Papers

We develop a finite-tree formalism for sharp inequalities on real-valued, mean-centered $\operatorname{L}^1$-BMO over homogeneous $q$-ary filtrations. A reduction theorem identifies local and global sharp constants with the supremum of finite-depth constants. Finite anchoring gives compact optimization problems and rational extremizers under rational piecewise-linear hypotheses, while stable centering shows that imposing zero root average preserves the supremum over all depths. The principal application is the absolute-value map: we construct centered examples in which taking absolute values increases the BMO seminorm and obtain bounds strictly below the universal factor $2$. Jensen-defect Carleson functionals and martingale-difference envelopes provide further applications, illustrating distinct anchoring mechanisms and the differences between finite-depth rationality, stabilization, and global attainment.

The Ramanujan bound states that the degree of the trace of the $\mathfrak{p}$-Hecke operator on a space of weight $k$ Drinfeld cusp forms is at most $d(k-2)/2$, where $\mathfrak{p}$ is a prime of degree $d$ in $\mathbb{F}_q[T]$. We prove the strong Ramanujan bound, which states that at level 1, this bound can be improved to $d(k-(q+1))/2$. In addition, we classify when the strong Ramanujan bound is sharp; surprisingly, this turns out to depend on $\mathfrak{p}$ only through the parity of its degree. We also prove results related to the smallest slope of the Atkin operator and mod $p$ distributions of Frobenius polynomials of Drinfeld modules over finite fields.

We establish inverse inequalities in Gaussian Sobolev norms for shallow $\mathrm{ReLU}^k$ trial spaces. The estimates identify the sharp regularity threshold and quantify the dependence on parameter separation and bias magnitude. Based on these estimates, we prove convergence of weighted least squares discretizations of linear PDEs on $\mathbb R^d$ using shallow networks with prescribed inner parameters and finitely many residual samples in an expanding ball. Under suitable stability and regularity assumptions, we obtain an error bound of order $h^{k+(d+1)/2-r}\delta_\pm^{-(S-r)}$ in the Gaussian Sobolev norm of order $S$, up to logarithmic factors, where $r$ is the PDE order, $h$ is the parameter fill distance, and $\delta_\pm$ is the antipodal separation. The convergence results cover deterministic and random sampling and extend to fixed bounded regions by restriction. We construct parameter sets with $\delta_\pm\asymp h$, yielding a convergence rate in terms of the network width that matches a lower bound up to logarithmic factors. Numerical experiments support the theoretical results and provide practical guidance on balancing accuracy and stability.

We study a density heat flow associated with normalized Liouville systems on a closed connected Riemannian surface, and we show that this flow leads to the existence of solutions for the corresponding subcritical Liouville systems. The coupling matrix is symmetric and entrywise nonnegative, with no invertibility, irreducibility or inverse-matrix sign assumption. Under the strict subcriticality conditions $8\pi\sum_{i\in J}\rho_i>\sum_{i,j\in J}a_{ij}\rho_i\rho_j$ for every nonempty set of components $J$, every smooth positive initial density of masses $\rho_i>0$ generates a unique global smooth positive solution converging in every $C^m$ norm to a stationary density. The limiting potentials solve the stationary Liouville system. Our proof uses entropy coercivity and a gradient inequality in the density variables, which also yields algebraic or exponential convergence rates.

Ordinary angles and unsigned areas discard orientation, and much of the case analysis in plane geometry is the price of that loss. Signed quantities (directed angles, oriented areas and directed distances) keep the signs and so let one argument serve for every configuration. We illustrate this with results that hold uniformly for convex, concave and crossed quadrilaterals: the diagonal area formula, an identity expressing one diagonal in terms of the sides and the other diagonal, a test for a pair of parallel sides, a theorem on equidiagonal quadrilaterals (such a quadrilateral has a pair of equal opposite sides if and only if it has a pair of parallel opposite sides), and a single ratio of directed distances that decides whether the vertices lie on an ellipse, only on hyperbolas, or on a circle.

