Papers

Geometry of quantum states is a fundamental research area with applications ranging from topological materials in condensed matter to variational algorithms in quantum information. Due to their relative simplicity, pure states are often studied, while mixed states are generally needed, for instance to have nonzero temperatures or other statistical mixtures. Mixed-state geometry, however, is much more challenging and has so far lacked any overall structure that relates distance and curvature, like the Kähler structure does for pure states. This work introduces that missing structure. We show that it has the physical meaning of fluctuation-dissipation—by identifying distance with fluctuations and curvature with linear response—and becomes the known structure for pure states, but is non-Kähler otherwise. This paints a simple picture that unifies fluctuation-dissipation and geometry of quantum states. It also applies on parameter spaces, as encountered in condensed matter or estimation theory. Our picture implies that the geometry contains all linear responses and fluctuations, proves the usual fluctuation-dissipation relation, explains that quantum geometry is present in any sum rule, and allows one to geometrically characterize generic transport behaviors.

The supermoduli stack ${\mathfrak M}_g$, parametrizing smooth unpunctured super Riemann surfaces of genus $g$, is known to be non-projected for $g \ge 5$. We extend this result to the even-spin component of ${\mathfrak M}_4$. This is done by exhibiting an appropriate compact curve in the spin moduli space ${\mathcal S\mathcal M}_4^{+}$, and computing the restriction to it of the class obstructing the projectedness of ${\mathfrak M}_4^{+}$. This restriction turns out to be non-zero, proving that ${\mathfrak M}_4^{+}$ is non-projected.

We establish a uniqueness theorem in four-dimensional spacetime, demonstrating that a nonlinear spinor system naturally constrains the background geometry to the Kastor-Traschen spacetime, which describes dynamical, cosmic multi-black-holes. By analyzing a nonlinear field equation where the super-covariant derivative of a Dirac spinor is sourced by intrinsic fermion currents, we evaluate its integrability under two physical requirements: a phase-locking condition enforcing a real hermitian product $\overline{\psi}_R \psi_L$, and a pure-electric source condition. We show that the field equation uniquely compels the spinor to be an eigenvector of the super-covariant Dirac operator, with the eigenvalue dynamically identified as the Hubble constant of the de Sitter background. These results provide a novel geometric mechanism wherein both the cosmological constant and dynamical black hole configurations spontaneously emerge from fermion condensates.

We study the Ising model on $\mathbb{Z}^d$ with $d\geq 4$ and derive near-critical bounds on the truncated two-point function $\langle\sigma_0;\sigma_x\rangle_{\beta,h} := \langle\sigma_0\sigma_x\rangle_{\beta,h} - \langle\sigma_0\rangle_{\beta,h}\langle\sigma_x\rangle_{\beta,h}$ at parameters $\beta\leq\beta_c$ and $h\geq 0$. As a corollary, we obtain that the associated mass (or exponential decay rate) is equal to \begin{equation*} \max\bigl((\beta_c-\beta)^{1/2},h^{1/3}\bigr)^{1+o(1)}, \end{equation*} where $o(1)$ tends to $0$ as $(\beta,h)$ tends to $(\beta_c,0)$. The proof combines the corresponding result at $h=0$, recently established by Duminil-Copin and Panis, with an interpolation argument inspired by Aizenman and Fernández and carried out via the random current representation of the model.

For $N\ge2$ and $\beta>0$, consider the canonical 2D Coulomb gas ensemble \begin{equation} \mathrm{d}\mathbb{P}_{N,\beta}(Z) =\mathsf{Z}_{N,\beta}^{-1}e^{-\beta|Z|^2/2} \prod_{i<j}|z_i-z_j|^{\beta/N}\,\mathrm{d}Z \qquad\text{on }\mathbb{C}^N. \end{equation} The factor $N^{-1}$ in the pair exponent places the ensemble at the diffusive, or high-temperature, scale. We prove that, for every fixed inverse temperature $\beta>0$, this Gibbs measure satisfies a logarithmic Sobolev inequality (LSI) on the closed Dirichlet-form domain generated by collision-free compactly supported smooth functions, with a positive constant independent of $N$. The LSI constant may be chosen uniformly when $\beta$ ranges over a compact subset of $(0,\infty)$. A second main result establishes Poincaré and logarithmic Sobolev inequalities for planar Gaussian measures weighted by finite products of positive powers of distances to points, with constants depending only on the total variance of the underlying Gaussian measure and the total exponent, not on the number or locations of the points. To prove the full labeled Poincaré inequality, we decompose variance into permutation-invariant and label-dependent parts, controlled respectively by a weighted $\bar\partial$ estimate and by relative-coordinate conditioning with random transpositions; separately, one-site logarithmic Sobolev inequalities, a conditional entropy inequality, and a repulsive partition estimate give a defective logarithmic Sobolev inequality. Rothaus tightening combines the resulting full labeled Poincaré inequality with the defective logarithmic Sobolev inequality. No uniformity as $\beta\to0$ or $\beta\to\infty$ is asserted.

