Papers

A set $S$ of vertices in a graph $G$ is a dominating set of $G$ if every vertex not in $S$ has a neighbor in $S$, where two vertices are neighbors if they are adjacent. The domination number, $\gamma(G)$, of $G$ is the minimum cardinality among all dominating sets of $G$. Given a set $S$ of vertices of a graph $G$, two vertices are located by $S$ if they have distinct sets of neighbors in $S$. Moreover, if $S$ locates every pair of vertices not in $S$, then it is called a locating set of $G$. A locating dominating set of $G$ is both a dominating and a locating set of $G$. The locating domination number, $\gamma^{\rm LD}(G)$, is the minimum cardinality among all locating dominating sets of $G$. A notable conjecture in the study of locating dominating sets is to show that the locating domination number of an isolate-free and twin-free graph of order $n$ is at most $\frac{1}{2}n$. So far, the best approximation to this upper bound conjecture is known to be $\left \lceil \frac{5}{8}n \right \rceil$. Much in line with the conjecture, an even stronger reformulation proposed in the literature asks if it is possible to partition the vertex set of an isolate-free and twin-free graph into two locating sets. However, such partitions into locating sets may not exist if the graph is also allowed to have twins. Continuing with this line of research, we show that if $G$ is an isolate-free (and not necessarily twin-free) graph, then the vertex set of $G$ can be partitioned into a dominating set and a locating dominating set. As a consequence, we infer that every isolate-free graph $G$ of order $n$ satisfies $\gamma(G) + \gamma^{\rm LD}(G) \le n$, and we show that the last bound is tight. Moreover, our proof of the existence of a partition of the vertex set of an isolate-free graph into a dominating and a locating dominating set also provides a polynomial-time algorithm to construct such a partition.

We consider the problem of uniqueness of Ricci flows, starting from singular initial data. Even when the initial data is smooth and smoothly obtained, in the complete noncompact case uniqueness is not known in full generality. When starting from nonsmooth initial data, very little is known, even in the compact case. We obtain a uniqueness result for Ricci flows with scaling-invariant curvature bounds, a suitable noncollapsing assumption, and sectional curvature lower bounds, coming out of (possibly singular) Reifenberg spaces. As a consequence of our result, and existence results obtained by Simon–Topping and Lai, there is a canonical way to define a smooth structure on any Reifenberg uniformly volume-noncollapsed PIC2 limit space. Further, we obtain a diffeomorphic version of Perelman's stability theorem for these limit spaces.

Reeb Transforms offer a compact representation of how shapes in $\mathbb{R}^n$ change when sliced along varying directions. We define the Reeb Transform as the family of Reeb graphs induced by height functions along all directions in the unit sphere. In this paper, we develop a rigorous treatment of Reeb Transforms for o-minimal definable sets, with particular emphasis on one-dimensional stratified spaces embedded in $\mathbb{R}^d$ and on surfaces in $\mathbb{R}^3$. We establish injectivity of the Reeb Transform in multiple settings, including compact surfaces in $\mathbb{R}^3$, capturing the essential topological features needed to uniquely reconstruct such surfaces. However, in dimensions above three, the Reeb Transform ceases to be injective, indicating the limitations of this descriptor in higher-dimensional settings.

We prove Ehrhart positivity for series-parallel matroids and extend this result to all series-parallel extensions of multi-path matroids. A formula for the Ehrhart coefficients of series-parallel matroids is derived and used to prove strict positivity. An extension criterion based on weighted lattice-point counts is established. Recursive formulas for lattice-point counts are derived, and the recursion is shown to preserve a class of polynomials with nonnegative coefficients. We finally construct a connected transversal matroid with a negative Ehrhart coefficient, disproving the conjecture of Ferroni, Morales, and Panova that every connected matroid whose base polytope admits a subdivision into series-parallel matroid base polytopes is Ehrhart positive.

