Papers

Let $k$ be an algebraically closed field of characteristic zero and $G$ a finite group. We realize the based Grothendieck ring $R_G$ of ${\mathcal{R}\mathit{ep}}(D(G))$-module categories as the degree-two cocycle-decorated double Burnside ring and derive an explicit Clifford formula for multiplication and for the action of $R_G$ on the Grothendieck group of ${\operatorname{\operatorname{\mathsf{Vec}}}}_G$-module categories. We determine the extremal based ideals, study factorization through smaller groups, and prove a Mackey theorem for standard two-sided subgroup inductions. We prove that $\mathbb C\otimes_{\mathbb Z}R_G$ is semisimple exactly when $G$ is cyclic. We study the Brauer–Picard action on indecomposable ${\operatorname{\operatorname{\mathsf{Vec}}}}_G$-module categories and show that it is transitive exactly when $G$ is abelian of square-free exponent. For abelian $G$, we determine the possible abstract group types of Lagrangian subgroups of $G\oplus\widehat G$, apply the known orthogonal classification in the homocyclic case, and exhibit same-type nonconjugate Lagrangians for mixed exponents.

Scanning-probe x-ray tomography is useful for imaging nanoscale structures of a sample. The image resolution, which can reach down to 10 nm by the increased brightness and coherence of the x-ray optics, however, is highly susceptible to experimental error. Failure to address these errors can lead to a smeared image and, in the worst case, to misinterpretation of the imaged object's structure. In this work, we present a novel optimization-based approach to calibrate a common yet challenging source of experimental error, the drifts of the scanning positions, while simultaneously reconstructing the object. This approach uses the coupled and complementary information from different measurements to enforce consistency between the measurements and the reconstruction. We illustrate the proposed approach on both synthetic and real tomography images and show its superior performance compared with reconstruction without explicit error calibration.

Outpatient appointment scheduling must balance provider idle time, overtime, and patient waiting under uncertain service times, punctuality, and attendance. These factors may follow patient-and-time-dependent probability distributions, which existing models rarely incorporate simultaneously because of modeling and computational complexity. We propose a stochastic programming model that captures exponentially many scenarios with polynomially many variables and constraints, without sampling. It accommodates patient-dependent service times and patient-and-time-dependent no-shows and arrival times. We also discuss estimating these distributions from electronic health records. We examine how personalized reminders and changes in attendance due to incentives or negative clinic experiences affect scheduling. Our model optimally solves instances of up to 14 patients in reasonable computational time. Its schedules outperform classical Bailey-type rules. Even the best-performing variant has 97% higher expected cost on average. Incorporating patient-dependent service times reduces total costs by 34% on average. Accounting for patient-and-time-dependent unpunctuality and no-shows reduces costs by 12% and 67%, respectively. Personalized reminders can reduce costs by 23%, and sensitivity analyses show that these gains withstand moderate errors in estimated distributions. These findings show how modeling individual patient behavior and tailoring communication can reduce waiting, idle time, and overtime.

Public health measures at mass gatherings work only if people follow them, and adherence depends both on how instructions are communicated and what others do. We link these behavioural factors to pathogen transmission in a local-scale, agent-based exposure model combining pedestrian dynamics with airborne transmission. Rather than fitting a parametric behavioural model, agents draw their adherence at run time from survey respondents (2,170 attendees of UK sports and music events) who reported the same context: we vary stewards' communication effectiveness and adherence of role models, as well as agents' shared social identity with each, assuming that strong shared social identity increases their influence. In a ticket-checkpoint queue of 101 agents, effective communication combined with strong shared social identity with stewards reduces the number of highly exposed agents by 82% for mask wearing. In contrast, strong shared social identity with non-adhering role models gives almost nine times as many highly exposed agents as with adhering role models. Physical distancing was not monotonically beneficial, because adherence changes how agents move: an infectious agent that kept distance moved along the edge of the queue, leading to similar results across conditions. Our pipeline transfers to any survey that includes behavioural measures and contextual variables.

