Papers

The limiting Hodge structure relates the geometric degeneration of varieties to its Hodge-theoretic behavior. We study degenerations of cubic hypersurfaces whose singular central fibers are secant cubics. The pertinent degenerations of cubic threefolds and fourfolds have been extensively studied for compactifying the associated moduli spaces. In the two cases, the secant cubics are constructed by the rational normal quartic and the Veronese surface, respectively. The Veronese surface is a Severi variety, and the rational normal quartic is a hyperplane section of it. In eariler work, we investigated degenerations to the secant cubics of higher-dimensional Severi varieties. Here we study the limiting Hodge structures for degenerations to their hyperplane sections.

Recently, Mohan et al. [Results Math. 80 (2025), No. 4, 122] answered Freiman's $3k-4$ conjecture in right-ordered groups under certain restrictions. In this paper, we take a step further by investigating the structure of nonempty subsets $S$ of a right-ordered group satisfying the small doubling condition $|S^2| = 3|S|-3$. Moreover, we provide a complete characterization of all nonempty finite subsets $S$ of the Baumslag-Solitar group $\mathrm{BS}(1,q)$ (with $q \in \mathbb{Z}$ and $q \neq -1$) for which $|S^2| = 3|S|-3$ and the identity element is the minimum of $S$.

Galatolo and Kim proved that the dynamical Borel-Cantelli property for decreasing sequences of balls is tightly connected with the waiting time problem. In systems where all such sequences are Borel-Cantelli, the time needed to enter a small ball $B$ for the first time scales as $\mu(B)^{-1}$, and conversely, waiting time estimates yield Borel-Cantelli results for sequences of balls whose radii decrease in a controlled way. We extend this correspondence to the exponent-$s$ setting introduced by Tseng. For $s\ge 1$, the $s$-exponent monotone shrinking target property ($s$MSTP) requires the Borel-Cantelli conclusion only for decreasing sequences of centered balls satisfying the stronger divergence condition $\sum_n\mu(B_n)^s=\infty$. We prove that $s$MSTP forces the lower waiting time exponent, measured on the scale of $-\log\mu(B(y,r))$, to lie in the interval $[1,s]$ almost everywhere. That a quantitative ($s$-strong) form of the property bounds the upper exponent by $s$ and that, conversely, an exponent-$s$ waiting time estimate implies the Borel-Cantelli property for decreasing sequences of centered balls whose radii obey the calibrated decay condition matching the critical divergence exponent $s$. We also obtain the corresponding quantitative orbit approximation statement $\liminf_n n^{\beta}\,d(T^nx,y)=0$ for $\beta<1/(s\,\underline{d}_\mu(y))$, show that the universal lower bound with exponent $1$ pins the theory to $s\ge 1$, and discuss sharpness on circle rotations, where by results of Kurzweil, Kim-Seo and Tseng the picture is governed by the Diophantine type of the rotation number.

We establish two distinct zero-one laws for the uniform Diophantine approximation of complex numbers by quotients of Gaussian integers and by quotients of Eisenstein integers. Using tools from homogeneous dynamics, we study this problem by reducing it to a shrinking target problem on certain homogeneous spaces of $\mathrm{SL}_2(\mathbb{C})$. The main novel ingredients include measure estimates on a certain family of neighborhoods of the corresponding critical loci, as well as new disjointness statements to control the short-range mixing contribution. Due to the different nature of the critical loci in the Gaussian and Eisenstein cases, these measure estimates are obtained by rather different arguments.

