Sophie Germain (1776-1831) was the first woman we know who did important original research in mathematics, specifically in elasticity theory and number theory. Celebrating her semiquincentennial year, we outline Germain's recently unearthed number theory results on Fermat's Last Theorem, in the context of her life, work, and interactions with Lagrange, Legendre, and Gauss. For two centuries her accomplishment on Fermat's Last Theorem was thought to consist of a single theorem attributed to her in a publication by Legendre, the first general result towards proving Fermat's Last Theorem. But recent discoveries in her handwritten manuscripts and correspondence with Legendre and Gauss show that she accomplished much more, albeit forgotten. In particular, she had a grand plan for proving Fermat's Last Theorem in its entirety, and carried this plan a long way, using then new tools, e.g., congruence, modular primitive roots, and permutations.

We point out that a simple late-time cosmological model where our Universe can absorb "baby universes" explains the exponential expansion of our universe without the need of a cosmological constant and leads to a z-dependence of the parameter w(z) in the equation of state. In this model w(z) is less than -1 for z sufficiently large.

We say that a $d$-regular graph is a $γ$-expander if for every not too large set of vertices $S$, there are at least $γd |S|$ edges leaving $S$, and we say that a graph $G$ is $γ$-far from bipartite if at least $γe(G)$ edges need to be removed to make it bipartite. We prove that there exists an absolute constant $K$ such that any $n$-vertex $d$-regular $γ$-expander with $d \ge (γ^{-1} \log n)^K$ is Hamiltonian, provided that it is bipartite or $γ$-far from bipartite. As applications, we obtain highly robust versions of recent important results on the Hamiltonicity of Cayley graphs and Kneser graphs. As part of our proof, we prove a random connecting lemma for sublinear expanders which might be of independent interest.

For each integer $n\geq 3$, let $W_n$ denote the wheel graph obtained by connecting a single vertex to all vertices of a cycle of length $n$. In particular, $W_6$ is obtained from the Petersen graph by contracting three edges incident with a common vertex. In this paper, we determine all $4$-connected graphs that do not contain $W_6$ as a minor.

For a proper immersed minimal disk in $\bf{R}^N$ with quadratic area growth, we show that any harmonic function whose negative part grows at a slow sub-linear rate is constant. This leads to a higher codimensional Bernstein theorem for minimal disks contained in a sub-linearly growing cone. The catenoid, helicoid and Enneper's family of surfaces together show that this result is optimal. We also show uniform Hölder regularity of harmonic functions.

For every adjunction of stable $\infty$-categories -- or more generally, in any locally stable $(\infty,2)$-category -- we give a simple procedure for inverting the twist and cotwist functors associated to this adjunction. As a consequence, we obtain an explicit construction for a left and right adjoint to the inclusion of the $(\infty,2)$-category of spherical adjunctions of stable $\infty$-categories into all adjunctions. We utilize these adjoints to give a description of the walking spherical adjunction, a locally stable $(\infty,2)$-category which classifies spherical adjunctions, and to provide a synthetic proof of the fact that every spherical functor admits infinitely many left and right adjoints.

We derive closed-form propagators for any $K$-qubit subsystem of a closed $N$-qubit network with a single conserved excitation. A single transition amplitude simultaneously controls excitation flow between subsystems, the positivity and complete positivity of every propagator, the entanglement entropy of every subsystem, and the quantum Fisher information for global parameters. Positivity and complete positivity coincide, determined solely by the direction of excitation flow, independently of subsystem size, coherence, or entanglement structure. A propagator is positive and completely positive if and only if it contracts the subsystem state toward its fixed point. The ensemble of propagators collectively constrains global properties inaccessible to any single subsystem. For single-qubit subsystems, we characterize the ensemble's fixed-point distribution and domain of positivity, finding a band of states that lies inside the positivity domain of every propagator yet is never visited by the physical dynamics. The quantum Fisher information decomposes into state and process contributions over any observation window $[t_1,t_2]$, with the state contribution bounded while the process contribution grows secularly. The total Fisher information is minimal when all future propagators are nonpositive and not completely positive, and near its maximum when they are positive and completely positive.