We generalize Christol's theorem—which characterizes algebraic power series over a finite field in terms of automatic coefficient sequences—to completions of domains of finite type over a finite field~$\mathbb{F}_{q}$. Let $A$ be such a domain of dimension~$d$ and $\mathfrak{a}$ an ideal of depth~$d$. In the $\mathfrak{a}$-adic completion $\hat{A}$, choose an $\hat{A}$-regular sequence $\boldsymbol{x}=\{x_{1},\dots,x_{d}\}\subset\hat{\mathfrak{a}}$ and a set of representatives $E$ of $\hat{A}/(\boldsymbol{x})$ that is an $\mathbb{F}_{q}$-vector subspace consisting of algebraic elements. Every element $y\in\hat{A}$ then admits a unique expansion $y=\sum_{u\in\mathbb{N}^{d}}a_{u}x^{u}$ with coefficients $a_{u}\in E$. We prove that $y$ is algebraic over $A$ if and only if the coefficient sequence $\{a_{u}\}_{u\in\mathbb{N}^{d}}$ is $q$-automatic. The proof relies on a careful study of henselizations, which are shown to coincide with the algebraic closures inside the completions under certain conditions. As an application, we obtain a complete characterization of the automaticity of $\beta$-expansions for formal Laurent series over $\mathbb{F}_{q}$, answering questions posed by Hbaib–Mkaouar and Scheicher–Sirvent.

We develop and analyze a symmetric two-level preconditioner for the method of reflections for Laplace problems in perforated domains. Using constant boundary modes, we define a coarse correction through a symmetric coercive dual formulation. Combining this coarse correction with parallel reflection corrections gives a symmetric positive definite preconditioner for solving the dual system by the conjugate gradient method. Under a geometric separation condition, we prove a condition number bound independent of the number and sizes of the holes and the diameter of the ambient domain. We also derive geometry-dependent convergence estimates for the one-level methods that explain the need for a coarse correction.

We establish a compact-localization theorem for finite iterates of the Smyth power space. Successive union maps assign to every finite-level Smyth object a canonical least compact saturated carrier in the original space, and co-sobriety at each positive finite level is equivalent to co-sobriety of the corresponding iterates over all compact saturated carriers. Consequently, if every compact saturated subspace of a $T_0$-space $X$ is stably compact—in particular, if every such subspace is Hausdorff—then $\PS^n(X)$ is co-sober for every finite $n\geq 1$. This applies to every $k_1$-Hausdorff space and hence to every Hausdorff or metrizable space. For the first iterate, we also give a direct full-Vietoris argument, independent of stable compactness, which shows that each $k$-irreducible compact saturated family is generated by its union.

We construct a chiral algebra $U$ of central charge eight inside the lattice vertex operator algebra $V_{E_6\oplus A_2}$. It is generated by two commuting affine $\mathfrak{sl}_2$ current algebras at levels $1$ and $39$, the stress tensor of the lattice theory, and a single weight-two field built from three lattice exponentials. We announce that $U$ is strongly rational and completely unitary, and that $Rep(U)$ is equivalent, as a unitary ribbon category, to the Drinfeld center $Z(\mathcal{H}_3)$ of the Haagerup fusion category $\mathcal{H}_3$. Its characters are the $\gamma=0$ Evans–Gannon vector. From $U$ we obtain a diagonal full CFT, and also a holomorphic $E_8$ extension on which $\mathcal{H}_3$ acts by topological defects.

Symbol-pair distances of simple-root cyclic codes are important for evaluating error correction performance under overlapping pair read channels. In this paper, we construct three families of simple-root cyclic codes over finite fields of odd characteristic via generator polynomials. For length \(2q+2\), the minimum symbol-pair distance is \(10\) for every odd prime power \(q\ge 7\), established by excluding invalid support configurations. For length \(4q-4\), when \(q\equiv 3\pmod 4\) and finite field constraints hold, it is \(9\) or \(10\). For length \(2(q-1)\), reciprocal-polynomial analysis shows it is \(11\) or \(12\). All obtained codes are MDS or AMDS symbol-pair codes.