After Katok, a homeomorphism $f\colon M\to M$ of a compact metric space is said to be cohomologically $C^0$-stable if its space of real $C^0$-coboundaries is closed in $C^0(M)$. Kocsard proved that this is the case if and only if $f$ is periodic. We extend the classification to actions of arbitrary finitely generated groups: an action $\alpha : G \to Homeo(M)$ is cohomologically $C^0$ - stable if and only if the image $\alpha(G)$ is a finite group. In particular this settles the case of $\mathbb Z^k$-actions generated by finitely many commuting homeomorphisms. Notably, no amenability assumption is needed: we explain why spectral-gap phenomena for non-amenable actions, which do produce cohomological stability in Hölder, Sobolev and $L^2$ categories, are invisible to the uniform norm. We also discuss the genuinely different smooth category and state a conjecture regarding cohomological $C^\infty$-stability of $\mathbb Z^k$-action by smooth circle diffeomorphisms without periodic orbits, connecting the problem with works of Moser, Fayad-Khanin, Avila-Kocsard and Petković.

We determine the structure needed in a tangent category in the sense of Rosický and Cockett-Cruttwell to construct the Cartan calculus on all objects. The missing ingredient is a scalar multiplication by a commutative ring object $R$, playing the role of the smooth real line, which equips the tangent bundle of every object with the structure of an $R$-module compatible with the tangent structure. We show that under these axioms the Lie algebra of vector fields acts by derivations on the ring of $R$-valued functions and satisfies the Leibniz rule. In other words, the tangent bundle is an abstract Lie algebroid, so that the Lie algebra of vector fields is a Lie-Rinehart algebra over the ring of functions. Consequently, every object carries a Cartan calculus of Lie-Rinehart forms, given by the Chevalley-Eilenberg complex together with its differential, inner derivative, and Lie derivative. Examples include the tangent categories of smooth manifolds, $G$-manifolds, Lie groupoids, log manifolds, pro-manifolds, elastic diffeological spaces, affine and general schemes, graded manifolds, and affine $C^\infty$-schemes.

We consider the inhomogeneous nonlinear heat equation \[ \partial_t u-\Delta u=|u|^{p-1}u+f(x),\qquad x\in\mathbb{R}^3,\quad p>5, \] where \(f\in L^\infty\cap C^{0,1}(\mathbb{R}^3)\). For every sufficiently large integer \(n\), we construct a codimension-\(n\) Lipschitz manifold of non-radial initial data whose corresponding solutions blow up in finite time and whose rescaled profiles converge to the prescribed self-similar profile \(\Phi_n\) of the homogeneous equation. The main novelty is to show that the finite-codimensional stability mechanism for self-similar blow-up, developed in the work of Collot, Raphaël and Szeftel [Mem. Amer. Math. Soc. (2019)] for the homogeneous equation, is robust under the addition of a bounded, Lipschitz spatially inhomogeneous source term. In contrast with the homogeneous problem, the equation considered here has no exact scaling invariance, which is a key ingredient in many previous constructions. We expect that the framework developed in this work may also be useful for related problems in which exact scaling invariance is broken.

Joint communication and sensing has been a key technology in 6G. By integrating sensing into maritime communications, ships can communicate with the base station while sensing the surrounding environment to ensure safe navigation. In this paper, we introduce an integrated satellite-terrestrial maritime system (ISTMS) with joint communication and sensing based on the same radio-frequency signals. Specifically, the terrestrial base station (TBS) and low Earth orbit (LEO) satellite provide communication services for near-shore users (NSUs) and off-shore users (OSUs), respectively, while simultaneously performing target sensing. Based on a differential evolution method (DE), we propose a sensing algorithm, which can enhance the location accuracy and reduce resource consumption. Furthermore, we derive the key performance metrics for both communication and sensing. Through joint beamforming optimization of the TBS and LEO satellite, we maximize the sum rate of maritime users while satisfying target localization accuracy requirements and transmit power constraints. Finally, extensive simulation results demonstrate the effectiveness of the proposed algorithms in terms of location accuracy and transmission rate compared with the baseline algorithms.

