Papers

These are the lecture notes for a course on diffusive quantum measuring instruments. They were prepared and delivered at the Perimeter Institute on Mondays and Thursdays, from 2:30 to 4:00 PM, beginning October 27th, 2025 and ending December 11th, 2025. These lectures were recorded and can be found at https://pirsa.org/c25038.

We show the existence of an acylindrically hyperbolic group that has non-uniform exponential growth and Kazhdan's property $(T)$. The group is constructed as a common quotient of automorphism groups of free groups of increasing finite rank.

Multi-constraint safety filters based on control barrier functions for linear systems with affine state constraints yield continuous piecewise-affine closed-loop dynamics and may introduce boundary equilibria and unstable active-set modes. Although these filters guarantee forward invariance, they can change nominal stability, and it remains unclear when unstable modes cause divergence versus bounded, convergent behavior. This paper develops a geometric framework to analyze these cases: leveraging explicit active-set realizations, we show that equilibria associated with nonempty active sets lie on the corresponding constraint faces and that any unstable directions are tangent to those faces due to exponential enforcement of the active constraints. We characterize mode stability via a minimum-phase test, provide a sufficient recession-cone certificate for divergence under a fixed active set, and derive tractable linear-matrix-inequality conditions for global exponential stability or boundedness using Lyapunov and LaSalle arguments.

Let $(X,0)$ be a germ of a reduced and irreducible complex surface embedded in $(\mathbb{C}^k,0)$. In this paper, we give a complete invariant of the inner Lipschitz geometry of complex surface germs, extending the result of Birbrair–Neumann–Pichon \cite{BNP} to the non-isolated case. This invariant is expressed in terms of numerical invariants associated with the coordinate functions $f_1,\dots,f_k$ of the normalization map $n:(\overline{X},0)\to (X,0)\subset(\mathbb{C}^k,0)$, together with the combinatorics of a suitable good resolution of $(\overline{X},0)$.

We consider a one-parameter family of nonlinear wave equations on the $d$-dimensional torus, with polynomial nonlinearities of arbitrary degree $q+1$, where $q\geq 1$. We investigate the long-time behavior of high Sobolev $H^s$-norms of solutions in different settings. In the one-dimensional case, and for almost any value of the mass parameter $\mathtt{m}>0$, we prove exponentially long stability times for small initial data. The proof relies on normal form techniques together with suitable weak Diophantine conditions. In higher space dimensions, for initial data $u_0\in H^{s}$, $s \geq s_1 + 1$, satisfying suitable smallness conditions on the low Sobolev norm $H^{s_1}$ and on the $L^2$-norm, we prove a polynomial upper bound on the possible growth of the high Sobolev $H^{s}$-norm, over finite but exponentially long time scales in the regularity parameter $s_1$. The key ingredient consists in establishing suitable a priori tame estimates for the solution. The result applies in any space dimension $d\geq 1$ and for all values of the mass parameter $\mathtt{m}\geq 0$.

We develop a localised wavelet formulation of multifractal random walk unwrapping based on the local multiplicative modulation freezing. The framework is motivated by the observation that finite-support wavelet localisation may induce approximate local factorisation of multiplicatively modulated stochastic fields, allowing the modulation component to become effectively frozen within sufficiently localised probing domains. Within this regime, logarithmic wavelet amplitudes admit an approximate additive decomposition linking local wavelet statistics directly to the underlying modulation field. This viewpoint reformulates covariance-based MRW unwrapping as a localised multiscale operator problem in which wavelet coefficients act as finite-support probes of multiplicative organisation. The validity of the approximation depends explicitly on support geometry, scale-dependent overlap, and residual multiscale mixing generated by internal modulation variability. We show that these effects naturally produce finite-scale deviations from ideal logarithmic covariance scaling and lead to structured covariance distortions whose form depends on the interaction between the modulation field and the geometry of the wavelet representation. In the resulting framework, localisation itself becomes the operational mechanism enabling multiscale probing of local stochastic organisation. Numerical investigations using orthonormal wavelet decompositions support the proposed interpretation and demonstrate the emergence of scale-dependent freezing regimes, residual covariance mixing, and finite-support breakdown effects consistent with the theory. The proposed framework suggests a broader connection between wavelet localisation, local regularity organisation, and finite-support multiscale stochastic operators. Wavelet localisation becomes an operational mechanism for probing localised multiscale structure.