We show that every smooth manifold $M$ of positive dimension admits fully supported distributions $u,v\in \mathcal D'(M)$ with $\mathrm{WF}(u)\cap({-}\mathrm{WF}(v))=\emptyset$ and $uv=0$. Neither factor can be continuous on any non-empty open set, but $u$ may be chosen of optimal Sobolev order $\dim M/2$, with wavefront set contained in any prescribed closed conical set $\Gamma\subset T^\ast M\setminus 0$ with non-empty and non-maximal fibers. Simultaneously, the wavefront set of $v$ may be confined to any compatible open conical set or, in geometric situations, to closed conical sets such as conormal bundles of foliations or ray bundles generated by locally conformally closed one-forms. We give positivity criteria and show that full-support zero divisor pairs accumulate at $(1,0)$ in Sobolev-microlocal topologies. The one-dimensional case uses a Fourier null series of Kozma-Olevskiĭ.

In this work, we investigate rapid prediction of the dominant $n{=}0$ vertical instability growth rate in C-Mod and SPARC equilibria, where nonrigid free boundary response models are too slow for control cycle use. Using a Physics Attention Transformer trained on MEQ-FGE-L labels, we predict both the scalar growth rate and the associated two dimensional perturbed toroidal current density. We find mean absolute errors of 5.4~s$^{-1}$ on held out C-Mod equilibria and 12.7~s$^{-1}$ on synthetic SPARC cases, with spatial eigenfunction errors near 5%. We also compared PAT with operator based ML models : FNO2D and DeepONet, where we found PAT predicts a much lower normalised growth rate error and improved spatial reconstruction. These results indicate that PAT can reproduce MEQ-FGE-L outputs at control relevant latency and could support future studies of growth rate headroom monitoring and proximity aware shape control.

We study a thermodynamically consistent bulk-surface Navier–Stokes–Cahn–Hilliard system describing two-phase flows exhibiting moving contact lines, variable contact angles, and mass transfer. We establish the existence of global weak solutions for non-degenerate mobility functions and singular free-energy potentials. The proof is based on an implicit time-discretization scheme and a subdifferential characterization of the convex part of the bulk-surface free energy. Finally, we study the simultaneous high-friction and decoupling limit, and prove that corresponding weak solutions converge, up to a subsequence, to a weak solution of the Abels–Garcke–Grün model. One key ingredient of the proof is the Mosco convergence of the convex part of the bulk-surface free energy, which might be of independent interest.

Identities asserting that an antisymmetrizer admits a determinantal expression are central to several proofs of counting formulas for alternating sign matrices. Antisymmetrizers appear also frequently in symmetric function theory as, for instance, the Hall-Littlewood polynomials can be realized via antisymmetrizers. In this paper, we establish a new identity of this type, thereby proving a conjecture of Lukas Riegler and one of the authors. We also elaborate on a related antisymmetrizer that may be pivotal in resolving a beautiful conjecture of Colomo and Pronko, and we formulate a version of their conjecture tailored to the suggested approach.

Fu and Nijhoff's direct linearization scheme for the KP hierarchy and its reductions employs a system of evolution equations of an infinite matrix $U$ with quadratic nonlinearity. Part of the matrix elements of $U$ can be identified with affine coordinates $w_{ij}$ of the top cell of the Sato Grassmannian. The evolution equations of these matrix elements are identical to the evolution equations of $w_{ij}$ representing the KP hierarchy in geometric terms. This geometric interpretation can be extended to other matrix elements of $U$ by introducing negative flows to the evolution equations of $U$. The extended system turns out to be substantially equivalent to the two-component KP hierarchy. The Cauchy matrix approach to the KP hierarchy can be explained in this geometric perspective. Multi-component generalizations of Fu and Nijhoff's nonlinear system are related to the AKNS and ASDYM hierarchies. Multi-component Sato Grassmannians show up therein as the relevant geometric structure.