In this paper we introduce the group action-stabilizer graph $GAS(G)$ of a group $G$ on a set $X$ with vertex set as the collection of all the group actions of $G$ on $X$, and any two vertices $\phi$ and $\psi$ are adjacent if and only if the non-trivial subgroups $\cap G_x^\phi$ and $\cap G_x^\psi$ of $G$ intersect non-trivially, where $G_x^\phi$ and $G_x^\psi$ are two stabilizers of $x$ with respect to the actions $\phi$ and $\psi$, respectively. We characterize a special subgraph $gas(G)$ of $GAS(G)$ in which the vertex set contains the actions $\phi$ of $G$ on $X$ such that $\cap G_x^\phi$'s are distinct. We determine the number of group actions within some specific groups and find certain conditions under which $GAS(G)$ is equal to $gas(G)$. We also examine the conditions under which $gas(G)$ and its complement are derived graph for a finite nilpotent group $G$.

Zernike moments are widely used rotation-invariant descriptors for shape and image analysis, but their standard computation relies on a pixel-based quadrature that treats each pixel as a point mass located at its center. This approximation introduces spatial aliasing that increases with moment order, degrading image reconstruction and reducing the discriminative power of high-order moments. We present an edge-based formulation that eliminates this source of error by applying Green's theorem to transform the two-dimensional area integral defining a Zernike moment into a sum of one-dimensional integrals along image boundaries. The resulting framework applies equally to polygonal shapes, binary images, grayscale images, and color images. We derive recurrence relations for the required radial primitives and show that the transformed edge integrands are polynomial functions, allowing their exact evaluation using Clenshaw–Curtis quadrature. The proposed method computes Zernike moments from polygonal image representations without the spatial discretization errors inherent to conventional pixel-based approaches and remains computationally practical for high-order moments. Numerical experiments on image reconstruction, shape analysis, and character classification demonstrate that the proposed formulation matches the accuracy of classical methods at low orders while remaining stable at orders for which pixel-based moments suffer from significant aliasing and numerical degradation.

Motivated by applications to the study of problems for partial differential equations with Bessel operator, we introduce a new reproducing kernel Hilbert spaces (RKHS) which is fractional weighted Sobolev space with positive definite kernel. As an application, we obtain a simple and practical approximate solution to the singular Poisson equation.

In this paper, we present an integrated terrestrial and satellite maritime communication system, where a shorebased terrestrial base station (TBS) and a low Earth orbit (LEO) satellite cooperatively provide wide-area communication services to maritime users. We conduct performance analysis for the integrated satellite-terrestrial maritime communication system. Specifically, we analyze the transmission rate and coverage probability of near-shore and off-shore users respectively according to the maritime communication environment. Besides, in order to better understand the impact of some key parameters, we also make asymptotic analysis in some special cases. Further, we design an optimization algorithm to maximize the coverage probability of near-shore users by adjusting the transmission power of TBS and the LEO satellite, while ensuring the both off-shore and near-shore users can meet the minimum communication rate requirements. Finally, extensive numerical analysis results verify the accuracy of the theoretical results and the effectiveness of the proposed optimization algorithm in the maritime communication system.

Let $G$ be a locally compact Abelian (LCA) group with dual group $\Gamma$, and let $\mu$ be a probability measure on (the Borel sets of) $G$. Given a unitary representation $\{U(t): t \in G\}$ in a complex Hilbert space $H$, we study the spectrum of the $\mu$-average $V:=\int_G U(t)d\mu(t)$ (defined in the strong topology of $H$). We prove that $\sigma(V) \subset \overline{\hat\mu(\Gamma)}$ and give a sufficient condition for equality. Using the spectral measure $E(\cdot)$ given by the general Stone theorem, we prove a (weak) spectral mapping theorem for the operators $U(\nu):=\int_GU(t)d\nu(t)$, where $\nu$ is any bounded complex measure on $G$. For a unitary representation of $\mathbb Z$, defined by the powers of a unitary operator $U$, we prove that $\sigma(V)={\widehat{\mu}}(\sigma(U))$. For a unitary representation of $\mathbb R$, given as $U(t)={\rm e}^{itB}$ ($t\in\mathbb R$), we show that $\sigma(V)=\overline{{\widehat{\mu}}(\sigma(B))}$.