We introduce the notion of local pseudofunction spectrum $\sigma_{PF}(A)$ for the infinitesimal generator $A$ of a bounded $C_0$-semigroup $\mathcal{T} = (T(t))_{t \geq 0}$ on a Banach space $X$ and show it is the right spectral concept to deliver a full characterization of the strong stability of $\mathcal{T}$: \[ \forall x \in X : ~ \lim_{t \to \infty} \| T(t) x \|_X = 0 \quad \Longleftrightarrow \quad \sigma_{PF}(A) = \varnothing. \] We demonstrate how this yields a quick proof of the well-known Arendt-Batty-Lyubich-Vũ theorem and establish novel stability results for semigroups whose local pseudofunction spectra are contained in a null subset of the imaginary axis through localization of range density conditions to the potential null singular set. We also obtain similar stability characterization theorems for individual orbits and for semi-uniform stability. As an application of our results, we provide spectral characterizations of asymptotically almost periodic $C_0$-semigroups with countable local pseudofunction spectrum. In addition, we prove optimal Tauberian theorems of Katznelson-Tzafriri type and discuss connections with Wiener kernels.

Compatible-strain mixed finite elements (CSMFEs) use the differential complex of nonlinear elasticity to construct discretizations that preserve the underlying topological structure. Existing CSMFEs have focused on simplicial meshes for compressible and incompressible nonlinear elasticity. In this paper, we develop compatible-strain mixed formulations for quadrilateral elements applicable to both compressible and incompressible solids. For general quadrilateral elements, the Piola transformation preserves vector fields tangent and normal to element edges only for special geometries, such as rectangles, parallelograms, and trapezoids, and therefore cannot be used to construct compatible shape functions. To overcome this limitation, the shape functions are computed directly in the physical space using numerical integration. Nedelec shape functions of the first kind are employed to interpolate the displacement gradient, and a new class of stress shape functions compatible with the displacement and displacement-gradient discretizations is introduced. The compressible formulation follows the framework previously developed for simplicial CSMFEs, whereas a new incompressible formulation employing element-level condensation of the pressure field is proposed, thereby avoiding additional global degrees of freedom. These developments extend the compatible-strain mixed finite element framework from simplicial to general quadrilateral meshes for both compressible and incompressible nonlinear elasticity. Numerical examples demonstrate that the proposed quadrilateral elements can solve problems that require second-order simplicial CSMFEs while using fewer degrees of freedom.

We develop a comprehensive, gauge-covariant geometric framework for non-Hermitian quantum systems in the quasi-Hermitian regime, that is, the region of parameter space where the non-Hermitian Hamiltonian admits a real spectrum and a positive-definite metric operator. We build this framework by elevating the Dyson map to a central geometric object. This map is the transformation that converts a non-Hermitian Hamiltonian into an equivalent Hermitian one. From it we construct the Dyson connection and decompose it into Hermitian and anti-Hermitian parts, identified respectively as stretching and rotation components. This decomposition cleanly separates the genuine physical metric deformations from the unitary gauge redundancies. Working with manifestly gauge-covariant states, we then derive the complex non-Hermitian Berry phase and the quantum geometric tensor (QGT), and show that the non-Hermitian geometric curvature originates from the non-commutativity of the stretching components at the operator level. We further analyse the geometric singularities near an exceptional point (EP) and uncover a distinct hierarchy of divergences. For a general two-level non-Hermitian model, the quantum metric tensor (QMT) exhibits a leading-order divergence $\sim |\epsilon_\mu|^{-2}$, while the Berry curvature shows a weaker, subleading divergence $\sim |\epsilon_\mu|^{-3/2}$, with $\epsilon_\mu$ denoting the parameter displacement from the EP along an individual parameter axis $\mu$. Finally, we examine physical realizations of this model, including the non-Hermitian Su–Schrieffer–Heeger (SSH) and Hatano–Nelson (HN) models, where exact analytical results confirm the predicted critical scaling laws and illustrate the metric-deformation-driven non-Hermitian geometries.