We first present an introduction to the differential geometry of noncommutative algebras that carry a representation of a triangular Hopf algebra. Their noncommutativity is encoded in the universal R-matrix. The differential geometry is canonically constructed from these data. We then develop a new approach to the Bianchi identities for curvature and torsion of arbitrary connections (not necessarily bimodule connections). Using the Cartan calculus for connections, we prove the equivalence of their global formulations in terms of exterior forms and tensor fields. In particular, we obtain the noncommutative analogues of the first and second Bianchi identities familiar from general relativity.

A formula for the $\alpha$-K-energy functional for the Kähler-Yang-Mills (KYM) equations is provided in this paper. Using this formula and Chen's $\epsilon$-geodesic equation on the space of Kähler potentials, we prove that if a solution exists to the KYM equations for a simple vector bundle on a Kähler manifold with discrete automorphism group, then it is unique and the $\alpha$-K-energy is bounded from below. Inspired by the study of constant scalar curvature Kähler metrics, we introduce a coupled J-equation to study the $\alpha$-K energy functional.

We study rank-one delta interactions supported by small normal deformations of an \(n\)-dimensional hypersphere \(S_R^n\subset\mathbb{R}^{n+1}\). The interaction is defined by the normalized surface measure on the support and therefore corresponds to the rotationally invariant sector of the hyperspherical shell problem. For the undeformed hypersphere, we derive the bound-state equation for \(E=-\nu^2\) in terms of the modified Bessel product \[ I_{\frac{n-1}{2}}(\nu R)K_{\frac{n-1}{2}}(\nu R). \] We then consider an inward normal deformation \[ X_\varepsilon(\omega)=(R-\varepsilon h(\omega))\omega, \qquad \omega\in S^n, \qquad 0<\varepsilon\ll1. \] Our main result is that, to first order in \(\varepsilon\), the bound-state energy depends only on the average normal displacement \[ \langle h\rangle = \frac1{|S^n|}\int_{S^n}h(\omega)\,d\Omega_n(\omega). \] Equivalently, the deformed hypersphere is spectrally equivalent, up to \(O(\varepsilon^2)\), to a round hypersphere with effective radius \(R-\varepsilon\langle h\rangle\). In particular, mean-zero deformations do not change the rank-one bound-state energy at first order.

In this paper, we study the finite-dimensional, homogeneous, all-to-all coupled Kuramoto model. We begin by performing a complete spectral analysis of all equilibria of the system. Motivated by this analysis, we then derive an explicit description of the unstable manifolds associated with the family of incoherent equilibria. Subsequently, we establish the convergence of this family of unstable manifolds to the Ott-Antonsen manifold $\mathcal{M}_{\mathrm{OA}}$, with respect to the Hausdorff distance induced by the $p$-Wasserstein metric. We further carry out an analogous analysis for the corresponding counterpart of $\mathcal{M}_{\mathrm{OA}}$ in the continuum limit. Moreover, we establish the uniform-in-time convergence of trajectories of the finite-dimensional Kuramoto model on these invariant manifolds towards their corresponding mean-field limit trajectories. Our results provide a direct geometric link between finite-dimensional particle systems and their mean-field, or continuum, limits.

Starting from an arbitrary conformal net, we construct a genus zero functorial conformal field theory whose Hilbert space is the vacuum sector of the net. Specifically, we construct an algebra for the operad of little discs and conformal embeddings with values in the category of Hilbert spaces and bounded linear maps. We work with closed discs, and our conformal embeddings explicitly allow the image of an incoming disc to overlap with the boundary of the outgoing disc. As an application, we show that all conformal nets satisfy the trace class condition: if $L_0$ is the conformal Hamiltonian of the net, then the operators $r^{L_0}$ are trace class whenever $0 \le r < 1$. In particular, the $L_0$-eigenspaces of a conformal net are automatically finite-dimensional.