Agents in social networks with threshold-based dynamics change opinions when influenced by sufficiently many peers. Existing literature typically assumes that the network structure and dynamics are fully known, which is often unrealistic. In this work, we ask how to learn a network structure from samples of the agents' synchronous opinion updates. Firstly, if the opinion dynamics follow a threshold rule in which a fixed number of influencers prevent opinion change (e.g., unanimity and quasi-unanimity), we provide an efficient PAC learning algorithm provided that the number of influencers per agent is bounded. Secondly, under standard computational complexity assumptions, we prove that if agents' opinions follow the majority of their influencers, then there is no efficient PAC learning algorithm. We propose a polynomial-time heuristic that successfully learns consistent networks in over $98\%$ of our simulations on random graphs, with no failures for some specified conditions on the numbers of agents and opinion diffusion examples.

Shared multipartite entanglement defines a ``whatever channel'', i.e., a latent communication substrate that does not determine a priori which end-to-end entangled links are activated, but can be configured to support different entanglement-connectivity graphs through Local Operations and Classical Communication (LOCC). Building on this, we propose a resource-driven framework in which multipartite entanglement is treated as a programmable resource that induces a space of admissible entanglement-graph configurations. Within this framework, connectivity provisioning emerges as a particular instance of a more general resource reconfiguration process. To support this paradigm, we introduce a set of structural design parameters that characterize the operational degrees of freedom of the resource and define the admissible transformations independently of the specific mechanism used to realize them. We then formalize Entanglement Rolling as a measurement-based protocol that operates over the induced configuration space, enabling the systematic reconfiguration of the shared resource across a family of multipartite states. Finally, we analyze the proposed framework under realistic noise conditions. Leveraging the Noisy Stabilizer Formalism (NSF), we derive closed-form noise maps that characterize the effect of noise on the resource transformations and show that the proposed approach maintains reliable performance under relevant noise processes.

History matching is a central inverse problem in reservoir engineering, where uncertain reservoir parameters must be calibrated against observations. Although automated history matching can reduce manual effort, practical deployment remains difficult because engineers must still configure heterogeneous workflows involving parameter selection, physically admissible bounds, optimizer choice, hyperparameter tuning, simulator execution, and diagnostic reporting. We propose PetroGraph, a multi-agent framework for intelligent reservoir history matching that decomposes this workflow into specialized agents for model review, experimental planning, parameterization, optimization, simulation, and summarization. The system combines large language model agents with domain-specific tools, retrieval-augmented access to simulator documentation, validation of modified ECLIPSE input decks, human-in-the-loop checkpoints, and an OPM Flow-based simulation backend. This design enables users to initiate and steer history matching through natural language while preserving explicit control over selected parameters and optimization settings. We evaluate PetroGraph on three reservoir models of increasing complexity: the synthetic SPE1 model, the faulted SPE9 benchmark, and the real-field Norne model. Using weighted normalized root mean square error as the objective, PetroGraph reduces the mismatch by 95% on SPE1, 69% on SPE9, and 13% on Norne. These results demonstrate that multi-agent orchestration can automate key decisions in history matching, lower the expertise barrier for operating complex simulation workflows, and provide a flexible foundation for extensible, domain-aware reservoir model adaptation.

In this note, we establish the periodicity of chains of affine standard Lyndon words in all types and determine tight bounds on that periodicity, greatly generalizing the $A$-type results of arXiv:2305.16299. Our approach crucially utilizes the convexity and monotonicity of arXiv:2505.15432 together with the new idea to consider the polarization of the root system given by increasing and decreasing chains.