We study the long-time behavior of Wasserstein gradient flows for interaction energies \[ \mathsf E[\mu] = \frac12\iint_{M\times M}K(x,y)\,\mathrm d\mu(x)\,\mathrm d\mu(y) \] on a closed manifold $M$. For kernels diagonal in a Laplace eigenbasis with nonnegative spectral coefficients, we prove a differential inequality relating the relative entropy to the energy gap. Consequently, for any nonnegative initial density $u_0\in L^p(M)$, $p>1$, the energy gap is integrable in time and satisfies \[ \mathsf E[\mu_t]-\mathsf E_{\min}=o(t^{-1}). \] If all spectral coefficients are positive, the flow converges weakly to the constant measure. These interaction energies need not be geodesically convex in Wasserstein space, and the associated flows contain no diffusion; their global convergence therefore does not follow from standard Wasserstein gradient flow theory. The kernels covered by our results include zonal kernels on spheres, kernels arising in transformer models, regularized Riesz kernels, and inverse fractional Laplacian kernels. We also investigate the sharpness of the $o(t^{-1})$ rate. For any smooth kernel in this class with infinitely many positive spectral coefficients and any $\delta>0$, we construct a solution of the linearized flow whose energy is comparable to $t^{-1-\delta}$ along a sequence of times tending to infinity. Moreover, for any $\delta>0$, by choosing a suitable inverse fractional Laplacian kernel on the flat torus, we construct an exact solution of the nonlinear Wasserstein gradient flow whose energy is comparable to $t^{-1-\delta}$. The nonlinear construction is based on uniform-in-time estimates for the evolution of the dyadic Fourier coefficient blocks and a blockwise energy-persistence argument. These estimates also yield a uniform-in-time quantitative comparison between the nonlinear Wasserstein gradient flow and its linearization.

Nonlocal models have found great success in recent years. In particular, models that replace spatial derivatives with strongly singular integral operators involving finite interaction radii are used in a wide range of applications, including image processing, continuum mechanics, and many other areas. However, from the perspective of mathematical analysis, difficulties arise when considering the appropriate notion of “boundary” conditions. (The nonlocal analogue of the boundary is often called a “collar” and is typically not a lower-dimensional set.) For instance, merely enforcing homogeneous Dirichlet-type constraints on the collar fails to guarantee a number of essential properties used in the analysis and numerical simulation of these models, as we show in the present work. In the local setting, working in a Sobolev space of functions that are zero on the boundary allows one to integrate by parts with no boundary term, approximate by test functions, extend by zero to the full space, and apply the Hardy and Poincaré inequalities. We prove that the nonlocal analog of each of these fails if one only requires the functions to be zero on the collar, and that one must move to a smaller, more restrictive space to enjoy these properties. In addition, we introduce a new lifting operator, correct an error in a previously published result on a nonlocal Green's theorem, and establish how the properties listed above relate to existing notions of fractional Sobolev spaces.

We establish an exact parametrized reduction of the classical Willmore flow of Hopf tori in the round three-sphere to the spherical elastic flow. In Willmore-flow time the reduction is $(\partial_t\gamma)^{\perp}=-4\nabla_{L^2}\mathcal E(\gamma)$; its global reconstruction uses a pullback-circle-bundle normalization and does not require preservation of simplicity along flow lines. Combining the known global subconvergence of the spherical elastic flow with Pozzetta's full convergence theorem then yields full smooth convergence of flow lines, modulo domain diffeomorphisms, for every simply parametrized Hopf torus at arbitrary initial energy. We then obtain quantitative conclusions for the unreparametrized trajectory: finite normal $L^2$-metric length, total-variation convergence of the pullback area measures, bounded-Lipschitz convergence of the image measures, and explicit tail estimates. The density of the full varifold endpoint equals the covering multiplicity of the limiting elastic profile, and Pinkall's marked flat lattices converge. In particular, a simple initial Hopf immersion with energy arbitrarily close to $4\pi^2$ from above can smoothly converge to some parametrization of the Clifford torus with multiplicity two, although its limiting conformal class in moduli space is exactly the square class.