In a billiard table with $C^2$ boundary, trajectories are critical points of the negative length functional on the space of admissible paths in the billiard table. The goal of this paper is twofold: first, to prove theorems which aid in computing the Morse index of trajectories; second, to use the Morse index to determine the stability of periodic trajectories. To the first end, we prove discrete analogs to the Morse index theorem for arbitrary trajectories, which are critical points with respect to fixed-endpoint variations, and for periodic trajectories, which are critical points with respect to periodic variations. To the second end, we prove a criterion for linear stability of periodic trajectories depending on the Morse index of the second iterate of the trajectory, and prove a sufficient condition for a billiard table to carry a linearly stable 2-periodic trajectory.

Let $Q$ be a monic polynomial of degree $M+1$. We study the algebraic system \[ \sum_{j\ne i}\frac{1}{x_i-x_j}=Q(x_i),\qquad i=1,\ldots,N, \] for pairwise distinct complex numbers $x_1,\ldots,x_N$, modulo permutations of these numbers. The case $M=0$ is, after a translation, the classical Stieltjes system for the zeros of a Hermite polynomial. We prove that, for arbitrary $Q$, the number of solutions is at most $\binom{N+M}{N}$, and that the coefficient equations for the associated monic Stieltjes polynomial have total intersection multiplicity exactly $\binom{N+M}{N}$. Consequently the bound is attained for all $Q$ in a non-empty Zariski open subset of the affine space of monic polynomials of degree $M+1$. We also describe the solutions when the coefficient of the linear term of $Q$ is large: the system splits into $M+1$ weakly coupled classical Stieltjes systems, one near each zero of $Q$.

Discrete conjugate nets (also known as quad-surfaces) are polyhedral surfaces made of quadrilaterals connected in the combinatorics of the square grid. A generic discrete conjugate net is rigid. The present article contains counterexamples to an erroneous statement that a non-degenerate discrete conjugate net with flexible $3 \times 3$-subnets is flexible and proves a corrected version of it under stronger non-degeneracy assumptions.

For arbitrary $m\geq 2$ and $d\geq 1$, we establish square-root cancellation for sums of Fourier coefficients of $\mathrm{GL}(m)$-automorphic forms averaged over values of random degree-$d$ polynomials.

Let $(\Omega,g)$ be a smooth compact 3D Riemannian manifold with the smooth boundary $\Gamma$, $\tau(x):={\rm dist\,}(x,\Gamma)$, $x\in\Omega$; $\Omega^\tau:=\{x\in\Omega\,|\,\,{\rm dist\,}(x,\Gamma)<\tau$\}, $\Gamma^\tau:=\{x\in\Omega\,|\,\,{\rm dist\,}(x,\Gamma)=\tau$\}, $\tau\geqslant 0$. For the sake of technical simplicity, we deal with $\Omega$ diffeomorphic to a ball in $\Bbb R^3$. Let $\mathscr P:=\{\nabla p\,|\,\,p\in H^1(\Omega)\}$ be the space of the potential vector fields, and let $\mathscr L_\lambda:=\{\varkappa\nabla\tau\,|\,\,\varkappa\in L_2(\Omega)\}$ be the space of the vector fields parallel to $\nabla\tau$. The N-transform is a map from $\mathscr P$ to $\mathscr L_\lambda$ defined layer-wise (in accordance with $\Omega=\cup_{\tau\geqslant 0}\Gamma^\tau$) by $$ Nh\,\big|_{\Gamma^\tau}:=(P^\tau h)\big|_{\Gamma^{\tau-0}}, \qquad\tau>0, $$ where $P^\tau$ are the projections in $\mathscr P$ onto the subspaces $\mathscr P^\tau:=\{h\in\mathscr P\,|\,\,{\rm supp\,}h\subset\overline{\Omega^\tau}\}$. We show that $N$ is a unitary operator. Let $p=p^f(x)$ be a solution to the Dirichlet problem: $\Delta_g p=0$ in $\Omega\setminus\Gamma$, $p=f$ on $\Gamma$. The DN-map $\Lambda$ is defined by $\Lambda f:=-\langle\nabla p^f,\nabla\tau\rangle$ on $\Gamma$. We show that the N-transform provides a certain factorization $\Lambda^{-1}=V^*V$ and discuss its possible usefulness for determination of $(\Omega,g)$ from $\Lambda$.