We answer negatively the question whether the product of two co-sober spaces must be co-sober. For a nonempty closed subset $A$ of a $T_1$-space $(X,\tau)$, we consider the topology consisting of the co-finite open sets together with the $\tau$-open sets disjoint from $A$. We determine its $k$-irreducible compact subsets and derive a criterion that produces two compact co-sober spaces whose product contains a non-singleton $k$-irreducible compact saturated subset of the diagonal. Applying the criterion to a specific Kovář–van Douwen space gives two countable compact Fréchet–Urysohn $T_1$ anti-Hausdorff co-sober spaces whose product is not co-sober. A further construction yields a countable compact Fréchet–Urysohn $T_1$ anti-Hausdorff co-sober space whose square is not co-sober. The failure of product closure also gives an alternative proof of the known non-reflectivity of co-sober spaces in the category of $T_0$-spaces.

We study Wasserstein-$p$ goodness-of-fit and two-sample testing under differential privacy. For general compact metric spaces, we establish upper and lower bounds on the minimax separation radii in terms of covering numbers, matching up to logarithmic factors. On the hypercube $[0,1]^d$, these bounds match up to constant factors for two-sample testing in all dimensions and for goodness-of-fit testing whenever $d\neq 4p$, thereby closing logarithmic gaps in existing non-private results. At the critical case $(d,p)=(4,1)$, we further show that a logarithmic factor is unavoidable by establishing matching minimax bounds for non-private Wasserstein-$1$ goodness-of-fit testing under a bounded-density assumption. Finally, for unbounded domains, we establish minimax-optimal separation rates up to logarithmic factors under an exponential tail condition.

In this work, we focus on deriving the absolute moments of the general ized hyperbolic (GH) distribution via its mean-variance mixture structure. Conditioning on the GIG mixing variable reduces the problem to Gaussian moment calculations and known GIG power moments, avoiding direct integration with Bessel functions. The variance gamma (VG), normal inverse Gaussian (NIG), hyperbolic, Normal, and Student t distributions emerge as natural special or limiting cases of the generalized hyperbolic family. This approach offers a substantially simpler and more transparent derivation, facilitating both implementation and verification.

Identities of spanning forest polynomials can come from determinantal identities such as the classical Dodgson identity. They can also be useful in quantum field theory calculations, particularly the quadratic ones. To better understand the structure of spanning forest polynomial identities, we look at when identities on a given graph are generic for all graphs, we ask about dimensions of spaces of spanning forest identities indexed by certain parameters, and we set up a framework that we hope will be helpful for further progress in these directions. As part of this framework, we identify a family of maps naturally given by the all-minors matrix-tree theorem, and we show that their ranks are counted by the Narayana numbers via explicit construction of bases.

Let $\mathcal{T}$ be a tensor triangulated category with a compact tilting generator $M$ and abelian heart $\mathcal{H}_M = M^{\bot_{\neq0}}$. We define a triangulated flat dimension $\mathrm{flatdim}_{\mathcal{T}}X$ for $X\in\mathcal{T}$, and give criteria in terms of tensor vanishing and resolutions by the class of pure quotients of $\mathrm{Add}(M)$. We further introduce $M$-ghost length and $M$-ghost dimension, and show that the global dimension of $\mathcal{H}_M$ controls the $M$-ghost length of all objects in $\mathcal{T}$, and that the $M$-ghost dimension of $\mathcal{T}$ coincides with the weak global dimension of $\mathcal{H}_M$ and the supremum of $\mathrm{flatdim}_{\mathcal{T}}X+\mathrm{inf}X$ of bounded objects, and the relative triangulated dimension of $\mathcal{T}^c$ with respect to $\mathrm{add}(M)$. This extends the classical Hovey-Lockridge theorem, which is recovered for derived categories of rings.