We record a finite-field census of the split-cubic elliptic curves $E_{a,b,c}^{(n)}:y^2=(ax+n)(bx+n)(cx+n)$ attached to positive integral $D(n)$-triples. For $n\in\{-3,3,5,8,12,20\}$ there are $164$ such triples with $1\le a<b<c\le100$, allowing zero squares. Their reductions at the ten primes $\operatorname{NextPrime}(10^k)$, $3\le k\le12$, give $1640$ records, of which $45$ have cofactor four relative to the largest prime divisor of the group order. We collect the standard split-cubic and quadratic-twist identities in the explicit normalization $\lambda=b(c-a)/(c(b-a))$ and $\delta=cn(b-a)$, and use them to audit the point counts. The archive records group orders, factorizations, cofactors, embedding degrees and transfer-field bit lengths. A worked example compares the main curve with its nonsquare twist. The census is deterministic and descriptive; it provides neither an asymptotic density estimate nor cryptographic parameter recommendations.

We show that the conformal class of a compact spin manifold with boundary is uniquely determined by its boundary values of harmonic spinors if the metric is locally conformal to a real-analytic metric in some coordinates. More precisely, we consider the boundary conjugation (BC) map for the Dirac operator corresponding to MIT boundary conditions, and prove that if two locally conformally real-analytic (LCRA) metrics have equivalent BC maps over an open subset of the boundary, then the two manifolds are conformally equivalent. The class of LCRA manifolds includes conformally Einstein manifolds, as well as all smooth surfaces. We also show that a real-analytic unitary connection on an LCRA manifold is uniquely determined up to local gauge equivalence by the BC map of its twisted Dirac operator in dimensions greater than 2.

We study the scaling limit of the collision measure of two i.i.d. discrete-time random walks on $\mathbb Z^2$, which records their collision sites and times. This is a critical regime for collisions: the walks collide infinitely often, whereas the limiting Brownian motions do not collide at positive times. Hence, existing general results on the convergence of collision measures do not apply. Assuming that the jump distribution has mean zero and finite second moment, we prove that, under a logarithmic scaling, the collision measure converges in distribution to a non-trivial random measure. We also give an explicit representation of the limiting measure in terms of a Poisson point process. To prove convergence, we introduce a new cluster decomposition of the Laplace transform of the collision measure, revealing the cluster structure of collisions.

In this paper, we first expand fixed diagonals of the Jacobi-Stirling numbers in a binomial basis. For the second kind, the expansion coefficients are polynomials in $z+1$ with nonnegative integer coefficients. We give a recurrence and a signed-partition interpretation for these coefficients. The same holds for the differences between corresponding unsigned first-kind and second-kind coefficients. We then prove that every nonzero nonnegative linear combination of the descent polynomials over Jacobi-Stirling permutations with a fixed number of deleted barred letters has only simple negative zeros when that number is one, two, or three. The same holds when exactly one or two barred letters are retained. Thus we verify five infinite families in a conjecture of Gessel, Lin and Zeng. Finally, using insertion operators, we find that every descent polynomial over Jacobi-Stirling permutations with a fixed number of deleted barred letters is top heavy and has an increasing left half.

We establish a weak type $(1,1)$ estimate for the Bochner–Riesz mean operator at the critical index $\delta=\frac{d-1}{2}$ on the quantum torus $\mathbb{T}^d_\theta$, where the associated kernel is rough.

We construct degree-three isogenies of Hesse elliptic curves over an arbitrary field of characteristic different from three, which correspond to a step of the cubic arithmetic-geometric mean. As an application, we describe the Serre-Tate canonical lift of an ordinary elliptic curve in characteristic three by the 3-adic cubic arithmetic-geometric mean. We also discuss the relations of the Hesse curves with complex and finite hypergeometric functions.