In this paper, we investigate how AdS scalar propagators are represented in the celestial basis. Starting from the standard bulk-to-boundary propagator in Euclidean AdS space, we express the propagator in a Schwinger parametrization and construct the corresponding boundary-to-boundary propagator. We then transform the resulting propagators to the celestial basis using conformal primary wavefunctions for both massless and massive scalar fields. For the massless case, the celestial propagator reduces to an effectively two-dimensional boundary-to-boundary object on the celestial sphere dependent on the AdS/CFT conformal dimension $Δ$. For the massive case, the celestial propagator exhibits a nontrivial kernel involving modified Bessel functions, closely resembling the momentum-space radial structure of AdS bulk-to-boundary propagators. The results suggest a structural translation from AdS propagators and celestial propagators.

A search for new physics in the production of three massive gauge bosons (VVV, where V is a W or Z boson) is presented. The event selection is most effective in the Lorentz-boosted regime in which all three bosons have a transverse momentum ($p_\mathrm{T}$) above 200 GeV. Standard model (SM) processes contribute few events in this regime. When a boosted W or Z boson decays hadronically, the decay products tend to form a large-radius jet with substructure that reflects the presence of two quarks from the decay; such jets are called V-tagged jets. Special techniques to reconstruct and select V-tagged jets are applied. Events are categorized according to the number and kinematic features of charged leptons and V-tagged jets. Event yields are obtained in bins of a suitable kinematic variable such as the scalar $p_\mathrm{T}$ sum of the reconstructed objects in the event. No excess over SM expectations is observed. Bounds are placed on Wilson coefficients for a set of mass dimension-6 and -8 operators in the framework of SM effective field theory. The two most stringent bounds placed by this analysis are $-$0.13 $\lt$ $c_\mathrm{W}/Λ^2$ $\lt$ 0.12 TeV$^{-2}$ and $-$0.24 $\lt$ $c_\mathrm{Hq3}/Λ^2$ $\lt$ 0.21 TeV$^{-2}$ at 95% CL, where $c_\mathrm{W}$ and $c_\mathrm{Hq3}$ are dimension-6 Wilson coefficients in the Warsaw basis and $Λ$ is the mass scale of new physics.

We propose a geometric mechanism for fractional quantum Hall states based on impurity-induced correlations within a Landau level. A correlated distribution of ionized impurities partially modifies the Landau-level degeneracy through coherent coupling between cyclotron orbits, generating fractional energy sublevels. The odd-denominator hierarchy emerges naturally from the intrinsic guiding-center quantization and the correlated cyclotron motion. The resulting spectrum reproduces the principal experimentally observed fractional sequences and predicts a strong dependence of fractional-state stability on impurity geometry and layer separation. The absence of an incompressible Hall plateau at filling factor 1/2 follows from cancellation of the geometric correlations responsible for odd-denominator states. These results suggest that impurity-induced geometry may constitute an additional organizing principle in quantum Hall systems.

For a graph $G$ and $e\in [0,1]$, denote by $I_G(e)$ the supremum of densities of $G$ over $n$-vertex graphs with edge density $e$ as $n$ goes to infinity. Liu, Mubayi and Reiher asked if there exists a graph $G$, where $I_G(e)$ has a non-trivial local maximum. In this note we resolve their problem by showing that $I_{K_{2,2,1}}(e)$ has at least two local maxima in $(0,1)$. Additionally, we determine $I_{K_{2,2,1}}(e)$, when $e=(k-1)/k$ for every integer $k\ge 3.$

Algorithmic fairness research often assumes a tradeoff between fairness and accuracy. Yet this tradeoff may not be universal. We test this assumption in the context of U.S. property tax assessment - a setting in which the output of predictive algorithms directly determines the distribution of tax obligations among homeowners. Currently, systematic assessment errors cause owners of lower-valued properties to face disproportionately high tax burdens, creating regressivity in the property tax system. Using data on 26 million property sales spanning 95% of U.S. counties, we conduct three complementary analyses. First, we find that assessment accuracy and fairness - measured using domain-relevant metrics - are strongly correlated across counties under status quo practices. Second, in simulated assessment models, we show that adding property features improves accuracy in most cases, and that when accuracy improves, fairness almost always improves as well. Third, we show that incorporating publicly available Census data into assessment models - a feasible reform in most counties - would significantly improve both accuracy and fairness relative to status quo assessments. Together, these results challenge the presumed universality of the fairness-accuracy tradeoff and demonstrate that well-designed modeling improvements can advance both fairness and accuracy in large-scale public sector systems.