We study the problem of enumerating symmetric chain decompositions (SCDs) of the minuscule lattices $L(m,n)$ of partitions in an $m$ by $n$ box and $M(n)$ of partitions into distinct parts at most $n$. We shift the focus from constructing a single SCD to analyzing the global structure of the set of SCDs. Let $\#SCD(P)$ be the number of symmetric chain decompositions of $P$. We give an explicit formula for $\#SCD(L(2,n))$ based on inversion sets of permutations and conjecture that for fixed $m>1$ both $\#SCD(L(m,n))$ and $\#SCD(M(n))$ grow super-exponentially. These conjectures are supported by data produced by AlphaEvolve, an evolutionary coding agent from Google DeepMind, and are in the same vein as a recent paper of Tomon on the growth rate of $\#SCD$ for the Boolean lattice and hypergrid. We make connections with crystal bases and show that the Lusztig involution (evacuation) extends to an involution on SCDs, which we use to show that $\#SCD(M(n))$ is even for $n>2$. We use skew tableaux sequences, which are equivalent to SCDs, and describe a potential path forward for finding SCDs through a notion of tableaux avoidance. We discuss implications of the conjectures for the problem of computing plethysm coefficients and discuss connections to physics and geometry. We end with a list of conjectures, questions and open problems.

torch_vn_algebra is a PyTorch library for numerical work with finite-dimensional von Neumann algebras, i.e. direct sums of matrix algebras. These algebras are central to operator algebra theory and, via commutants of symmetries, describe quantum systems with conserved charges or superselection rules. To our knowledge, no numerical software treats them as a data type with their centre, family of traces, conditional expectations, inclusions and functional calculus. The library fills the gap between computer algebra (exact but small-scale) and quantum simulators (large-scale but limited to Hilbert-space matrices): it handles batches of operators on CPUs and GPUs with automatic differentiation, and converts operators, states and channels. Key features: Compact tensor representation: 4D tensor (B, C, kmax, kmax), where B is batch size (e.g. Monte Carlo samples), C is the number of summands, and kmax is the max block size, enabling batched GPU processing. Lazy evaluation: matrix data materialized only on access, avoiding memory waste. Random operator generation: custom eigenvalue distributions; Haar, SU(n), COE, CSE unitary ensembles. Functional calculus and spectral methods: matrix absolute value, square root, inverse, entropy via SVD; automatic switch between exact diagonalization (block size up to 256) and power iteration for extreme eigenvalues. Three trace functionals: Tr_blunt (sum of block traces), Tr_norm (dimension-normalized), tau_vN (faithful normal tracial state). Performance: on a Tesla P100 GPU, processing 20,000 samples of 100x100 operators yields up to 30x speedup over traditional CPU workflows. Availability: open-source at https://gitlab.com/a.hobukob/von_neumann_type_i/

In this work we study the differentiability for the Sobolev norm of $W^{k,1}(M)$ in the sense of Gâteaux , where $(M,g)$ is an arbitrary closed Riemannian manifold.