We prove that the generalized Fermat equation $x^3+y^5=z^7$ has no solution in nonzero coprime integers, and we make explicit how the proof extends the prior work cited below. Dahmen-Siksek had established the signed local descent and eliminated the three cases in which the associated degree-seven algebra is reducible. Putz had then proved that the remaining irreducible case can involve only seven septic fields: six pure fields and one exceptional field. The missing step was to exclude those seven fields. For the six pure fields we construct an explicit Fano resolvent, descend to a smooth plane quartic of genus three over $\mathbb{Q}(\sqrt{-7})$, and prove that its rational-parameter locus consists only of five points above the branch values. For the exceptional field we combine the modularity and conductor results of Pacetti-Villagra Torcomian with level lowering over $\mathbb{Q}(\sqrt{5})$ and a finite Hecke comparison at $29$. We also reconstruct the three reducible-sector arguments of Dahmen-Siksek as independently replayable calculations, including their database-free identification of the quadratic field, and add an intrinsic formal-group treatment at the ramified prime. The paper labels every major step as a literature input, reconstructed input, or new argument. All project-specific finite calculations are supplied as exact certificates, programs, inputs, and authenticated logs.

The problem of differentiating Lipschitz curves in Banach spaces goes back to Tamarkin in the early 1930s and is one of the origins of the Radon–Nikodým property in classical Banach space theory. In this article we characterize those quasi-Banach spaces $X$ for which every Lipschitz curve $F\to\mathbb{R}\to X$ admits a point of differentiability and prove that this property holds if and only if $X$ is isomorphic to a Banach space with the Radon–Nikodým property, thus substantiating a conjecture of Kalton, communicated personally to the first-named author in 2008. Equivalently, every nonlocally convex quasi-Banach space admits a nowhere differentiable Lipschitz curve. Our result is obtained from a quantitative characterization of local convexity in terms of the infinitesimal oscillation of Lipschitz curves. Motivated by this obstruction, we then investigate positive differentiability phenomena in $\ell_p$, $0<p<1$, and identify natural decoupled-coordinate constructions for which Lipschitz regularity nevertheless implies almost-everywhere differentiability. We also contrast this behavior with metric differentiability for the natural metric and the quasi-metric on $\ell_p$.

We introduce a magnetic coherent–state frame over the Landau levels of the two–dimensional Landau Hamiltonian, indexed by a discrete metric space combining spatial and energy degrees of freedom. Although overcomplete, the frame is almost–orthogonal, and we use it to endow the CAR $C^*$–algebra over the one–particle Hilbert space with an asymptotic quasi–local (AQL) structure, in which graded commutators of observables localized on disjoint index sets decay exponentially rather than vanish exactly. By means of a family of conditional expectations onto the resulting local subalgebras, we construct a Fréchet $*$–subalgebra of almost–local observables, invariant under the discrete group of lattice translations, which act on it by continuous automorphisms. For the full continuous group of magnetic translations we establish a quantitative control on how local observables spread through this almost–local algebra, and we show that this control suffices to extend graded asymptotic abelianness from the lattice to the entire continuous translation group. The results provide a discrete–continuum dictionary intended as the starting point for extending Lieb–Robinson–type techniques from lattice systems to interacting continuum electrons in a magnetic field.

Let $R$ be the closed linear span of the Rademacher functions in $L^1(0,1)$. We prove that $L^1(0,1)/R$ embeds isomorphically into $L^1(0,1)$, answering affirmatively a question of González and Martínez-Abejón \cite[p.~140]{GM97}. We then use this embedding to show that the weak Calkin algebra of the quotient is uniformly incompressible. \end{abstract

We prove that the irrationality exponent of $\zeta(3)$ satisfies $\mu(\zeta(3)) < 5.5138800$, improving the bound $5.513891$ obtained by Rhin and Viola in 2001, which has been the best known for twenty-five years. The proof is an application of their Theorem 5.1 at a new integral direction. That direction is not reached by enlarging the search range: its eight parameters are 52000 times theirs, shifted by at most 23. The mechanism is the one recently used by Bai for $\pi$. In particular the Rhin-Viola point, optimal in every region covered by enumeration, is not optimal.