In this paper, we establish a connection between joint moments of characteristic polynomials and their derivatives in the Circular Jacobi Ensemble, a generalisation of the Circular Unitary Ensemble, and solutions of nonlinear Painlevé equations. For finite $N$, we show that these joint moments are characterised by a solution of the $\sigma$-Painlevé V equation for all real moment exponents in their admissible range. Under an appropriate large-$N$ scaling limit, we further prove that the limiting joint moments admit a representation in terms of a solution of the $\sigma$-Painlevé III$'$ equation for a certain range of moment exponents. As applications, we answer a question posed by Assiotis et al. in [Math. Physics. Anal. Geom. 25 (2022), no.2, Paper No. 15, 24pp, Remark 1.7] by showing that the characteristic function of a distinguished random variable is connected with the $\sigma$-Painlevé III$'$ equation for complex parameters. Furthermore, we investigate joint moments involving higher-order derivatives of characteristic polynomials in the Circular Jacobi Ensemble. As a consequence, we extend a result of Assiotis et al. in [Comm. Pure Appl. Math. 79 (2026), no. 7, 1771-1827, Theorem 1.11] concerning the joint moments of a sequence of random variables arising from the ergodic decomposition of Hua-Pickrell measures, from real parameters to complex parameters.

Mathematical elasticity has a long history and a huge body of literature. Surprisingly, at its heart elasticity exhibits a rich and transparent structure that, to our knowledge, is rarely presented in an explicit and unified form. Using standard tools from functional analysis, this article reveals four orthogonal decompositions of Hilbert spaces that characterize the static equilibrium of an elastic body at small deformations, universally for arbitrary spatial dimension, arbitrary boundary conditions, and classical as well as data-driven formulations. Regarding data-driven continuum mechanics, it highlights the intrinsic structure that remains unchanged when constitutive laws are replaced by material data sets.

We study boundary free energies and half-space responses in nearest-neighbor disordered Ising models. First, we prove the existence of fixed-depth tangential pressure and identify its derivative almost everywhere with the limiting boundary response. Second, under quenched exponential boundary-response mixing, we prove Gibbs-state selection independence and exponential convergence of finite-depth pressures. We further show that uniform one-spin mixing yields DLR uniqueness and Gaussian normal localization, and verify the required mixing conditions whenever $(2d-1)\mathbb E\tanh(\beta\vert{}J\vert{})<1$. Finally, we establish an all-face surface limit in the bounded Dobrushin regime and provide counterexamples that delimit general surface and stiffness claims.

We prove an optimal Hardy inequality for every aperiodic, symmetric random walk in $\mathbb{Z}^2$ with finite variance. In particular, we verify null-criticality, and thus, optimality of the underlying Hardy weight. Under suitable moment conditions, we use fine asymptotics of the potential kernel due to Fukai and Uchiyama in order to derive the asymptotics of the weight. For the standard Laplacian, we recover the expected first order term in the asymptotics but also show that next order term is negative. Thus, our result shows that the constant in the Hardy inequality proven by Kapitanski and Laptev cannot be larger than $1/4$, which is the optimal constant in the continuum. The proof of null-criticality rests on a new criterion for general graphs beyond the locally finite case. We also recover the situation of $\mathbb{Z}^d$ with $d \geq 3$ which can also be also treated by our new method.

Dual active galactic nuclei (DAGN) mark a critical phase in the evolution of merging galaxies and the pairing of supermassive black holes, yet they remain difficult to identify in large imaging surveys because of projection effects and limited spatial resolution. Compact foreground stars and unresolved substructure can mimic dual nuclei through chance superposition, complicating automated detection. We revisit the 46,061 galaxies flagged but rejected as DAGN candidates by the GOTHIC pipeline, primarily because the two nuclei fell within the SDSS fibre aperture or exceeded its separation threshold. We train a supervised deep-learning framework based on the YOLOv11 oriented-bounding-box architecture on annotated SDSS imaging to separate genuine dual nuclei from foreground stellar contaminants and other spurious alignments. The final model attains a validation precision of 0.919, recall of 0.905, and $F_1$ of 0.912 for the dual-nuclei class, and yields 29,605 dual-nucleus candidates after removing star-dominated and blended detections. Structured visual inspection indicates that $54.5$–$62\%$ are consistent with genuine dual nuclei, implying $\sim(1.4$–$1.8)\times10^{4}$ plausible systems. Cross-calibrating the YOLO separation against the deterministic GOTHIC centroid measurement and restricting to the compact regime ($d \le 6.87''$) gives a conservative subset of $\sim 13{,}672$ candidates, reaching calibrated separations of $\sim 0.56''$. Spectroscopy of the most compact ($\le 1$~kpc) systems shows they are dominated by passive, absorption-line galaxies with no resolved double-peaked emission, so confirmation requires higher-resolution follow-up. The catalogue is a statistically refined list of candidates, not confirmed DAGN. Nonetheless, deep-learning detection substantially reduces contamination and expands the plausible DAGN census.