A connected undirected graph $G = (V,E)$ is lower conformally rigid if uniform edge weights maximize the second smallest Laplacian eigenvalue $λ_2(w)$ over all normalized edge weights $w$, and upper conformally rigid if uniform edge weights minimize the largest eigenvalue $λ_n(w)$ over all normalized edge weights; $G$ is conformally rigid if it is lower or upper conformally rigid. This paper establishes a new framework for conformal rigidity through the language of subdifferentials, unifying the variational perspective on eigenvalue optimization with the geometry of edge-isometric spectral embeddings, which are known to characterize conformal rigidity. This subdifferential framework lends itself naturally to techniques of symmetry reduction that motivate the notion of an orbit-isometric embedding - a weaker condition than edge-isometry that accounts for the symmetries of $G$ while remaining sufficient for conformal rigidity. The notion opens the door to tools from representation theory: for a large class of graphs, including all vertex-transitive ones, we show that conformal rigidity is certified by a single eigenvector, resolving an open question and explaining the conformal rigidity of previously unexplained graphs. This extra structure enables a new, algebraically exact certification method for conformal rigidity, bypassing the numerical difficulties of prior approaches. In many cases, the problem reduces to a check of linear feasibility, and in general, to solving a system of quadratic equations via Gröbner bases.

Millimeter continuum emission and self-scattering polarization from protoplanetary disks are widely used to constrain dust properties. Interpreting these observations requires practical prescriptions for the disk emission. However, only approximate formulae are available for the continuum emission, and no widely applicable formula has yet been established for the polarized emission. We aim (i) to assess the validity of commonly used analytic approximations for the (sub)millimeter continuum emission from protoplanetary disks, and (ii) to derive realistic prescriptions for the disk emission for both the continuum and the polarization. We numerically solve the radiative transfer equation in an isothermal, constant-density plane-parallel slab, including dust absorption, emission, and self-scattering with full Stokes parameters. We find that commonly used analytic approximations for the continuum emission are systematically about 10 to 15% lower than our numerical solutions. Consequently, SED analyses of (sub)millimeter observations that adopt these formulae are likely to overestimate the optical depth (and thus the disk mass) and the dust temperature, and underestimate the albedo (and thus altering the inferred constraints on grain size). We also provide empirical fitting formulae that reproduce our numerical results for the continuum emission and polarization fraction. These formulae will enable observational data analyses to be carried out more accurately and efficiently than with the conventional approaches. For the analysis of (sub)millimeter observations, we recommend using our new empirical formulae or interpolation of our numerical results, rather than commonly used approximations.

We present the first systematic measurement of the tilt of the pion emission source in relativistic Au+Au collisions at center-of-mass energies per nucleon pair, $\sqrt{s_{NN}}$ = 7.7, 14.5, 17.3 and 27 GeV, using data from the STAR experiment. The tilt angle and final freeze-out eccentricity are extracted through azimuthally sensitive femtoscopy of identical pion pairs. Our results reveal a strong dependence of the tilt parameter on the pair transverse momentum, indicating that the apparent source geometry is strongly coupled to expansion dynamics. Moreover, we observe a rapid decrease of the tilt magnitude with increasing collision energy, consistent with the emission source approaching longitudinal boost invariance at higher energies. These findings demonstrate that the commonly assumed boost-invariant geometry is insufficient and highlight the necessity of exploring the spatial structure of a tilted source, which is required in hydrodynamic models to reproduce features of the longitudinally expanding system, such as the slope of the directed flow. Comparisons with the UrQMD transport model show that it reproduces the overall energy dependence of the tilt magnitude qualitatively, but not quantitatively.