What distinguishes the solution of a difficult mathematical problem from a conceptual revolution? Some mathematical innovations do more than establish new results: they reorganize the relations among problems. We propose to describe such events through changes in the transfer structure of mathematics, namely the pattern determining which problems can naturally inform one another, which methods can move between them, and which new questions become accessible. The eighteenth-century controversy over the vibrating string provides our central historical case. D'Alembert found a famous model for wave propagation, Bernoulli proposed a modal interpretation, Euler broadened the class of admissible profiles, and Lagrange constructed a finite-dimensional strategy based on modal decomposition. Later, Fourier analysis gave the trigonometric language in which all the previous descriptions can interact. This motivates our Rewiring Thesis: a conceptual revolution is a persistent, large-scale reorganization of transfer relations among mathematical problems. The productive representation is one mechanism capable of producing such rewiring, but some stress tests (involving Galois, Riemann, Lebesgue-Schwartz and Wiles) show that it is not the only one. The musical notation provides an independent case in which the representation becomes operationally generative without being informationally complete. The resulting framework suggests a stronger criterion for artificial mathematical creativity than problem solving alone. The relevant question is not merely "whether AI can solve a famous conjecture", but "whether it can recognize that an inherited conceptual organization is itself an obstruction and produce a new one that reorganizes several problems at once". In this sense, the decisive achievement may be the invention of "the next Fourier transform".

This paper describes a 'model' for the space of convex quadrilaterals, that is, a set $\mathbb{Q}$ of convex quadrilaterals in the plane which gives a 'cross-section' of the equivalence classes of similar convex quadrilaterals, in the sense that every convex quadrilateral is similar to exactly one member of $\mathbb{Q}$. It is also has the property that quadrilaterals which have nearly the same vertices are close to each other in the model in the Hausdorff metric. We also parameterize our model with the 4-cell and use that to describe and investigate various classes of quadrilaterals. For example, we show that the trapezoids form a 3-dimensional closed subset of the model which separates the model into 3 disjoint open sets, each homeomorphic with a 4-cell. We used SageMath to compose the latex for this note, and to make the Sage Cell Interacts to explore the model, available at sagelets.org.

We prove that every bounded-to-one action of a finitely generated commutative monoid has asymptotic dimension at most the torsion-free rank of its group completion, with control uniform for a fixed monoid, generating set, and generator fiber bound. The estimate is sharp. Together with the equality of classical and Borel asymptotic dimensions for these actions, it gives the sharp Borel rank bound and recovers hyperfiniteness for bounded-to-one Borel actions of countable commutative monoids.

We introduce Rolling Conformal Prediction (rolling-CP), a distribution-free predictive inference method for the setting of sequential model training. Specifically, given a data stream $(X_1,Y_1),(X_2,Y_2),\dots$, at each time $n$ the trained model may depend on the observed history $\{(X_i,Y_i)\}_{i<n}$. This setting arises naturally in modern sequential training, including one-pass training over massive datasets and continual fine-tuning or test-time adaptation of language models during deployment. Rolling-CP first calibrates each incoming observation against the current predictor and then rolls it into future training. In this way, we avoid the need for data splitting. Remarkably, although the models at times $n=1,2,\dots$ may have entirely different properties and accuracy levels, for exchangeable data it is nonetheless possible to establish a guarantee of marginal coverage, with a familiar universal factor-two guarantee (a worst case guarantee of $1-2\alpha$ coverage, as compared to the target level $1-\alpha$), without any assumptions of stability or any restrictions on the model training process. For i.i.d. data streams, we further prove high-probability training-conditional validity uniformly over time; under stability conditions, coverage guarantees sharpen towards $1-\alpha$. Numerical experiments on sequential regression, multiclass SGD, and one-pass neural-network training further demonstrate the practical effectiveness of rolling-CP.