This Master's thesis develops and applies the multisymplectic field-theory framework to (pure) Yang-Mills theory. Building from the geometric foundations of jet bundles and the Poincaré-Cartan forms, the work constructs the Lagrangian multisymplectic formalism for first-order field theories, recovering the usual Euler-Lagrange field equations and extending the Poisson brackets of symplectic mechanics to a full graded Lie algebra. It then establishes the Legendre correspondence with the multisymplectic Hamiltonian formalism (in which Hamilton equations are retrieved) including the case of singular Lagrangians (where a constraint algorithm is reproduced). Breaking Lorentz covariance via a Cauchy surface slicing, the thesis then relates the covariant multisymplectic picture to an instantaneous presymplectic (infinite-dimensional) formulation in which the instantaneous De Donder equations and presymplectic brackets are obtained in a compatible way. A comprehensive treatment of symmetries then leads to multisymplectic and instantaneous momentum maps generalizations and to several versions of Noether's theorems. Specializing the formalism to principal-bundle gauge theory, the thesis identifies the multisymplectic configuration space with the bundle of connections, gauge symmetries with symmetries in the multisymplectic language and builds the Yang-Mills Lagrangian. Lastly, among many computations, the standard tensorial Yang-Mills equations and the Bianchi identity in local trivializations are retrieved, Yang-Mills conserved currents and charges are obtained and the graded algebra is derived for a special class of Hamiltonian forms.

Let $T$ be the theory of the valued field of complex Laurent series $\mathbb{C}(\!(t)\!)$ in the language of rings expanded by a constant for $t$ and by predicates for $n$-th powers. We introduce a $t$-adic analogue of Kochen's operator, namely $\gamma(x)=x/(1+tx^{2})$, and the associated Kochen ring $\Lambda_{L}$ of a field extension $L$ of a model $K$ of $T$. Our main result characterizes the field extensions of $K$ which embed into a model of $T$ (the formally $T$-adic} fields): $L$ is formally $T$-adic if and only if $1/t\notin\Lambda_{L}$. As an application we prove a Nullstellensatz for $\mathbb{C}(\!(t)\!)$: a polynomial $f\in K[X_{1},\dots,X_{n}]$ vanishes on the zero set of an ideal $I$ if and only if $f^{m}=\sum_{i}\omega_{i}f_{i}$ for some $m\geqslant 1$, generators $f_{i}$ of $I$ and coefficients $\omega_{i}$ in the ring $\Lambda_{K(X)}\cdot K[X]$. We derive the resulting correspondence between $R$-ideals and algebraic sets, and we compare natural incarnations of the associated $T$-spectrum: prime ideals, morphisms into models of $T$, and the space of types of $T$.

Binary orthogonal optimization arises in semantic hashing, orthogonal binary projection, and Hadamard matrix design. Existing methods often handle binarity and orthogonality separately, so their outputs need not satisfy both constraints simultaneously and generally come without a residual-based guarantee of exact feasibility. Using an exact box-constrained reformulation, we establish local and global Lipschitz error bounds and prove that the Lipschitz exponent is optimal. In particular, when the binary orthogonal feasible set is nonempty, any box-constrained matrix whose orthogonality residual is below $1$ admits a feasible sign rounding, and the rounding distance is bounded by this residual. These bounds also yield a globally exact nonsmooth orthogonality-residual penalty over the box and a uniformly sharp local penalty threshold. We further develop a smoothing penalty method for binary orthogonal optimization (SPBO). We prove that every accumulation point is stationary for the limiting nonsmooth penalty problem. Numerical experiments on orthogonal binary projection, supervised hashing, and Hadamard completion show that SPBO substantially reduces orthogonality violations while retaining competitive objective values or retrieval accuracy.