The Majorana stellar representation translates abstract quantum spin states into intuitive geometric constellations on the Bloch sphere, revealing symmetries, degeneracies, and correlations that traditional algebraic methods often obscure. Within quantum information science, this framework provides a powerful lens for characterizing symmetric multi-qubit and higher-spin systems. By encoding entanglement directly into spatial coordinates, the constellation geometry yields exact measures of concurrence, three-tangle, and genuine multipartite entanglement, while its dynamical evolution uncovers internal anomalous contributions to geometric phases. While interest in stellar representations has resurged, existing literature remains fragmented, lacking a unified treatment of these entanglement-specific metrics and their higher-dimensional dynamics. This review synthesizes the entanglement-centric perspective on Majorana representations, bridging discrete algebraic classifications (e.g., SLOCC orbits) with continuous geometric interpretations. Crucially, we highlight how this framework circumvents \#P-hard computational bottlenecks, leveraging polynomial-time tractability to evaluate multipartite invariants. We detail the interplay between constellation topology and higher-spin Berry/Hannay phases, explore extensions beyond pure symmetric states, and review applications in quantum metrology, state engineering, and condensed-matter physics. By foregrounding entanglement as the unifying theme, this comprehensive examination establishes Majorana stars as a fundamental geometric language, uniquely positioned to inspire new theoretical and experimental directions in quantum technologies.

A quasi-static filtration system, comprising a poroelastic solid coupled to an incompressible free-flow, is considered in 3D. Across a flat 2D interface, the Beavers-Joseph-Saffman coupling conditions are taken. The system constitutes a doubly elliptic-parabolic coupling and can be seen as a degenerate case of the inertial Biot-Stokes dynamics of recent interest. These dynamics cannot be easily recovered through a vanishing inertia limit, however, utilizing a viscoelastic regularization of the inertial Biot system allows us to construct weak solutions in the inertial limit; subsequently, we pass to the limit in the regularization parameter to obtain quasi-static weak solutions. This addresses an open singular/degenerate limiting problem in filtrations, and allows for subsequent analysis of uniqueness and regularity. This also provides a foundation for the incorporation of physically-motivated nonlinear poroelastic effects.

We construct a five-dimensional singly rotating near-horizon solution in Einstein-Gauss-Bonnet gravity. We show that the Gauss-Bonnet term removes the local curvature singularity, yielding finite curvature invariants throughout the spacetime, provided the rotation parameter remains below a certain value set by the Gauss-Bonnet coupling. To our knowledge, this is the first analytic example of a singly rotating five-dimensional solution in this framework with finite curvature invariants over a nontrivial region of parameter space. We analyze the geometry across this space, identifying regular, singular, and marginal regimes. Finally, we study the thermodynamic properties, finding that while higher-derivative corrections regularize the local curvature behavior, they also introduce unique challenges to the standard thermodynamic description of Killing horizons.

Context. The spatially coherent multipath propagation of pulsar radiation leads to a temporal and spectral interference patterns called scintillation. It is caused by density variations in the ionized interstellar medium, which often take the form of thin scattering screens filled with multiple subimages of the pulsar. PSR B1508+55 is known to be scattered by one or two such screens. Aims. We investigate appropriate methods to achieve precise astrometry for a scattering screen from simultaneous observations of only two telescopes on a very long baseline without forming visibilities. Methods. Two simultaneous observations of PSR B1508+55 were performed with the 100-m telescope at Effelsberg and the Five-hundred-meter Aperture Spherical Telescope (FAST). Using and improving existing scintillometry techniques, we leveraged the evolving, very long baseline to precisely measure the screen orientation, effective velocity, and scintillation arc curvature. We inferred the one-screen and two-screen model parameters and we imaged the closer screen. Results. Each single epoch leads to much tighter angular constraints than long-term monitoring of scintillation arcs, revealing an ongoing evolution of the orientation of the closer screen. Images of the scattered pulsar were obtained with a resolution on the order of 0.1 mas. These results confirm the highly anisotropic alignment of the scattered images, while also revealing small-scale deviations from a large-scale straight line. Conclusions. We demonstrate that simultaneous observations of scintillation can be used as a powerful substitute for very long baseline inferometry.