Let $\nu$ be a compactly supported Borel probability measure on $\mathbb R^{d}$ with $\nu(B(x,r))\leq Cr^{a}$ for some $a>0$. Refine a dyadic cube exactly when its mass is at least $t$, and let $\mathcal{L}_{\nu}(t)$ be the mean depth at which this refinement stops. We show that the lower and upper geometric-mean quantization dimensions of $\nu$ are the lower and upper limits of $\log(1/t)/\mathcal{L}_{\nu}(t)$. The dimension exists precisely when $(q-1)\sum_{Q}\nu(Q)^{q}$, summed over all dyadic cubes, converges as $q\downarrow1$, and it is then determined by this limit. The mass-threshold formula yields harmonic integral bounds in terms of the local dimensions and encloses entropy and quantization dimensions in a common spectral interval. Convergence in law of the local information rates is equivalent to convergence of the rescaled spectra in a window of width $1/k$ around $q=1$; the two dimensions are then the arithmetic and the harmonic mean of the limit law, and we quantify their difference by sharp bounds and variance identities. Without any convergence assumption, vanishing threshold variance still forces equality of the corresponding lower and upper dimensions. Bernoulli mixtures realise every local-dimension law with compact support in $(0,1]$, and a regime-switching example separates convergence in law from almost-everywhere convergence.

Every Dubrovin–Frobenius manifold provides a solution to the WDVV equation, that was initially formulated in the study of the moduli space of curves. During the last two decades special attention was given to the "open" version of the moduli space of curves. Its genus zero intersection theory is governed by the system of equations called open WDVV equation, that extends "classical" WDVV equation. In this note we study Dubrovin–Frobenius manifold structures on the Hurwitz spaces of genus zero. We construct explicitly the solutions to open WDVV equation for these Dubrovin–Frobenius manifolds and show that these solutions appear as the restriction of the Dubrovin–Frobenius structure to its discriminant.

We answer three questions from Boulabiar's survey on truncated Riesz spaces. First, the truncation band projection property coincides with the principal projection property, with no completeness or Archimedean hypothesis needed, and the band projection onto a truncation band is then given explicitly by the truncation itself. Second, a truncation on any Riesz space is unital exactly when its fixed-point set has a supremum. Hence bounded truncations on $KB$-spaces are unital, and a nonunital truncated Banach lattice with bounded truncation contains a closed sublattice copy of $c_0$ and is never an $AL$-space, while on the $AM$ side the smallest truncation unitization norm is always an $M$-norm, though the largest one need not be. Third, a truncation homomorphism whose domain truncation is Archimedean need not be a Riesz homomorphism, as an explicit counterexample built from $c_0$ and $c_0/c_{00}$ shows, but it is one modulo truncation infinitesimals whenever the codomain space is Archimedean.

We construct in ZFC a unital, simple, infinite-dimensional $C^*$-algebra whose nonzero irreducible representations form a single unitary equivalence class, but which admits a faithful representation on a separable Hilbert space. The algebra contains a unital copy of the canonical anticommutation relation (CAR) algebra, and the normalized CAR trace has a unique extension among all states. This extension is tracial, and its GNS representation is separable and faithful, with weak closure the hyperfinite $\mathrm{II}_1$ factor. In contrast, every nonzero irreducible representation acts on a Hilbert space of density $2^{\aleph_0}$. The construction separates the added unitaries into shell terms, controlled by finite CAR relations, and rank-one defect terms. These relations determine every irreducible representation of the generated algebra and every extension of the CAR trace. The algebra has norm density $2^{\aleph_0}$; consequently, the continuum hypothesis (CH) is equivalent over ZFC to the existence of a counterexample to Naimark's problem of norm density $\aleph_1$.