Determining whether one finite group is isomorphic to a subgroup of another is a fundamental problem in computational group theory. In this work, we propose a Siamese Graph Neural Network (Siamese GNN) for subgroup prediction using Cayley graph representations of finite groups. Each input group is represented by its undirected Cayley graph and encoded by one branch of a Siamese GNN to produce a graph embedding. The resulting graph embeddings are combined with algebraic features derived directly from the input groups to construct a joint feature vector, which is processed by a fully connected classifier to predict subgroup relations between finite groups. By integrating graph-based structural representations with algebraic features, the proposed framework provides a unified approach for learning subgroup relations from finite groups. Experimental results on an expanded and more diverse dataset of 308 finite-group pairs drawn from 11 group families demonstrate the effectiveness of the proposed architecture, achieving a test BA of 91.67% on an independent test set. Additional experiments evaluate generalization to unseen groups, robustness to different Cayley graph generating sets, the contribution of GNN message passing, performance relative to non-neural baselines, and comparison with exact computational methods. These results illustrate the potential of geometric deep learning for subgroup prediction.

We study the PBW filtration on covariant representations for the Lie superalgebra $\mathfrak{gl}(m|n)$. We prove for all covariant weights of the form $(\lambda|\mu,0^{n-1})$, that there exists a lattice polytope such that the lattice points of this polytope parametrize a basis of the corresponding associated graded space. As a consequence, we obtain degenerations of partial flag supervarieties for the supergroup $GL(m|n)$ into toric supervarieties.

In this paper, we consider the famous Brezis-Nirenberg equation \begin{eqnarray*} \left\{ \aligned &-\Delta u=\lambda u+|u|^{\frac{4}{N-2}}u,\quad&\mbox{in}\,\, \Omega,\\ &u=0,\quad&\mbox{on}\,\, \partial\Omega, \endaligned \right. \end{eqnarray*} where $N\geq3$ is the dimension, $\Omega\subset\mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$ and $\lambda>0$ is a parameter. By developing a refined blow-up analysis based on the inverse reduction argument developed in \cite{WW2019,WW2019-2}, we classify, for the fist time, the Struwe decomposition of the Brezis-Nirenberg equation in the one-bubble case as the parameter $\lambda$ varies for $N\geq4$. As applications, we prove that the $4d$ Brezis-Nirenberg equation has a nontrivial solution (least energy solution) for $\lambda\in\sigma(-\Delta)$ in general bounded domains, where $\sigma(-\Delta)$ is the spectrum of $-\Delta$ in $H^1_0(\Omega)$. Our result completes the existence theory of the Brezis-Nirenberg equation for $N\geq4$ in \cite{AP2025,CFP1985,CFS,CSS1986,CW2005,CSZ2012,SWW2009,TYZ2022} since 1984.

This paper establishes the direct scattering transform for the Calogero-Moser derivative nonlinear Schrödinger (CMDNLS) equation on the real line. For potentials in a weighted Hardy-Sobolev space, we prove the existence and uniqueness of the Jost functions associated with the Lax operator, and construct the corresponding scattering coefficients including the transmission coefficient and the reflection coefficient. Under the CMDNLS flow, we determine the time evolution of these Jost functions and scattering data, revealing a simple dynamics: the transmission coefficient is invariant, and the reflection coefficient acquires a purely rotational phase. Furthermore, we establish the invariance of a suitable weighted Sobolev space under the flow, which guarantees the persistence of the required regularity for the solutions.