The search for an atmosphere on the closest rocky M dwarf planet, LTT 1445 A b, has been the subject of intense investigation from both the ground and space. Here, we present the first JWST transmission spectrum of LTT 1445 A b using a single visit spanning 3-5~$μ$m using NIRSpec/G395H. We conduct two independent reductions of the data using both the Eureka! and ExoTiC-JEDI pipelines. Overall, we measure the NRS1 transit depths to a median precision of $\sim23$~ppm in 41 spectroscopic channels with uniform widths of 30 pixels ($\sim$ 0.02 $μ$m), and the NRS2 transit depths to $\sim36$~ppm precision in 65 spectroscopic channels, also with uniform widths of 30 pixels. We rule out any statistically significant spectral features at this precision and place limits on atmospheric metallicity using a grid of chemical equilibrium models with grey opaque clouds. Using NIRSpec/G395H alone, we can place limits on the atmospheric metallicity of $\gtrsim350~\times$ Solar when the opaque pressure level is greater than 0.01~bars. We also conduct a combined analysis of JWST/NIRSpec and HST/WFC3 transmission data and find our atmospheric limits can be extended $\gtrsim500~\times$ Solar when considering both datasets. Future analyses both in transit and emission will uncover whether there are detectable atmospheric features.

Because CDCL produces proofs in the Resolution proof system, problems provably hard for Resolution are also provably hard for CDCL. Exponentially shorter proofs can sometimes be found using stronger proof systems such as $\text{Res}(\oplus)$, a generalization of Resolution to XNF formulas, whose constraints are disjunctions of parity equations ("linear clauses") such as $(x \oplus y) \lor \lnot (y \oplus z)$. While some modern solvers like CryptoMiniSAT reason on Boolean clauses with separate parity equations, reasoning about more general linear clauses is less explored. We present $\text{CDCL}(\oplus)$, a generalization of CDCL to XNF formulas, and prove a bidirectional connection with $\text{Res}(\oplus)$: $\text{CDCL}(\oplus)$ not only produces $\text{Res}(\oplus)$ proofs, but also polynomially simulates $\text{Res}(\oplus)$ given nondeterministic decisions and restarts, mirroring the classical relationship between CDCL and Resolution. Our key technical tool is a new set of inference rules for $\text{Res}(\oplus)$ that helps us translate Resolution-based subroutines such as 1-UIP clause learning. Altogether, $\text{CDCL}(\oplus)$'s parity reasoning includes branching on arbitrary parity equations, linear-algebraic reasoning during unit propagation, and learning linear clauses through conflict analysis. We provide a proof-of-concept implementation of $\text{CDCL}(\oplus)$ called Xorcle, which includes adaptations of existing CDCL heuristics to XNF formulas and an extension of LRUP proof logging that we call $\text{LRUP}(\oplus)$. On a selected suite of benchmarks focusing on native XNF formulas, Xorcle outperforms existing solvers such as Kissat and CryptoMiniSAT. Additionally, on Tseitin formulas written in CNF, even without preprocessing, Xorcle's running time appears to scale nearly polynomially.

Cyber-physical systems are inherently complex due to their connection between software and the physical world. Iterative design reduces their complexity, but increases the need to repeatedly recheck their safety in full after every change. We introduce the refactoring-as-propositions principle in which refactorings are represented as propositions along with a method for proving that system refactorings preserve their required properties by transferring the proof along the respective modification. It is based on differential refinement logic (dRL), with which one can simultaneously and rigorously refer to properties of the systems and the relation between a refactored system and its original version. Refinements represent a uniform way of expressing different types of hybrid system refactorings, including those that introduce auxiliary variables. Furthermore, we show how these refactorings can be proved automatically, and/or reduce to a modular proof solely about the local change rather than about the whole system.