We compute the orbifold Euler characteristics of strata of $k$-differentials on the moduli space of smooth pointed curves. Our main result is a closed coefficient-extraction formula in terms of hyperbolic functions, valid for arbitrary strata and generalizing the Harer–Zagier formula for the Euler characteristic of the moduli space of curves. In particular, these Euler characteristics are polynomial in the prescribed orders at the marked points. The proof first uses Riemann–Roch and Serre duality to reduce the computation to the Hurwitz case $k=0$, which is then evaluated using the representation theory of the symmetric group and Fock-space techniques. We also give an intersection-theoretic expression for the same invariants as tautological integrals over double ramification cycles and derive a recursion which uniquely determines them from genus zero. For odd $k$ and odd orders, we introduce a spin-weighted refinement and formulate corresponding closed and intersection-theoretic conjectural expressions. Finally, we determine the large-genus asymptotics of the Euler characteristics.

We investigate the maximal total quantum dimension at a bounded categorical rank cutoff $R$ in three-dimensional Dijkgraaf–Witten theories. This optimization problem physically governs the maximal state-optimized torus topological entanglement entropy at a fixed torus ground-state degeneracy, while providing a quantitative constraint on the search space required to classify these topological order phases. We analyze how topological twists (3-cocycles) and gauge-group structures independently govern this trade-off. For elementary-Abelian gauge groups at a fixed prime $p$, we prove that introducing cocycle twists elevates the maximum total quantum dimension from square-root growth $\Theta_p(R^{1/2})$ to linear growth $\Theta_p(R)$ in the rank cutoff, driven by the geometric protection of cyclic holonomies. For untwisted theories, we establish sharp envelopes for symmetric and alternating gauge groups, and derive group-theoretic bounds across almost-simple, solvable, and radical-free families. In particular, almost-simple groups and their direct products obey a quadratic-logarithmic upper envelope: $\log\mathcal{D} = O((\log R)^2)$. Finally, we compare these scales against WZW models and discuss the status of the general quadratic-logarithmic envelope conjecture.

We study integral dependence over left ideals using G-left ideals, namely left ideals maximal with respect to the failure of integrality of a fixed element. We establish several characterizations of G-left ideals, and show that their intersection is precisely the upper nilradical. Motivated by the Köthe conjecture, we introduce the class of left KI-rings as rings in which sums of left ideals integral over a fixed left ideal remain integral over that ideal. It is shown that left artinian rings, PI-rings, algebraic algebras, and algebras of dimension smaller than the cardinality of the ground field are left KI-rings.

We develop a contraction-based perspective on impact-time guidance that augments a baseline homing command with a timing bias. The proposed perspective treats the prescribed schedule as a moving time-to-go isochron and regulates motion normal to that set through velocity-normal lateral acceleration while the interceptor's speed remains constant. We derive a transport equation that characterizes homing-compatible time-to-go coordinates and define a predictor defect that quantifies the mismatch of approximate maps. We show that the scalar timing channel induces a coordinate-invariant rank-one metric on the normal quotient. To account for bounded lateral acceleration, we formulate a robust scalar filter and derive a necessary and sufficient condition for pointwise feasibility. We then show that terminal calibration and funnel invariance establish first interception at the prescribed time under the stated assumptions. We also develop a preterminal alignment and homing handover that avoids singular inversion as lateral timing authority vanishes near collision-course alignment. The proposed perspective accommodates analytic, numerical, and learned time-to-go maps that satisfy the required calibration and regularity conditions.

We give a complete classification of algebraically primitive Teichmüller curves in $\Omega\mathcal M_g(2g-2)^{\mathrm{hyp}}$ for $g\ge5$. They are precisely the curves generated by the double regular \((2g+1)\)-gon when \(2g+1\) is prime, and by the regular \(4g\)-gon when \(g\) is a power of two.