In this study, we present a thermodynamically consistent framework for modelling polyatomic gases by integrating the general equation for non-equilibrium reversible-irreversible coupling (GENERIC) with an 11-moment description. Our formulation unifies rational irreversible thermodynamics (RIT) and extended irreversible thermodynamics (EIT) into a single GENERIC-compatible structure that guarantees compliance with the second law of thermodynamics through the explicit construction of Poisson and friction matrices. By including translational and internal temperatures, as well as a temperature tensor, the model offers a well-structured and physically meaningful representation of non-equilibrium behavior in polyatomic gases. The entropy production is inherently non-negative, and all closure relations emerge naturally from thermodynamic principles. As a test of applicability, we compute the shock wave structure in nitrogen gas, validating the framework against Direct Simulation Monte Carlo (DSMC) simulations and experiments up to moderate Mach numbers.

We study modular transformation formulas from the viewpoint of Mellin space. The basic observation is that the functional relation between the Hurwitz zeta function and the polylogarithm can be used to organize modular transformations as reflection identities of Mellin kernels, while the accompanying polynomial anomalies arise from contour deformations. We first illustrate this mechanism for the $q$-$\theta$ function and then extend it to the elliptic Gamma function. In the latter case, independently Mellin transforming the two elliptic directions leads to a trilinear reflection identity relating the three elliptic Gamma functions appearing in the SL$(3,\mathbb{Z})$ modular formula, while the associated contour deformation reproduces the cubic Bernoulli polynomial. Further reflection formulas lead to a weighted $q$-Pochhammer type function with a modular transformation analogous to that of the $q$-$\theta$ function, as well as a transposed trilinear reflection identity analogous to the Mellin space structure underlying the SL$(3,\mathbb{Z})$ transformation. Our results suggest that Mellin space reflection identities provide a useful organizing principle for constructing and studying modular special functions beyond the standard multiple elliptic Gamma hierarchy.

Starting from the cyclic duality tensors introduced by Vernier, Miao, and Yamazaki (VMY), whose endpoint-traced matrix-product operators obey $\mathbb Z_N$ Tambara–Yamagami fusion and realize topological defect lines of the compactified-boson conformal field theory, we retain the virtual endpoint as an $N$-state dynamical degree of freedom and construct an exactly movable seam in the spin-$\tfrac12$ XXZ chain at $q=\mathrm e^{\mathrm i\pi M/N}$ with $\gcd(M,N)=1$. The local movement identity holds for any unitary $q$-Weyl pair; in the finite cyclic realization, all output phases are locally gauge equivalent and share the same four seam eigenvalues. On a ring, the output gauge becomes a directed twist on a single bond, while endpoint conjugacy reduces the $2N$ labels to two relative $\mathbb Z_2$ flux classes. An explicit antiunitary symmetry protects the class $\eta\equiv M-1\pmod 2$: for even $N$ it pairs distinct charge sectors isospectrally, whereas for odd $N$ it fixes one sector and squares to $-I$ there. Hence every many-body energy eigenspace in the protected class has even multiplicity. The source value $\eta_{\rm VMY}=1-M$ satisfies this condition for every coprime root, whereas the ungauged value $\eta=0$ does so only for odd $M$. Exact finite-size controls show that the opposite class can contain simple levels, so a global flux invisible to local gauge equivalence distinguishes the two classes spectrally.