Determining whether temperate rocky exoplanets orbiting M stars retain atmospheres is currently a central goal of exoplanet astronomy. To this end, the James Webb Space Telescope has begun searching for atmospheres on these worlds with MIRI secondary eclipse spectroscopy and photometry. Here, we develop a novel climate-constrained Bayesian inference framework that yields atmospheric pressure and composition constraints from these datasets, while accounting for planetary, stellar, and model uncertainties. Our approach fits observations with model spectra derived from self-consistent pressure-temperature profiles at radiative-convective equilibrium, thus maximizing the information extracted from the data and providing more robust inferences than retrievals that use parameterized pressure-temperature profiles. We demonstrate the framework on the existing MIRI LRS eclipse spectrum of LTT 1445A b (1.34 $R_\oplus$ and $T_{\mathrm{eq}} \approx 431$ K). An atmosphere does not need to be invoked to explain the data, meaning a bare rock model produces an adequate fit. If the planet has an atmosphere, the $2σ$ upper limits on surface partial pressures are $\lesssim 1$ bar for an optically thin gas like O$_2$, N$_2$ or CO, $\lesssim0.1$ bar for CO$_2$, $\lesssim 10^{-3}$ bar for H$_2$O, and $\lesssim 10^{-4}$ bar for SO$_2$. Scheduled MIRI F1500W observations could detect one of the thicker atmospheres permitted by the existing data (1 bar O$_2$ and 0.01 bar CO$_2$), if a precision of 20 ppm or better is achieved. This case study demonstrates that climate-constrained Bayesian inversion can turn rocky-planet eclipse spectra into the quantitative constraints necessary to test population-level atmospheric retention hypothesis, like the cosmic shoreline.

We establish a formula expressing Miyazawa's 2-knot invariant $|\mathrm{deg}|$ in terms of the Lefschetz number of a map on ordinary (i.e., not real) monopole Floer homology. As an application, we deduce that $|\mathrm{deg}|=1$ for any 2-knot in $S^4$ which has a punctured $L$-space as a Seifert solid. In the course of the proof of the main theorem, we show how Francesco Lin's construction of monopole Floer homology with $\operatorname{Pin}(2)$-equivariant perturbations can be made to work with integer coefficients.

We prove that the maximum eigenvalue of the (both signed and unsigned) Laplacian of level $k$ Kikuchi graph of any graph $G$ with $m$ edges is at most $m+k$. This confirms four recent conjectures of Apte, Parekh, and Sud. As applications, we obtain that tensor products of one and two qubit product states achieve an approximation ratio of $5/8$ for Quantum Max Cut and $5/7$ for the XY Hamiltonian. Moreover, combining our bounds with the algorithms analyzed by Apte, Parekh, and Sud, yields efficient algorithms achieving an approximation ratio of $0.614$ for Quantum Max Cut and $0.674$ for the XY Hamiltonian. Finally, we also make modest progress on Brouwer's conjecture and improve Lew's bound on the sum of the top-$k$ eigenvalues of a Graph Laplacian.

Rydberg atoms provide a powerful platform for exploring strongly interacting quantum systems, both in free space and in structured electromagnetic environments, with growing applications in quantum technology. Accurately modeling their single-atom properties and mutual interactions is essential for interpreting experiments and designing new architectures. We present a unified theoretical framework for Rydberg atoms and their interactions based on multi-channel quantum defect theory (MQDT) and static electromagnetic Green's tensors. MQDT provides a precise description of Rydberg states of divalent atoms such as strontium and ytterbium, while the Green's tensor formalism provides a general and flexible approach for calculating interactions between two Rydberg atoms in arbitrary geometries, including modifications induced by nearby surfaces. We implement this framework in an updated version of the open-source PairInteraction software [Weber et al., J.~Phys.~B~50 (2017)]. The implementation leverages high-performance libraries and achieves speedups of one order of magnitude for pair-potential calculations compared to prior software. We demonstrate the capabilities of the framework through example applications to divalent atoms and show excellent agreement with experimental data for an exemplary Stark map of $^{174}$Yb. The modular software architecture enables the community to extend it further.