Modern learning problems increasingly leverage data from heterogeneous sources to improve robustness and generalizability under distribution shift, often with high-dimensional parameters admitting low-rank structure. Existing group distributionally robust optimization (group-DRO) approaches optimize against adverse source mixtures, but extending this principle to low-rank matrix estimation with a broad class of losses leads to a challenging nonconvex-concave problem for which methods and theory remain limited. We propose Prior-Informed generalizable Low-rank representation Learning via Adversarial Reweighting (PILLAR), which smooths the group-DRO objective by regularizing adversarial source weights toward a user-specified reference distribution while learning a shared low-rank representation. By tempering extreme adversarial reweighting, PILLAR improves estimation stability and reduces the potential conservativeness of group-DRO. Using Burer-Monteiro factorization, we optimize the nonconvex-concave objective by bifactored gradient descent with mirror ascent. We establish linear convergence to a statistical neighborhood of the population saddle point and derive an error bound that separates low-rank estimation from adversarial reweighting. We further show that PILLAR improves on vanilla group DRO when the latter's excess conservativeness dominates source heterogeneity and prior misspecification. Simulations and an electronic health record study of suicide-attempt risk demonstrate improved worst-source and transported predictive accuracy.

We prove the persistence of conormal analyticity through the $H^r$ norm, with $r \geq 4$, for the 3D incompressible Euler equations in the half-space $\Omega = \mathbb{R}_+^3$. We also establish a lower bound on the radius of conormal analyticity which depends on $\|u\|_{H^r}$ and initial data. In particular, persistence is established in both of the spaces $X_\tau^{\rm co}$ and $A_\tau^{\rm co}$, the former being based on the conormal operator $Z_k = \phi^k \partial_3^k$ and the latter being based on $L^k = (\phi \partial_3)^k$. Finally, we show that the norms $\|\cdot \|_{X_\tau^{\rm co}}$ and $\|\cdot \|_{A_\tau^{\rm co}}$ are equivalent up to a small loss in radius, the proof of which uses explicit identities relating the two conormal operators.

We study a family of special numbers and polynomials introduced by Simsek whose exponential generating function has as denominator a falling factorial in the exponential variable. Using the partial fraction decomposition of this denominator, each member of the family is expressed by a single finite sum involving Bernoulli polynomials, Apostol Bernoulli polynomials and two variable Fubini polynomials evaluated at reciprocals of integers. A second expansion, around the point where the denominator vanishes, expresses the same numbers through Stirling numbers of the second kind and generalized harmonic numbers, and yields closed formulas for the first few members. We construct a function, built from the Hurwitz zeta function and Lerch s transcendent, which interpolates the polynomials at the negative integers and gives a closed form of a multiple series recently introduced by Pokaz, Ribicic Penava and Simsek, we give an Euler type integral representation for it, and show that the numbers are signed combinations of moments of geometric random variables. Recurrence relations, asymptotics and an extension to arbitrary distinct nodes are also given.

We extend Mark Kac's kinetic framework from single-species Euclidean spaces to a two-species, non-linear interaction space on the open cone $(0, \infty)^2$ governed by the Matsumoto–Yor rational collision involution. We identify a non-linear conserved energy $E(x,y) = x + y + 1/x$ and an invariant conic metric measure $d\mu = \frac{dx \, dy}{x(x+y)}$. By establishing a two-species $H$-theorem, we prove that entropy production vanishes if and only if post-collision velocities are statistically independent ($I(U;V)=0$). In contrast to the transform-domain proof of Letac and Wesołowski~\cite{letac2000}, which relies on joint Laplace transforms of general measures, our approach yields a direct physical derivation in state space via the transformation law for logarithmic gradients and separation of variables. This uncouples the functional-differential system to uniquely characterize the Generalized Inverse Gaussian (GIG) and Gamma distributions as the non-equilibrium thermodynamic equilibria.

Andrews, Ganchev, Kuyper, Lempp, Miller, A.\ Soskova, and M.\ Soskova conjectured that skip-cototality coincides with weak cototality. We show that the conjecture is false by constructing a $\Pi^0_2$ set whose skip does not have a cototal degree. We further adapt this construction to obtain additional results. We prove that for each odd $n$, there is a $\Pi^0_{n+1}$ degree whose $n$-iterated skip does not have a cototal degree. We show that there is a $\Pi^0_2$ degree which does not contain an introenumerable set.