We derive long-time asymptotics in different space-time regions for the Sawada-Kotera (SK) equation with admissible data involving continuous spectrum and finite discrete spectrum. Our result proves the soliton resolution conjecture, yields the asymptotics governed by the F-XVIII Painlevé transcendent, and establishes the asymptotic stability of \(N\)-soliton solutions. More importantly, this extends the long-time asymptotic theory to the coexistence of solitons and radiation, whereas previous results were restricted to the purely continuous spectrum. One key ingredient in our argument is a new meromorphic Riemann-Hilbert framework that incorporates the discrete spectrum through complete six-point pole orbits while preserving cofactor analyticity and the singular structure at the spectral origin. A second ingredient is the combination of an orbitwise, region-dependent pole reduction mechanism and the \(\bar{\partial}\)-nonlinear steepest descent method, which allows us to select the solitons contributing in each region and separate them from the continuous radiation. Our framework also resolves the coalescing stationary points at the singular spectral origin through the regular modified SK problem and the Miura map.

We prove that the perverse filtration of the canonical $\mathrm{SL}_n$ Hitchin system is multiplicative when the rank and degree are coprime. Over the elliptic locus of an even-degree twisted Hitchin system, we compare the Fourier projectors for compactified Pryms with the endoscopic correspondence. We then use the square-zero property of the variant cohomology and a parabolic Higgs moduli reduction to pass to the full canonical moduli space.

We develop a direct paradifferential approach to nonlinear graph integral operators arising in contour dynamics. The main methodological result is an abstract structure-preserving paralinearization theorem, valid in arbitrary dimension, for a class of homogeneous kernels whose nonlinear dependence on the graph is expressed through normalized finite differences. The theorem covers both critical principal-value kernels and subcritical locally integrable kernels, gives explicit formulas for the principal symbols, and separates the finite-order symbolic contribution generated by the homogeneous singularity from order-zero far-field terms and arbitrarily smoothing remainders. A structural feature of the reduction is the cancellation of the intermediate symbolic order; at the critical endpoint the construction also preserves the exact skew-adjoint structure of the original singular integral. As a main application, we consider the three-dimensional two-phase free-boundary Euler equations with constant interfacial background vorticity. We first derive an autonomous contour-dynamics equation in canonical surface variables and then apply the abstract theorem directly to the singular integral operators generated by the Birkhoff–Rott formulation. This yields the complete paradifferential structure of the system without taking the Dirichlet–Neumann reduction as a starting point. After introducing an Alinhac good unknown and diagonalizing the resulting system, we obtain local well-posedness for small Sobolev perturbations in the stable Kelvin–Rayleigh–Taylor regime, uniformly on compact subsets of the stable parameter set.

Primal heuristics for finding high-quality feasible solutions are an important component in mixed-integer optimization (MIO) solvers. Recent advances in GPU-accelerated optimization algorithms show the potential of GPU acceleration for continuous optimization. In this paper, we introduce the Parallel Node Generation and Exploration Algorithm (PaNGEA), a GPU-friendly MIO primal heuristic. PaNGEA explores restricted subproblems by combining linear-relaxation solves with a local-search procedure designed for efficient batched execution on GPUs. In addition, instead of relying on a single heuristic for iterative variable fixing, we leverage GPU batching capabilities to generate and explore multiple subproblems in parallel. PaNGEA leverages GPU capabilities in two ways. First, on 283 MIPcc26 and MIPLIB instances, implementing a single-node primal heuristic on the GPU reduces the average gap integral by 8-18% relative to its CPU counterpart. Second, generating and exploring multiple nodes in parallel further reduces the average gap integral by 12-19%.

Every monotone Boolean function on the continuous cube admits a coalition of $O(n/(\varepsilon\log n))$ coordinates that can force a fixed output (either zero or one) with probability at least $1-\varepsilon$. This old conjecture of mine was recently proven by Chattopadhyay and Gurumukhani~\cite{CG}. This writeup contains a simplification of their proof, generated by AI after suggesting the use of the junta theorem of~\cite{F98} and the discretization procedure of ~\cite{F04}.