The perfectly matched layer-based boundary integral equation (PML-BIE) method (Lu et al., SIAM J. Appl. Math. 78 (2018)) has become an effective tool for wave scattering problems in unbounded domains. Despite its successful applications, a rigorous convergence theory has remained incomplete in many physically relevant settings. In this paper, we present a general framework for establishing convergence of PML-BIE methods, using acoustic scattering in an impedance half-space as the first illustrative example. The framework separates the analysis into three components: convergence of the PML truncated partial differential equation, equivalence between the PML problem and an exact PML-BIE, and convergence of a computable PML-BIE obtained by replacing the exact kernel with a stretched free-space kernel. For the impedance half-space problem, we prove convergence of both the exact and computable PML-BIE formulations. The final error bound decays exponentially as the PML absorption power increases. The resulting theory provides a rigorous foundation for the PML-BIE method and gives a reference model for more general scattering problems in layered and inhomogeneous media.

In this paper, we study properties of Rhaly operators acting on weighted Hardy spaces $H^2(\omega)$. This problem is intimately related to the study of factorable matrices on $\ell^2$. In particular, our main contributions are in the study of Schatten class properties of such operators.

Let $N(\Delta,D)$ denote the maximum number of vertices in a simple undirected connected graph with maximum degree at most $\Delta$ and diameter at most $D$. Determining $N(\Delta,D)$ is known as the degree/diameter problem. Although the problem has been studied for many years, the exact value of $N(\Delta,D)$ is unknown for most pairs $(\Delta,D)$. In this paper we construct explicit graphs, using a construction discovered through interaction with ChatGPT via its standard web interface, showing that $N(12,5)\ge 34{,}992$ and $N(16,5)\ge 147{,}456$. These improve the corresponding recorded lower bounds 29,621 and 132,496. The search was conducted without an external orchestration layer around ChatGPT: no custom agent framework, automated evaluator-driven search loop, problem-specific search engine, or formal proof assistant was set up in advance by the author. As far as the visible transcript shows, the author did not prompt the model with the concrete components or candidate families of the construction. After presenting the mathematical result, we describe the discovery process on the basis of the visible transcript. We focus on the meta-level interventions made during the approximately six-day search, and we identify the stage at which the abstraction underlying the construction first appeared.

In this paper, we investigate the existence and qualitative properties of ground state solutions for the nonlocal Born-Infeld-Choquard problem \begin{equation*} \begin{cases} -div\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+ \omega u=\big(I_\alpha\ast |u|^{p}\big)|u|^{p-2}u, & \hbox{in }\mathbb{R}^N,\; N\geq 3, \\[5mm] u(x)\to 0, &\hbox{as }|x|\to +\infty. \end{cases} \end{equation*} where $p>\frac{N+\alpha}{N}$, $\omega=0,1$ and $0<\alpha<N$. The equation is driven by the mean curvature operator in Lorentz-Minkowski space, motivated by the Born-Infeld nonlinear electromagnetic theory, and is coupled with a Choquard-type nonlocal nonlinearity. Due to the inherent relativistic gradient constraint $|\nabla u| \le 1$, the associated energy functional lacks standard $\mathcal{C}^1$ regularity, preventing the direct use of classical variational techniques. We employ a non-smooth critical point theory on appropriate Pohožaev-type manifold to establish the existence of ground state solutions. Such non-smooth critical point theorem is abstract and we further show that it can be employed for strongly indefinite problem as well. We also demonstrate that these solutions are radially symmetric, and monotonously decay to zero at infinity.

Operational disturbance monitoring in power networks requires decisions to be made from waveform windows as they arrive, rather than from completed records after the event. This study evaluates full-vector Wigner–Ville Distribution Slice (WVDS) spectra for sequential anomaly-onset detection in high-voltage grid-voltage waveforms. The approach keeps the bilinear midpoint interaction structure of the Wigner–Ville distribution and represents each 128-sample voltage window by a 128-dimensional slice spectrum, avoiding manually selected fault-frequency markers. WVDS is used with a baseline-normalized deviation (BND) score and is compared against the BND of Fast Fourier Transform (FFT-BND), raw-window autoencoders, FFT autoencoders, and WVDS autoencoders under the same thresholding and three-window persistence rule. A synthetic autoencoder–clustering teacher is used to select RTE fault records that start from an initially normal region and then transition to anomalous behavior. On the filtered test set, FFT-BND achieves the highest sensitivity, whereas WVDS-BND provides the lowest false-alarm operating point, reducing record-level pre-onset false alarms to 0.69%. The autoencoder comparison follows the same selectivity pattern: WVDS reconstruction decreases false alarms relative to FFT reconstruction but misses more examples. The results indicate that preserved WVD cross-term information can form a selective representation for online grid-waveform anomaly monitoring when false alarms are costly.