In this work, we develop and analyze a splitting mixed finite element method, combined with a fully implicit Euler time discretization, for a stochastic Keller–Segel system with general polynomial logistic growth and multiplicative noise. The proposed splitting formulation allows the use of standard finite element spaces without requiring the LBB inf–sup condition associated with traditional mixed finite element methods and, at each time step, avoids the need for solving a coupled nonlinear system. We first establish the well-posedness of the variational solution and derive new high-moment, higher-order regularity, and exponential stability estimates. These estimates provide the key analytical foundation for the strong convergence analysis of the numerical method. A principal contribution of this work is the derivation of error estimates directly in full expectation, without the use of localization arguments. By combining the new stability and regularity estimates with a stochastic Gronwall framework, we overcome the difficulties arising from the interaction between the nonlinear chemotactic coupling and the multiplicative noise. As a result, we establish strong convergence in high-moment norms and obtain optimal convergence rates with respect to both the temporal and spatial discretizations. In particular, the proposed method achieves the optimal rates of order $k^{\frac{1}{2}}$ in time and $h$ in space in the relevant $L^2$- and $H^1$-based error norms. These results provide, to our knowledge, the first optimal strong convergence analysis in full expectation for a splitting mixed finite element approximation of this class of stochastic Keller–Segel systems.

We propose two new decoding algorithms for low-density parity-check (LDPC) codes. The Consensus and Return residual belief propagation (C&R-RBP) algorithm reduces the number of iterations required to achieve a target decoding error rate over an AWGN channel by exploiting the availability of updated information, thereby mitigating the performance degradation caused by the greedy behavior of RBP-based algorithms and reducing error propagation. The Consensus and Delayed Return RBP algorithm (C&DR-RBP) achieves better performance in the early decoding iterations than C&R-RBP, making it particularly suitable for low-latency applications. Both proposed methods maintain the same computational complexity as the original RBP algorithm in terms of residual calculations. We also provide a formal analysis showing that the Consensus mechanism increases the probability of correct bit decisions under initial decoding conditions and in the presence of severe bit transmission errors, providing insight into the performance gains observed in the simulations. Numerical results based on the rapidly growing 5G New Radio (NR) standard, considering different code lengths, code rates, and channel models, demonstrate that both methods outperform competing RBP-based algorithms with similar overall computational complexity in terms of frame error rate (FER) and bit error rate (BER).

This article offers an overview of recent developments in deep learning from a control vantage point. It reviews several classes of models including feedforward and residual neural networks, transformers, large-language models, and selective state space models while explaining how control theory plays an important role in understanding their training, properties, or architecture design. The article explains how residual neural networks can be interpreted as the discretization of a continuous-time control system thereby enabling the use of control theoretic techniques for its analysis. To illustrate the power of this bridge between control and deep learning, two technical results are discussed: 1) the ability of a residual neural network to approximate arbitrary continuous functions is a non-standard controllability problem that can be solved using geometric nonlinear control and 2) transformers, the key architectural component of most large language-models, can be seen the discretization of a consensus algorithm and thus large language-models exhibit consensus. Another point of contact between control theory and deep learning is the connection between backpropagation, the algorithm used to train neural networks, and optimal control. The article addresses how the older connection between these problems, established in the 80's using discrete-time models can be reconciled with the more recent approach using continuous-time models. Several open problems and directions for future research are discussed, including selective state-space models, a recent class of models inspired by the models used in control.

In this paper, we establish the stability of general entropy wave solutions of the compressible Euler equations (that is, profiles with constant velocity and pressure) in the class of vanishing dissipation limits from the Navier–Stokes–Fourier system. If the initial data of our sequence of solutions to the Navier–Stokes–Fourier system converges to the initial data of the entropy wave solution, we further have convergence of the solutions to the entropy wave. This gives uniqueness and stability of the entropy wave in this class. The proof is based on the $a$-contraction theory.