The linearity of the Stokes equations organises the hydrodynamic response of a rigid particle into a hierarchy of resistance operators, coupling successive truncations of the ambient-flow jet to moments of the surface traction. Since the work of Kelvin and Larmor, it has been known that this response does not resolve particle geometry faithfully: bodies with discrete rotational symmetry may be indistinguishable from bodies of revolution (Brenner's helicoidal symmetry) and a chiral body may respond isotropically, as in Kelvin's isotropic helicoid. We regard the resistance operators as elements of finite-dimensional O(3)-representation spaces and use character formulae to determine, at every level of the hierarchy, which point-group symmetries are hydrodynamically distinguishable and the dimension of each invariant space. This yields an explicit nested sequence of hydrodynamic symmetry-group sets, from the translation-force level to the quadratic-flow level. The framework reveals hydrodynamic classes that no shape can realise geometrically, gives helicoidal symmetry a level-dependent definition, and shows that polyhedral symmetry becomes visible in a strict order: tetrahedral symmetry in shear, octahedral symmetry through the stresslet, and icosahedral symmetry in quadratic flow. Projecting the resistance operators onto force- and torque-free motion provides symmetry-based parameter counts and a constructive route to the corresponding dynamical normal forms. We thereby complete the Jeffery-Bretherton-Ishimoto classification, characterise all hydrodynamic classes producing Jeffery dynamics, and identify chiral tetrahedral and octahedral normal forms that can generate irregular full-attitude dynamics.

Using an analogue of the Mandelstam-Giddings-Wolpert diagrams, we introduce a new canonical representative of the conformal class of a bounded domain of arbitrary connectivity in $\mathbb{C}$ as a flat conical surface with geodesic boundary (a "truncated light-cone diagram", simply LC-diagram in the sequel). The space of diagrams carries natural coordinates. We derive variational formulas for the determinant of the Dirichlet Laplacian on a LC-diagram with respect to these coordinates. Then, passing to the Schottky double of the LC-diagram, making use of the Burghelea-Friedlander-Kappeler formula and the known variational formulas for determinants of Laplacians on the moduli space of holomorphic differentials, we compute the Dirichlet-to-Neumann (DN) conformal invariant $\frac{{\rm det}'\Lambda}{|\Gamma|}$ (here $\Lambda$ is the DN operator on the boundary, $\Gamma$, of a multiply connected domain and $|\Gamma|$ is the length of the boundary). The resulting formula (which uses the periods of the Schottky double of the domain only) provides an elementary counterpart to the formulas of Guillarmou and Guillopé who had expressed the DN invariant through the Ruelle and Selberg zeta-functions. Our formula agrees with the recent result of Wentworth on the asymptotics of the DN invariant as all but one boundary components shrink; we have used this result to fix the undetermined constant of integration in our formula. As a corollary, we derive an explicit formula for the determinant of the Dirichlet Laplacian in a multiply connected domain.

We study the minimisers of the Hartree–Fock functional for flat bands in the chiral model of twisted bilayer graphene introduced by Tarnopolsky, Kruchkov, and Vishwanath. Our model is a continuous version of the discrete model considered by Becker, Lin and Stubbs in previous works and the first result is a direct analogue of main theorems of those papers: at half filling of the flat band of multiplicity two, there are exactly two minimisers, obtained by occupying either one of the two sub-bands. (A more general version of this result, stressing its topological origins, was presented in Appendix A.8 of the Lecture Notes by Tao-Zworski.) The new contribution is the strict coercivity of the Hessian of the Hartree–Fock functional at the two minimisers. It is a consequence of the difference between the Chern numbers of the component bands and can be optimistically considered as a soft version of a “spectral gap”.

We develop a Monte Carlo lattice Boltzmann method to compute statistical solutions to the three-dimensional incompressible Navier–Stokes and Euler equations. Entropic space-time adaptive relaxation of the higher order kinetic moments yields stable numerical solutions with decreasing viscosity. We provide a convergence analysis that is conditional on four explicitly stated assumptions regarding the discrete dynamics. Under diffusive scaling, the laws of the discrete ensemble converge along a subsequence to a limit satisfying the Foias–Temam Liouville formulation of the Navier–Stokes equations. Consequently, provided the structure function scaling holds uniformly, the vanishing viscosity limit of these measures satisfies the multi-point statistical Euler hierarchy of Fjordholm, Mishra, and Weber. The limit measures inherit the known weak-strong uniqueness principle on the interval of existence of a strong Euler solution. Under explicit scaling assumptions, a Kuznetsov-type argument yields a fractional 1-Wasserstein convergence rate. We present three-dimensional computations of time-dependent statistical solutions along the inviscid limit of the incompressible Navier–Stokes equations together with convergence measurements in the Wasserstein metric. Numerical experiments on a randomized Taylor–Green vortex with 24-dimensional initial uncertainty recover Kolmogorov's K41 scaling for energy spectra and structure functions, exhibit the failure of pathwise strong convergence, and yield Wasserstein convergence rates of about $0.5$ at the onset of turbulence. Finally, error measurements with respect to spectral hyperviscosity computations indicate that the computed limit measure is independent of the numerical regularization.