We develop a cancellation mechanism for linear second-order kinetic equations in non-divergence form with measurable uniformly elliptic coefficients. We decompose the dynamics into spatial waves and follow two families of velocity paths along which the waves acquire nearly opposite phases. The parabolic Krylov–Safonov theory gives a common lower bound for the two velocity marginals. The corresponding contributions cancel up to a small phase error, producing a contraction \textità la Döblin}. Iterating this contraction yields exponential Fourier decay estimates with enhanced dissipation. We then present two applications. The first contribution is a kinetic Aleksandrov estimate: a maximum principle in which the source is measured in an $L^p$ norm. For rough coefficients $A(t,v)$ independent of position, we obtain the estimate for every $p>2n+1$, where $n$ is the dimension of position and velocity. The Fourier decay also gives spatial smoothness, and parabolic regularity gives Hölder continuity in time and velocity. For autonomous coefficients $a(x,v)$ in dimension one, position serves as time away from zero velocity. A Harnack comparison controls returns to small velocity intervals and yields the estimate for every $p>4$. Both thresholds are optimal among those valid for all ellipticity ratios. The second contribution concerns position on the torus and velocity on the sphere, with measurable uniformly elliptic diffusion coefficients $A(t,v)$ independent of position. We prove enhanced dissipation with the optimal square-root frequency power and Gevrey regularity in position. For autonomous coefficients $A=A(v)$, we also obtain exponential convergence to equilibrium in $L^2$ weighted by the invariant velocity measure, and a quantitative spectral gap. To our knowledge, these are the first results on decay established for laws of kinetic equations with rough coefficients.

Let $p$ be an odd prime, $\psi$ a nontrivial additive character of $\mathbb F_p$, $\psi_q=\psi\circ\mathrm{Tr}_{\mathbb F_q/\mathbb F_p}$ and $\lambda\in\mathbb F_p^\times$. For $h\in(\mathbb F_q^\times)^n$ let $\Sigma_n(h)$ be the sum of $\psi_q(u_1^{-1}+\cdots+u_n^{-1})$ over $u\in(\mathbb F_q^\times)^n$ with $h_1u_1+\cdots+h_nu_n=-\lambda$, an additively twisted average of products of Kloosterman sums. For every $n\ge2$, the cohomology computing $\Sigma_n(h)$ is concentrated in one degree and has dimension $2^n-1$. It has $r(n)=2^n-\binom n{\lfloor n/2\rfloor}$ eigenvalues of weight $n-1$, and its other eigenvalues are the integers $q^{k-1}$, $1\le k\le\lfloor n/2\rfloor$, with ballot numbers as multiplicities. The part of top weight forms a lisse sheaf $\mathcal W_n$ of rank $r(n)$ on the torus $\mathbb G_m^n$. We determine its local monodromy at the coordinate boundary and its determinant: $\det\mathcal W_n$ is the Artin–Schreier sheaf of the sum $-2^{n-1}\lambda^{-1}(h_1+\cdots+h_n)$ of the critical values of the phase, twisted by $p^{(n+1)r(n)/2}$. The geometric monodromy group of $\mathcal W_n$, and the arithmetic one after normalization to weight $0$, are equal to $\{g\in\mathrm{GL}_{r(n)}:\det(g)^p=1\}$. Consequently the normalized sums become equidistributed like traces of Haar-random elements of $\{g\in\mathrm U(r(n)):\det(g)^p=1\}$. The proofs use Laumon's Fourier transform, middle convolution, the Cayley cubic surface and Larsen's alternative, but no classification of finite groups.

