted conjugate-gradient iteration on a real symmetric positive definite problem either terminates or has normalised residuals that converge separately along the even and odd restart subsequences. Apart from the classical steepest-descent case, this asymptotic question remained unresolved in full generality for nearly six decades. We give a complete classification by restart length in the original finite-dimensional setting and identify a sharp threshold. For $s=2$ and $s=3$, every problem either terminates or has convergent even and odd residual directions. For every $s\ge4$, there is a diagonal positive definite counterexample of dimension $s+4$ which never terminates and whose even residual directions do not converge. Together with Akaike's theorem for $s=1$, this shows that the conjectured universal conclusion is true precisely for $s\in\{1,2,3\}$ and false for every $s\ge4$. The positive results follow from a degree-independent double-orthogonality identity and an analysis of the low-degree fixed-point sets. At restart length four, rational interval arithmetic and Sturm sequences certify a transverse Hopf point of the leading vector field of the rescaled squared-weight map. Analytic periodic-orbit and shadowing arguments yield the counterexample at restart length four, and degree elevation extends the construction to every larger restart length. The classification for all $s\ge2$ is also formally verified in Lean.

math.NA↗
arXiv2026

The Spectral Skeleton of Chaos: Koopman Wave Packets on Poincaré Sections

A Poincaré section replaces a flow by a return map, but for a chaotic system this map is usually known only from sampled crossings. We show that coarse transport can be read directly from Koopman spectral data, without fitting the map. Measure-preserving EDMD retains the isometric structure; riggedDMD then approximates spectral measures and constructs finite regularized wave packets. Packet phase supplies a finite-resolution transport coordinate; low modulus marks a singular skeleton where the phase becomes ill-conditioned. We demonstrate the idea on the Rössler system, a 32-mode Kuramoto--Sivashinsky Galerkin system, and the forced Duffing oscillator. The packets yield coarse symbolic models on sections ranging from an almost one-dimensional curve to a visibly thick set. Their graphs organize observed low-period orbits and guide targeted searches for others. In Duffing Regime~II, a seven-region rule accounts for $91\%$--$94\%$ of filtered one-step transitions, while failures in the lowest retained modulus decile occur at $5.08$--$5.20$ times the overall rate. The packets are not Koopman eigenfunctions, nor are the regions exact Markov partitions. Together these computations show how spectral information beyond isolated eigenpairs can expose chaotic transport directly from trajectories.

nlin.CD↗
arXiv2026

Deep greedy unfolding: Sorting out argsorting in greedy sparse recovery algorithms

Gradient-based learning imposes (deep) neural networks to be differentiable at all steps. This includes model-based architectures constructed by unrolling iterations of an iterative algorithm onto layers of a neural network, known as algorithm unrolling. However, greedy sparse recovery algorithms depend on the non-differentiable argsort operator, which hinders their integration into neural networks. In this paper, we address this challenge in Orthogonal Matching Pursuit (OMP) and Iterative Hard Thresholding (IHT), two popular representative algorithms in this class. We propose permutation-based variants of these algorithms and approximate permutation matrices using "soft" permutation matrices derived from softsort, a continuous relaxation of argsort. We demonstrate -- both theoretically and numerically -- that Soft-OMP and Soft-IHT, as differentiable counterparts of OMP and IHT and fully compatible with neural network training, effectively approximate these algorithms with a controllable degree of accuracy. This leads to the development of OMP- and IHT-Net, fully trainable network architectures based on Soft-OMP and Soft-IHT, respectively. Finally, by choosing weights as "structure-aware" trainable parameters, we connect our approach to structured sparse recovery and demonstrate its ability to extract latent sparsity patterns from data.

cs.LG↗
arXiv2026

A sharp isoperimetric inequality for the Neumann--Poincaré operator in every dimension