This paper investigates high-order congruences of reciprocal power sums and Lehmer-type products. Let $n\geq 1$ with $(n,6)=1$ and $e\in\{2,3,4,6\}$. For the reciprocal square sums \begin{equation*} S(n)=\sum_{\substack{r=1 \\ (r,n)=1}}^{\lfloor n/e \rfloor}\frac{1}{r^2} \end{equation*} we already know the form of the congruence modulo $n$. In this paper, motivated by the known congruences, we first extend these results to certain reciprocal sums of odd order and establish a uniform congruence modulo $n$ for \begin{equation*} S_m(n)=\sum_{\substack{r=1 \\ (r,n)=1}}^{\lfloor n/e \rfloor}\frac{1}{r^m} \end{equation*} We then study the generalized Lehmer-type product \begin{equation*} \prod_{d \mid n}\binom{kd-1}{\lfloor d/e \rfloor}^{\mu(n/d)} \end{equation*} Although congruences modulo $n^3$ for this product have previously been obtained, higher-order congruences do not admit a comparably simple closed form. To address this difficulty, we derive an explicit truncated expansion in terms of complete exponential Bell polynomials. The results provide a unified framework for explicit computation and algorithmic verification of higher-order congruences involving reciprocal sums and related product expressions.

Discovering the memory or nonlocal kernel governing an integro-differential equation (IDE) from sparse and noisy observations is an ill-posed inverse problem. Existing identification methods often rely on problem-specific analytical derivations, specialized observation requirements, or restrictive assumptions about the kernel, limiting their applicability across different classes of IDEs. In this work, we propose a differentiable-solver-based framework for discovering memory and nonlocal kernels directly from spatiotemporal observations. Within the solver, the unknown kernel is represented using a constrained Kolmogorov–Arnold Network (KAN) parameterization, with the physical constraints imposed through two different approaches: a Bernstein-polynomial-based Monotone–Convex KAN (MC-KAN), whose coefficient constraints enforce positivity, monotonic decrease, and convexity by construction, and a Chebyshev-based KAN (Cheb-KAN), in which the same properties are encouraged through soft penalty terms. After training, symbolic regression is applied to the learned kernels to obtain interpretable closed-form representations. We evaluate both methods on benchmarks spanning a one-dimensional Volterra equation, a one-dimensional viscoelastic wave partial integro-differential equation, and a two-dimensional nonlocal reaction-diffusion equation with an anisotropic coupled kernel. For the 1D problems, both methods recover the correct kernel functional form and achieve comparable solution-reconstruction accuracy. In contrast, for the sparse and noisy 2D nonlocal problem, the hard-constrained MC-KAN consistently achieves lower kernel reconstruction errors than the soft-constrained Cheb-KAN. Our results demonstrate that enforcing physically motivated shape constraints by construction provides greater robustness than soft penalties for multidimensional kernel discovery from sparse and noisy observations.

We consider the classical satellite and splice constructions of knot theory from the perspective of link-complement states in three-dimensional topological quantum field theory (TQFT). These states are prepared by a TQFT on a link-complement manifold, namely a manifold with multiple disjoint torus boundaries, obtained by removing a thickened link from a closed ambient three-manifold. In this setting, satellite and splice constructions are realized by gluing manifolds along torus boundaries, and the TQFT assigns to these building blocks multilinear operators acting on the corresponding torus Hilbert spaces. We give explicit prescriptions for these operators and show how they can be used to build link-complement states. This construction is closely related to the JSJ decomposition of link-complement manifolds. While splicing builds manifolds by gluing along tori, the JSJ decomposition cuts a manifold along essential tori into canonical pieces. From the TQFT point of view, this expresses a link-complement state as a network of elementary operators associated with the JSJ pieces. We illustrate this framework by studying the entanglement entropy of Seifert-fibered link complements, Hopf-chain complements, and complements of Whitehead-doubled Hopf chains, showing how their different JSJ structures lead to distinct entanglement patterns.

In this paper, we classify the category of $\mathfrak{so}_n(\mathbb{C})$-modules whose restrictions to the universal enveloping algebra of the abelian nilradical of a maximal parabolic subalgebra are free of rank $1$. We show that they generically are simple, and give an explicit description of submodules in the non-simple case. This category embodies a certain connection between weight modules and non-weight modules.