Neural network-based approaches have emerged as efficient alternatives to traditional optimization-based procedures for the calibration of stochastic volatility models. However, existing work has focused primarily on predictive accuracy, with comparatively little attention devoted to understanding the structure of the learned inverse calibration mappings. In this work, we analyze neural calibration mappings for the Heston and rough Heston models across multilayer perceptron, highway, and softmax-parametrized highway architectures, using complementary Shapley-based methods from explainable AI. Specifically, we consider SHAP and $\nu$SHAP explanations, which capture distinct, complementary notions of feature relevance, corresponding to sensitivity and sufficiency of feature subsets, respectively. Short maturities and smile wings consistently dominate parameter inference, and the dominant attribution structure remains qualitatively stable across architectures despite differences in predictive accuracy and parameter count. Parameter-specific differences between SHAP and $\nu$SHAP further reveal how distinct regions of the implied volatility surface contribute to parameter recovery and expose substantial redundancy in the calibration input. Building on this redundancy, we show that $\nu$SHAP explanations can guide a significant reduction in input dimensionality for the rough Heston model while matching calibration accuracy relative to the full implied volatility surface. These findings demonstrate that complementary Shapley-based methods provide structural insight into learned inverse calibration mappings beyond predictive error metrics, and offer a practical route to feature selection in neural calibration problems.

A group is conjugacy separable if every pair of non-conjugate elements remains non-conjugate in some finite quotient. Virtually polycyclic groups, and hence in particular all finitely generated nilpotent groups, are conjugacy separable. The conjugacy separability growth function measures the complexity of distinguishing non-conjugate elements in finite quotients by giving the smallest order of a finite quotient $Q$ that separates them. Recent work showed that this function admits polynomial upper and lower bounds for nilpotent groups, but these estimates are neither explicit nor optimal. We prove improved polynomial upper bounds whose degree is at most linear in the nilpotency class and at most quadratic in the Hirsch length. For $2$-step nilpotent groups, we obtain sharper estimates and show that these estimates are optimal among bounds depending only on the nilpotency class and Hirsch length. This gives a correction of an error in the existing literature. The main tool is a translation of conjugacy separability growth into Lie rings, generalizing previous work in the case of residual finiteness.

Chance-constrained programs (CCPs) optimize decisions under uncertainty by limiting the probability of constraint violation. Despite advances in traditional and learning-based approaches, optimizing non-convex or non-smooth objectives and adapting to different objectives under fixed chance constraints remain challenging. In this paper, we propose a Derivative-free Diffusion-based framework that Disentangles constraint modeling from objective optimization, termed \textbf{D$^3$Opt}. We learn the chance-feasible structure once, independently of any particular objective, by training a risk-conditioned diffusion model solely on constraint-filtered decisions and freezing it as a reusable prior for post-specified objectives. At inference time, we propose an annealed, particle-based Feynman–Kac correction along the frozen reverse diffusion process to optimize post-specified objectives using only function evaluations. This enables derivative-free optimization of non-convex and non-smooth objectives without objective-specific retraining. We prove that the correction preserves feasibility when this property holds for the frozen prior, and derive an optimization-error bound separating learned-prior coverage, finite-particle approximation, and finite-temperature effects. Experiments on linear Gaussian CCPs, objective-transfer tasks, and chance-constrained economic dispatch demonstrate effective optimization across smooth and non-smooth objectives, including non-convex cases, and objective generalization under fixed chance constraints without retraining.

Let $$ S_n=\left\langle \genfrac{[}{]}{0pt}{}{n}{1},\ldots,\genfrac{[}{]}{0pt}{}{n}{n-1}\right\rangle $$ be the numerical semigroup generated by the nontrivial unsigned Stirling numbers of the first kind in the $n$th row. Writing $a=\genfrac{[}{]}{0pt}{}{n}{n-1}=\binom n2$ and $b_j=\genfrac{[}{]}{0pt}{}{n}{n-2j}$, we determine the complete gcd filtration of the reduced generating system. More precisely, if \[ M_j=\frac12\operatorname{lcm}\{m\ge1:\varphi(m)\le2j\}, \] then $$ \gcd(a,b_1,\ldots,b_j)=\frac{a}{\gcd(a,M_j)}. $$ The proof uses prime-power block polynomials and exact $p$-adic support at threshold degrees. We then establish a divisor-support property for canonical mixed-radix reductions and combine it with an elementary-symmetric growth estimate to prove that, after inactive generators are removed, the resulting generating sequence is telescopic. Consequently, the Apéry set of $S_n$ is rectangular, yielding an explicit Frobenius formula. The same structure shows that $S_n$ is free and symmetric and gives explicit formulas for its genus and conductor, while its type is one.