Let $Ω\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,α}$, where $0<α<1$. For the adjoint Neumann--Poincaré operator $K^*_{\partialΩ}$, normalised so that its distinguished eigenvalue is $1/2$, let $λ_j^+(Ω)$ denote the upper min--max values on the mean-zero energy space. We prove $\sum_{j=1}^{d}λ_j^+(Ω)\ge \frac{d-2}{2}.$ It follows that $λ_1^+(Ω)\ge \frac{d-2}{2d},$ with equality if and only if $Ω$ is a ball. In dimension three this proves the $1/6$-conjecture of Miyanishi and Suzuki. The proof uses the coordinate boundary-charge densities induced by uniform applied fields. Their energy Gram matrix is the perfect-conductor polarization tensor $M_\infty$. Positivity of a $2d\times2d$ Gram matrix yields the endpoint Hashin--Shtrikman inequality $|Ω|\mathrm{Tr}(M_\infty^{-1})\le1,$ and bounds the trace of the compression of $K^*_{\partialΩ}$ to the applied-field space. Equality in the inverse-trace inequality makes the interior Newtonian potential quadratic, so a converse to Newton's theorem identifies $Ω$ as an ellipsoid. Equality in the spectral estimate also makes the Hessian of this potential isotropic, which forces the ellipsoid to be a ball.

math.AP↗
arXiv2026

A computer-assisted counterexample to the planar Berenstein conjecture

Recent work of Colbrook and Stepaniants produced the first counterexamples to the planar Pompeiu and Schiffer conjectures and introduced the conformal fixed-disc, disk-polynomial, and validated-tail machinery used here. By adapting this framework to the complementary Dirichlet endpoint, we disprove the unrestricted planar Berenstein conjecture. Specifically, we construct a bounded simply connected domain $Ω$ with real-analytic Jordan boundary, which is not a disc and for which there exist $k\in(27.4381178838,27.4381198839)$ and a nonzero real-valued function $u\in C^ω(\overlineΩ)$ satisfying $(Δ+k^2)u=0$ in $Ω$, with $u=0, \partial_νu=\text{constant}\ne0$ on $\partialΩ$. Thus the overdetermined Dirichlet--Neumann data do not characterize the disc without an additional sign assumption on $u$. The domain has dihedral symmetry of order $26$, but is neither a disc nor centrally symmetric, and the corresponding eigenfunction changes sign. Equivalently, its boundary arclength measure satisfies $\widehat{σ_{\partialΩ}}(kω)=0$ for $ω\in\mathbb S^1$. After conformally transferring to the unit disc, exact support identities and quantitative disk-polynomial estimates yield rigorous control of the infinite-dimensional tail. A Newton--Kantorovich argument then reduces existence to finitely many explicit inequalities, which are certified using interval arithmetic. The extension from the Pompeiu--Schiffer problem is not formal. The earlier construction absorbs both boundary conditions into a single inverse-Laplacian equation. At 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.

math.AP↗
arXiv2026

A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures

The planar Pompeiu problem, originating in 1929, and the associated Schiffer conjecture are long-standing rigidity questions linking rigid-motion integral transforms and Fourier zero sets to overdetermined Neumann eigenvalue problems. We construct a bounded simply connected noncircular domain $Ω\subset\mathbb{R}^2$ with real-analytic Jordan boundary and a nonconstant function $u$ such that $(Δ+k^2)u=0$ in $Ω$, $u=1,\partial_νu=0$ on $\partialΩ$ for some $k\in(31.967007261,31.967007293)$. Thus $u$ is a Neumann eigenfunction which is constant on the boundary, and $Ω$ is a counterexample to Schiffer's conjecture. Green's identity also gives $\widehat{\mathbf 1_Ω}(kω)=0$ $(ω\in\mathbb S^1)$, so $Ω$ fails the Pompeiu property and is also a counterexample to the planar Pompeiu conjecture for bounded simply connected Lipschitz domains. We obtain the domain as $Ω=ϕ(\mathbb{D})$, where $ϕ$ is a ten-fold symmetric conformal map close to an explicitly listed polynomial of degree $301$. On the unit disc, the analytic problem becomes a cubic operator equation on real coefficient spaces, $F(g,p)=g+|p|^2(1+Kg)=0$, where $K$, expressed in a disk-polynomial basis, is an explicit inverse of the Laplacian on the range compatible with zero Dirichlet and Neumann traces, and $p=kϕ'$. Positivity of the disk-polynomial linearisation coefficients, sharp bounds for $K$, and monotone control of the infinite tails establish an a posteriori contraction near the listed polynomial in a weighted coefficient algebra, and hence an exact zero of $F$.