We address a question posed by Herzog, Kumashiro, and Stamate regarding the upper bound for the multiplicity of any far-flung Gorenstein numerical semigroup. We answer this question in the affirmative by presenting a method for constructing certain far-flung Gorenstein numerical semigroups with maximal reduced type. Furthermore, using this method, we consider a question raised by Herzog, Hibi, and Stamate. More precisely, for any integer $t\ge5$, we construct a certain far-flung Gorenstein numerical semigroup with type $t$, which is a counterexample to the question.

Computing distances between probability distributions on non-parametric statistical manifolds is fundamentally intractable. The geodesicequations live in infinite-dimensional function spaces and admit no general closed-form solution. We develop a two-stage framework that produces a computable pseudo-metric on the Pistone–Sempi exponential manifold and apply it to two-sample hypothesis testing. In the first stage, an arbitrary distribution is projected onto a chosen finite-dimensional parametric exponential family via moment-matching. This projection is many-to-one, so the resulting object is a pseudo-metric rather than a true metric. In the second stage, the parametric family is embedded into the manifold of symmetric positive definite matrices via the expected outer product of the augmented sufficient statistics vector. The embedding is a smooth diffeomorphism. The induced pullback metric differs from the Fisher–Rao metric by an explicit correction involving third-order joint cumulants of the sufficient statistics; the correction vanishes for the Gaussian family, recovering the Calvo–Oller embedding as a special case. The affine-invariant Riemannian metric on the ambient matrix manifold then provides a closed-form lower bound for the pseudo-metric, computable directly from sample moments. Applied to two-sample testing, the framework produces a test statistic that is affine-invariant and requires no continuous tuning parameters such as a bandwidth. The choice of target exponential family determines which moments are compared. Critical values are obtained by permutation.

This paper investigates Rota–Baxter Lie $H$-pseudoalgebras. We develop a cohomology theory for $\lambda$-weighted relative Rota–Baxter operators via a Maurer–Cartan approach, constructing the underlying differential graded Lie algebra. We classify non-abelian extensions using second cohomology and derive the Wells exact sequence to address the inducibility of automorphisms. Furthermore, we explore the homotopy theory of these structures by introducing $2$-term skeletal and strict Rota–Baxter $L_\infty$-$H$-pseudoalgebras. In particular, we establish a one-to-one correspondence between strict $2$-term structures and crossed modules of Rota–Baxter Lie $H$-pseudoalgebras. These results establish a foundational framework for future advancements in the higher categorical theory of pseudoalgebras with algebraic operators. Ultimately, this work provides a robust foundation for the higher categorical study of pseudoalgebras equipped with algebraic operators.

We develop a theory of real sutured manifolds and a real Heegaard Floer theory for these manifolds. We develop a notion of real nice diagrams, and prove that our invariant is combinatorially computable. Our theory shares many structural properties with Juhász's sutured Floer homology, as does the topological theory of real sutured manifolds with Gabai's original sutured manifold theory. We also show that our invariant has several new structural properties differentiating it from sutured Floer homology.

In this paper, we discuss the validity of an approximation inspired by the Smoluchowski-Kramers approximation for a class of stochastic differential equations driven by fractional Brownian motion with additive noise. By rewriting such equations in the form of slow-fast systems and decomposing the fast component into three parts, we investigate the small mass limit of these equations and derive the corresponding convergence rates. Furthermore, under certain regularity conditions, we study the large and moderate deviation principles for a class of stochastic differential equations driven by fractional Brownian motion with small multiplicative noise via the weak convergence approach.