Azuma et al. showed that a cactus-expandable Boolean network $\Sigma$ is structurally oscillatory if all the simple cycles of $\Sigma$ contain an odd number of inhibiting edges. We show that we do not need the cactus-expandable condition. Namely, a strongly connected Boolean network $\Sigma$ is structurally oscillatory if and only if all the simple cycles of $\Sigma$ contain an odd number of inhibiting edges. Additionally, we provide a characterization of a structurally oscillatory Boolean network for the general case.

We prove that the multifractal spectrum D(h,omega) of a stochastic process is almost surely deterministic under a scale decorrelation condition on the HGDS scale derivative. The key difficulty is that the pointwise Hölder exponent lives in the germ sigma-algebra, where classical 0-1 laws do not reach. We get around this by working with the geometry accumulation integral G_Lambda, which is a genuine Lebesgue integral over scales and concentrates almost surely. The boundary case – log-correlated fields – is sharp: the variance summability condition fails exactly there.

In this note, we consider several quasiliniear minimization problems involving various Hardy type potentials for functions in $W^{1,p}(\Omega)$ with weighted mean zero. We prove that the strict inequality for the infimum yields the existence of a minimizer.

Continuing our previous work on the integral Arnold conjecture, we establish the foundation of Hamiltonian Floer theory over integer coefficients on any compact symplectic manifold. This package produces a well-defined chain homotopy class of Floer complexes, chain-level continuation maps, the Piunikhin–Salamon–Schwarz (PSS) isomorphism, and an associative pair-of-pants product. When the Hamiltonian is the $p$-th iteration for a prime number $p$, we also establish the $\mathbb{Z}/p$-equivariant Floer theory in characteristic $p$, including the equivariant Floer complex and the $\mathbb{Z}/p$-equivariant pair-of-pants product. Moreover, we define quantum Steenrod operations for any compact symplectic manifold and compare them with the equivariant pair-of-pants products via the $\mathbb{Z}/p$-equivariant PSS map. We also obtain chain-level invariants such as spectral numbers and barcodes, and prove quantitative properties of the equivariant pair-of-pants product. The counting theory is based on Fukaya-Ono's normally polynomial perturbation scheme and its realization by the authors, termed FOP perturbations, adapted in the abstract setting of flow categories, flow multimodules, and homotopies of flow bimodules. Along the way, we construct global Kuranishi charts for the relevant moduli spaces following Abouzaid-McLean-Smith's framework and its adaptation to the Hamiltonian Floer setting due to the authors, which may be of independent interest.

We study dynamical orthogonal (DO) approximations of stochastic differential equations and investigate their long-time behaviour. The DO formulation represents the solution by a low-rank decomposition and leads to a coupled system consisting of an evolution equation on the Stiefel manifold and a reduced stochastic process. We establish the well-posedness of the strong DO system and derive quantitative error estimates between the original stochastic differential equation and its low-rank approximation in the Wasserstein distance. Our main contribution is the analysis of invariant probability measures for the DO dynamics. Under suitable dissipativity, Lipschitz continuity, and non-degeneracy assumptions on the coefficients, we prove the existence of an invariant probability measure for the strong DO system. The proof combines uniform moment estimates, a Krylov–Bogoliubov argument for an associated frozen system, and a Kakutani-Fan-Glicksberg fixed-point theorem to recover the self-consistent dynamics. We further show that the induced low-rank process admits an invariant probability measure and discuss the structure of invariant measures through several illustrative examples. These results provide a rigorous foundation for the use of dynamical low-rank approximations in the approximation of long-time statistical properties of stochastic dynamical systems.