math.AP↗
arXiv2026

Finiteness and exponential growth of full graph $p$-spectra

We prove that the full spectrum of the graph $p$-Laplacian is finite for every finite graph and every real $p>1$. This resolves the problem of finiteness of the full graph spectrum posed by Amghibech (2003). More generally, for signed weighted graph $p$-Schrödinger operators with arbitrary real potentials, we obtain bounds in terms of the numbers of vertices and positive-weight edges, uniform in all coefficients and in $p$. On $n$ vertices with $m$ positive-weight edges, the logarithms of the spectral cardinality and of the total number of connected components of the normalised eigenvector sets are $O((n+m)^2)$; the sum over the full spectrum of their rational Betti numbers is at most $\exp(C(n+m)^2)$ for an absolute constant $C$. These topological bounds extend to homogeneous eigenproblems built from arbitrary finite families of linear forms, including generalised $p$-eigenvalue problems for matrix pairs. For the uniformly weighted $K_n$ and $p\ne2$, we identify the nonconstant eigenlines with the barycentres of the coordinate-hyperplane cell decomposition of $\mathbb RP^{n-2}$ and determine their local Morse data on either side of $p=2$. A renormalised logarithmic limit describes the transition at $p=2$. Positive integer weights then separate these critical values, giving at least $(3^n-2^{n+1}+3)/2$ distinct spectral values for every fixed $p\ne2$ and every $n\geq2$. By contrast, the maximal spectral cardinality at $p=2$ is $n$; at $p=4$, the maximal spectral cardinalities in the unsigned and general classes both have exponential growth rate exactly $3$. For $n\geq3$ the same examples resolve Amghibech's extremal question. The proof combines o-minimal and Pfaffian geometry with projective $L^p$-duality, Morse theory, critical groups, and tensor eigenvalue bounds.

math.SP↗
arXiv2026

No Spectral Invisibility for Dissipative Barrier Truncations in Any Dimension

The dissipative barrier method suppresses spectral pollution, but whether it can itself conceal genuine spectral points has remained open. Known as the graveyard problem in computational spectral theory, the higher-dimensional case has remained unresolved for more than a decade. We resolve it for Schrödinger operators in dimensions $d\geq2$; together with the known one-dimensional theorem, this settles the no-invisibility problem in all dimensions. Let $A=-Δ+V$ be a Dirichlet Schrödinger operator on a connected open set $Ω\subseteq\mathbb R^d$ with $V\in L^1_{\mathrm{loc}}(Ω)$ bounded below, and set $H=A+iS$, where $S\geq0$ and $S\in L^p(Ω)$, with $1 0}$ be a nested family of nonempty connected bounded open sets such that $Ω_R\nearrowΩ$ as $R\nearrow+\infty$. Denote by $H_R$ the Dirichlet truncation of $H$ to $Ω_R$. We prove that every spectral point of $H$ is detected by the truncations: for every $λ\inσ(H)$ and every neighborhood $U$ of $λ$, $σ(H_R)\cap U\neq\emptyset$ for sufficiently large $R$. Equivalently, $σ(H)\subseteq\liminf_{R\to\infty}σ(H_R)$. Thus the barrier method does not trade suppression of spectral pollution for spectral invisibility. No regularity of $\partialΩ$ is required, and the assumptions reach the critical Sobolev scale. The proof combines compactness of the dissipative form perturbation, Cwikel-type Schatten estimates for Birman--Schwinger operators, generalized strong resolvent convergence, and a reverse Hansmann--Weyl spectral-variation inequality due to Gil'. A two-dimensional numerical example illustrates the absence of spectral invisibility for the truncated dissipative operators.

math.NA↗