We find a new coupling between the Airy sheet $\mathcal{S}$ and the Airy line ensemble $\{ \mathcal{A}_n \}_{n \in \mathbb{N}}$, which we call the down-the-middle isometry. In this coupling, the Airy sheet is encoded by last passage values that start at $\mathcal{A}_k(0)$ for some $k \in \mathbb{N}$ and end on the top line $\mathcal{A}_1$. Using this coupling, we give a new construction of the (extended) Airy sheet as a scaling limit of Brownian last passage percolation and study coalescence of geodesics in the directed landscape. This coupling extends to discrete polymers and last passage percolation models, as well as to the KPZ line ensemble and KPZ sheet.

We study the static matter constraint associated with a conical deficit that varies along a string near a wormhole throat. A positive scalar-dependent factor multiplying the complete Abelian-Higgs Lagrangian couples a finite-width vortex to a canonical kink. At critical coupling, we obtain the leading longitudinally varying vortex tension. The flat-kink fluctuation operator has an even translational mode and an odd shape mode of mass $\sqrt{3}/L_\chi$. Reflection symmetry excludes the translational mode from the leading sourced response. We formulate the response with nonzero dressed-string boundary conditions at the longitudinal ends and derive a finite-range transverse halo. In the overlap $w\ll\rho\ll L_\chi$, its mixed stress satisfies $2\pi C\,\delta T^{\chi\,\rho}{}_l=\partial_l\mu_v^{(0)}$ to leading order, where $2\pi C$ is the azimuthal circumference. Together with the local weak-gravity deficit relation, this supplies the leading mixed Einstein constraint of a restricted varying cone. At fixed longitudinal position the sourced halo instead decays beyond $L_\chi$. Consequently, it cannot sustain that restricted cone throughout an outer overlap unless another source contributes the missing mixed stress. This obstruction is conditional on the response of the wormhole-supporting sector and is not a no-go result for general axisymmetric metrics. A finite-width calculation checks the linear-response crossover. We specify the perturbative domain and explain why a rotating extension requires an independent solution of Gauss's law. No complete backreacted wormhole solution or stability result is claimed.

Let $\bar{X}$ be a smooth projective connected complex curve, $D \subset \bar X$ be a finite set of reduced points, and $X=\bar X\setminus D$. We give a moduli-theoretic construction of an exact equivalence between logarithmic flat bundles on $(\bar X, D)$ with nilpotent residues and semistable logarithmic Higgs bundles of degree zero with nilpotent residues. This extends the tame Simpson-Mochizuki correspondence on polystable bundles.

The "$S_{n,n}$ conjecture" asserts that there does not exist a simple graph on $n$ vertices with Laplacian spectrum $\{0,1,2,\ldots,n-1\}$ for any integer $n \geq 2$. This conjecture has already been proved when $2 \leq n \leq 15$ and when $n \geq 6,649,688,933$. We prove all remaining cases and thus establish that the conjecture is true.

The quantum circuit complexity of an evolving many-body quantum system is believed to exhibit a sustained growth, maintained for timescales much longer than the onset of thermalization. Most previous works have focused on models which violate energy conservation, such as random unitary circuits. Here we study generic, local time-independent Hamiltonian dynamics. We unconditionally prove that for generic local Hamiltonians and typical initial product states at high effective temperature, the robust quantum circuit complexity must grow over a very long period of time, attaining an exponentially large value at late times. Our approach relies on two structural properties that we prove rigorously: (i) generic local Hamiltonians satisfy generalized spectral no-resonance conditions of arbitrary order, and (ii) typical high-temperature product states are effectively supported in exponentially many energy eigenstates. These properties have been widely assumed without proof in prior works. As corollaries of this result, we show the late-time state displays robust volume-law entanglement that is irremovable by polynomial-size circuits, and we establish a no fast-forwarding result for generic local Hamiltonians. We also lower bound the complexity of preparing sufficiently large subsystems with local quantum channels, in contrast with sufficiently small thermalizing regions which we show retain low complexity at late times.