We consider an infinite system of SDEs with Fleming-Viot noise indexed by $k=0,1,2,\dots$, whose parameters $\alpha,\lambda$, and $\nu$ are the (deleterious) selection coefficient, the (uni-directional) mutation rate, and a quantity which determines the size of the system's fluctuations. The SDE's unique weak solution $X(t) = (X_k(t))_{k=0,1,2,...}$ models what is known in population genetics as Muller's ratchet. Here, $X_k(t)$ stands for the frequency of individuals carrying $k$ deleterious mutations. Since the mutation process is uni-directional, $t\mapsto \inf\{k: X_k(t)> 0\}$ is non-decreasing for almost every path of $X$, and we refer to an increase as a click of Muller's ratchet. A long standing question concerns the clicking rate of Muller's ratchet. Using Duhamel's principle for semigroups, we give a partial answer by approximating $E(\sum_{k=1}^\infty kX_k(t) )$ and $E\big(X_0(t)\big)$ up to $O(1/\nu^2)$ for fixed $\alpha$, $\lambda$ and $t>0$. Our results suggest that $\psi:=\nu \alpha e^{-\lambda/\alpha}$ is a crucial quantity also when the mutation/selection ratio $\theta = \lambda/\alpha$ is moderately large: for large $\nu \alpha$, clicking of the ratchet on the time scale $\frac 1\alpha \log \theta$ becomes rare as soon as $\psi$ becomes large.

We study a dynamic inspection problem for infrastructure systems in which multiple vehicles are routed and scheduled to monitor components subject to heterogeneous and stochastic failures. A key challenge is endogenous uncertainty in which failure-time realizations are filtered through prior inspection actions and failure-propagation dynamics. This yields a multi-stage decision problem that integrates routing, scheduling, and decision-dependent reliability dynamics over time. We formulate the problem as a multi-stage stochastic mixed-integer program that jointly optimizes routing and scheduling decisions under endogenous uncertainty. We develop a stochastic dual dynamic integer programming (SDDiP) algorithm that integrates dual approximation, integer state reduction, and sampling-based forward simulation to approximate cost-to-go functions. In parallel, we propose a reinforcement learning framework that learns job clustering structures and routing policies through interaction with simulated system dynamics and failure processes. Numerical experiments on infrastructure networks with diverse topologies and failure patterns show that SDDiP yields high-quality solutions but faces scalability limitations, while the learning-based approach achieves strong scalability with competitive performance, highlighting a trade-off between optimality and tractability. This work advances the modeling of endogenous uncertainty in multi-stage stochastic routing problems and bridges stochastic programming with learning-based approaches. The results provide guidance on when to deploy optimization-based versus learning-based methods, enabling more effective and scalable inspection planning in practice. More broadly, the framework supports risk-aware allocation of inspection resources and improved reliability of critical infrastructure systems.

A single observed network reflects several mechanisms at once: communities, hubs, and clustering coexist in one graph, each a different model. We treat the network as a combination of candidate mechanisms and study, from a single graph, how strongly each mechanism contributes and how they combine. We address two questions. The first is how to measure each mechanism's contribution when the mechanisms must themselves be estimated from the graph: fitting the mechanisms and their strengths from the same data biases the strengths toward zero, and a correction removes this bias and yields valid confidence intervals. The second is whether the rule of combination is itself recoverable: when a graph is generated by two mechanisms acting together, the graph alone determines whether they combine additively or interact, exactly when the graph is dense enough, a sharp threshold below which no test can decide. The estimate calibrates the candidate mechanisms against the observed edges. We establish matching minimax rate, against a known-design benchmark and the estimated-design problem itself, confirm the methods in simulation, and apply them to real networks, where the signed coefficients recover known structure and, in one case, a confidence interval excludes any positive contribution from a candidate mechanism.