Let $S_1$ be the one-variation associated with the regular $n$-adic martingale filtration on $[0,1)$. We study the martingale isoperimetric profile \[ V_n(x):= \inf_{\substack{A\subset[0,1)\ {\rm measurable}\\ |A|=x}} \|S_1(\mathbbm 1_A)\|_1 . \] For the ternary filtration we determine this profile exactly. Namely, \[ V_3(x)=T_3(x):= \sum_{j=0}^{\infty}3^{-j}\psi_3(\{3^j x\}), \] where \[ \psi_3(t)= \min\left\{ \frac{1+2\left|t-\frac12\right|}{3}, \frac{2-4\left|t-\frac12\right|}{3} \right\}, \qquad 0\le t\le1 . \] Thus the sharp ternary profile is a Takagi-type Bellman function. It is, however, not the usual ternary Takagi–van der Waerden function $\omega_3$; for example, \[ T_3(1/3)=4/9, \qquad \omega_3(1/3)=1/3 . \] For general $n\ge2$, we prove that every measurable $A\subset[0,1)$ satisfies \[ \|S_1(\mathbbm 1_A)\|_1 \ge \omega_n(|A|^*) \asymp_n |A|^*\log\frac1{|A|^*}, \qquad |A|^*:=\min\{|A|,1-|A|\}. \] Moreover, this logarithmic order is sharp up to a constant depending only on $n$. Finally, for every $0<\alpha<1$, we prove the endpoint estimate \[ \|S_1(\mathbbm 1_A)\|_\alpha\ge |A|^*, \] and show that it is sharp up to a constant depending only on $\alpha$ and $n$.

This paper determines the exact frequency of all consecutive sign patterns of lengths two and three formed by the quadratic character modulo an odd prime $p$. Using basic properties of character sums over finite fields, we derive exact counting formulas for pairs $(n, n+1)$ and triples $(n-1, n, n+1)$ exhibiting specific sequence patterns in \(\{\pm 1\}\). The frequencies of pairs are classified entirely by \(p \pmod 4\), whereas triples exhibit a more complex dependency on \(p \pmod 8\) and the Jacobsthal sum $T_p=\sum_{x\in\mathbb F_p}\chi(x^3-x)$.

For every integer $n\ge27$, we construct a smooth metric on $\mathbb S^n$ that is invariant under the antipodal map and is not locally conformally flat. For this fixed background metric, the normalized $\sigma_2$-Yamabe equation admits a noncompact family of positive $\Gamma_2^+$-admissible solutions. The main difficulty is the possible loss of ellipticity of the linearized operator. This difficulty does not occur for the scalar Yamabe equation, whose linearization has a fixed Laplace-type principal part. In the $\sigma_2$ problem, the positive Newton tensor of a standard bubble decays in the far field, while the terms produced by the background metric need not decay at the same rate. Our construction provides the relative decay needed to keep the conformal metrics inside the ellipticity cone. A quartic profile in the finite-dimensional reduction yields the endpoint dimension $n=27$.

We determine the rate of convergence of the value functions of the $N$-particle stochastic optimal control problem to the value function of the corresponding mean field control problem, for mean field costs that are merely Lipschitz continuous in the 1-Wasserstein distance, a class that covers problems whose mean field optimizers are neither unique nor stable. For $d\geq2$, the optimal rate is the empirical-measure rate ($N^{-1/d}$ for $d\geq3$, $N^{-1/2}\sqrt{\log N}$ for $d=2$): this proves the rate conjectured by Daudin, Delarue and Jackson, and removes the semiconcavity hypothesis made there. In dimension one, we discover that the empirical-measure benchmark is not optimal: cooperating particles beat it, and the optimal polynomial exponent is $4/7$, strictly between the accuracy of independent samples and that of quantization by freely placed points. The proofs are control-theoretic: from each realization of an $N$-particle control we build a pathwise Fokker-Planck flow (a "shadow flow") which, repeatedly recoupled to the particles by optimal transport, shadows the empirical measure at the optimal rate. The one-dimensional rate requires additional constructions: we correct the shadow flow with a filter built on the future of the discarded noise, draw the cooperating particles from a Gibbs law, and prove the optimality of the exponent by a Schrodinger ground-state estimate. The empirical-measure rate also holds under additive common noise, uniformly in its intensity.