Let $\kappa_p(n)$ denote the maximum kissing number of a lattice $\mathcal{L} \subset \mathbb{R}^n$. We prove that $\kappa_p(n) \geq 2^{\Omega(\sqrt{n})}$ for all $p\ge 1$ and that $\kappa_p(n) \geq \Omega(\alpha_p^n)$ for all $p > 2$, where $\alpha_p > 1$ is a constant depending only on $p$. In particular, our former result improves on the previously best known lower bound of $\kappa_2(n) \geq 2^{\Theta(\log^2 n)}$ for the $\ell_2$ kissing number, which has stood for over $60$ years. (In concurrent and independent work, Laarhoven and Kominers [arXiv, 2026] showed a similar lower bound on $\kappa_2(n)$ using similar techniques.) In fact, our lower bounds follow from efficient (randomized) constructions of lattices with large kissing number. Our results use a general construction of lattices with large kissing number. Specifically, we show a simple equivalence between such lattices and a new object that we call an exact locally dense lattice. Closely related objects appear throughout the literature on computational lattice problems. Our constructions of lattices with large kissing number then follow as corollaries by showing that constructions due to Bennett and Peikert [RANDOM, 2023] (using Craig lattices) and Aggarwal and Stephens-Davidowitz [STOC, 2018] (using sparsification of $\mathbb{Z}^n$) in fact give exact locally dense lattices, as well as a new construction of exact locally dense lattices built from rotations of the $m$-dimensional checkerboard lattice $D_m$.

The new framework of semi-tautological extensions has allowed for explicit understanding of low-degree odd cohomology groups of moduli spaces of stable curves. Let $S^*_\omega(\overline{\mathcal M}_{g,n}) \subset H^*(\overline{\mathcal M}_{g,n})$ denote the semi-tautological extension generated by $H^{11}(\overline{\mathcal M}_{1,11})$, that is, the smallest system of subrings of $H^*(\overline{\mathcal M}_{g,n})$ containing $H^{11}(\overline{\mathcal M}_{1,11})$ and the tautological subrings, which is also closed under pullbacks along forgetful maps and pushforwards along gluing maps. Recently, it has been shown that $H^{11}(\overline{\mathcal M}_{g,n}) = S^{11}_\omega(\overline{\mathcal M}_{g,n})$ and $H^{13}(\overline{\mathcal M}_{g,n}) = S^{13}_\omega(\overline{\mathcal M}_{g,n})$ for all $g$ and $n$. Meanwhile, $H^{15}(\overline{\mathcal M}_{1,n}) \neq S_\omega^{15}(\overline{\mathcal M}_{1,n})$ for all $n \geq 15$. Nevertheless, we prove that $H^{15}(\overline{\mathcal M}_{2,n}) = S^{15}_\omega(\overline{\mathcal M}_{2,n})$ for all $n$. Moreover, we determine all relations in $S_\omega^{15}(\overline{\mathcal M}_{g,n})$ for all $g \geq 2$ and give explicit bases for this vector space in terms of decorated graphs. Conditional upon the conjectured dimension of $H^{15}(\overline{\mathcal M}_{3,14})$, we also prove that $H^{15}(\overline{\mathcal M}_{3,n}) = S^{15}_\omega(\overline{\mathcal M}_{3,n})$ for all $n$. As an application, we prove $H^*(\overline{\mathcal M}_{g,n}) = RH^*(\overline{\mathcal M}_{g,n})$ for several new small pairs $(g,n)$.

The interplay between a weighted volume measure and its naturally associated weighted Laplace operator is fundamental to the geometric analysis of weighted structures governed by gradient vector fields. Beyond the gradient setting, however, the theory has remained considerably less developed, largely because this natural correspondence is lost. In this paper, we identify invariant measures as the appropriate framework for extending the theory to general drifted Laplace operators. This approach yields a systematic treatment of parabolicity in the non-gradient setting, reveals its connection with volume growth, and leads to new volume finiteness results for non-gradient geometric structures.