We develop a general procedure for constructing time-dependent integrable chiral field theories from autonomous unitary difference-form $S$-matrices satisfying the Yang-Baxter equation. Spectral parameters are transported along the free right- and left-moving characteristics, defining a map from physical spacetime to spectral space on which the two-body scattering data are evaluated. Requiring spatially homogeneous right-left scattering forces the characteristic map to be affine. The inverse Cayley transform then determines the local contact interaction, while evaluation along the affine spectral trajectory fixes its time dependence. Thus, the nonautonomous interaction is determined by the autonomous scattering data together with chiral kinematics and spatial homogeneity, rather than being introduced independently. The Yang–Baxter equation supplies the factorized many-body transport, and on a spatial circle periodicity leads to quantum Knizhnik–Zamolodchikov (qKZ) equations whose compatibility defines a flat discrete transport in spectral space. Pulling the corresponding spectral-space amplitudes back to physical coordinates gives the time-dependent many-body wavefunctions. Rational $SU(N)$, trigonometric $U_q(\widehat{\mathfrak{sl}}_2)$, and rational $O(N)$ scattering illustrate how the same mechanism generates distinct nonautonomous interactions. The resulting framework gives a geometric route from autonomous factorized scattering data to time-dependent integrable field theories.

We investigate Bell-CHSH correlations for a chiral massless Majorana field localized in a double cone. The conformal modular flow of the associated light-ray interval is converted into translations by a rapidity-like coordinate, and the one-particle scalar product is derived in modular momentum space. Particular attention is paid to the fact that this scalar product carries a Fermi-Dirac weight. Requiring the modular conjugation to be antiunitary with respect to that weighted scalar product fixes the Fourier-space realization of the modular objects. The resulting Tomita-Takesaki operator and its adjoint generate Alice and twisted-dual Bob directions for which all four crossed CAR orthogonality relations follow automatically from modular localization. The Bell problem then reduces to a simple one-function functional. For a Gaussian modular-momentum profile $h_\sigma(k)=\exp[-k^2/(2\sigma^2)]$ the Bell parameter is larger than the classical bound for a broad range of widths and approaches the Tsirelson value analytically, \[ \lim_{\sigma\to 0^+}{\it B}(\sigma)=2\sqrt2 \] Thus near-maximal Bell violation is obtained from an explicit sequence of smooth rapidly decreasing profiles concentrated around zero modular momentum, corresponding to the modular spectral value $\lambda=1$.

We formulate split Galois descent for fusion categories through framed equivalences, reducing the existence of a split form to a group-theoretic splitting problem. The associated degree-two obstruction combines categorical coherence with stable descent of the simple objects, while fusion spaces impose index and parity restrictions. For quantum groups at roots of unity, we use the quasi-$R$-matrix to make the bar involution tensor-compatible and prove that the underlying fusion category $\mathcal C(\mathfrak g,k)$, associated with the simply connected root datum, has a split form over the maximal totally real subfield of the cyclotomic field generated by the quantum parameter. The same holds for all its cyclotomic Galois conjugates; in particular, every $\mathcal C(\mathfrak g,k)$ has a split real form. By contrast, no split real form exists for the standard braiding except on $\mathcal C(E_8,1)$ and $\mathcal C(D_{4m},1)$, $m \geq 1$. We further show that associative zesting can destroy real descent of $\mathcal C(\mathfrak g,k)$, while every braided zesting of $\mathcal C(\mathfrak g,k)$ still admits a split real form as an underlying fusion category. Applications to pointed and finite-group representation categories give criteria for real descent in terms of cohomology and Frobenius–Schur indicators.