The main result determines the zeroth integral Milnor-Witt stem of the motivic sphere spectrum in the Morel-Voevodsky motivic stable homotopy category of the integers. The component in weight zero is the Grothendieck-Witt ring of nondegenerate symmetric bilinear forms over the integers. Along the way, cellularity of connective Witt theory, as well as an absolute purity result for the {\eta}-inverted motivic sphere spectrum, is established over Dedekind domains of mixed characteristic.

We establish comparison Pogorelov estimates for admissible solutions of graphical $\sigma_k$-curvature equations, extending the two-surface estimate of Qiu and Yan for the graphical scalar curvature equation. The comparison graph is assumed only to be $k$-admissible, with a bounded slope and a positive interior gap that vanishes on the boundary. The estimates cover $2\le k<n$ for prescribed data $f(x,u)$, and $n/2\le k<n$ for general data $f(x,u,Du)$. The proof combines the two-surface localization and mixed Gårding comparison of Qiu and Yan with Yan's spectral concavity inequality and an angle-function weight.

In Rational Extended Thermodynamics, the monatomic limit of polyatomic gases is often singular. For the internal-state density $\phi(I)=I^a$, with $a=(D-5)/2$, we prove that the normalized equilibrium distribution of the molecular internal energy $I$ concentrates at $I=0$ as $D\to3$, providing a rigorous justification of the Dirac prescription and its first-order correction. In classical ET14, the internal-mode energy differs from equilibrium by $-3\Pi/2$; positivity yields $-p<\Pi<(D-3)p/3$, while the exact entropy develops a boundary corner at $\Pi=0$. Negative dynamic pressures remain admissible, but fixed negative values do not approach the monatomic state with the prescribed total energy. This explains the need for compatible initial data. In the relativistic case, the internal energy is conditionally gamma distributed at fixed momentum; the constitutive integrals follow directly, and we derive an exact energy identity and the first correction to the Synge energy.

We introduce the skew Schäffer type constant as an infimal counterpart of the recently developed skew James type constant \cite{RaoWuLiuNiWang} and establish its basic geometric theory in Banach spaces. The new constant admits sharp universal bounds and characterizes uniform nonsquareness through a strict gap condition. Exact values are obtained in Hilbert and Lebesgue spaces, where an exact relation with the skew James type constant is established. We further derive a James-modulus estimate leading to sufficient conditions for uniform normal structure.

Let $L$ be a line bundle of negative degree on an elliptic curve $E$. While the ruled surface $M := \mathbb P(\mathcal O_E \oplus L)$ is one of the simplest varieties in algebraic geometry, its tangent bundle $T_M$ is surprisingly complex: we first show that the ring of symmetric tensors is not finitely generated. We then construct a holomorphic function on the canonical extension $Z_M$ which allows us to show that $Z_M$ is not Stein.

We study a forward-backward McKean-Vlasov system with jumps and one-way coupling: the forward equation is independent of the backward variables, whereas the BSDE depends on the forward state and its law. A source generated by a maximal monotone operator is evaluated along the forward state and then accumulated in the backward equation. The operator is the subdifferential of a convex potential evaluated at the state translated by a function of the control, plus a term depending on the law of the state, so the control shifts the discontinuity interface of its minimal-norm selection. We establish modelwise well-posedness and strong stability, with estimates uniform over the model family. We then study the Yosida approximation: strong convergence, Minty-type strong-weak closedness when the state, its law, and the control (hence the moving interface) vary, and uniform convergence under an integrability condition. For an asymmetric loss-aversion prototype, a quantitative occupation estimate for a layer around the interface yields an explicit Yosida convergence rate. When the continuous martingale part is nondegenerate and the interface moves absolutely continuously, the occupation time is linear in the layer width. The same rate controls the backward components and the cumulative source.

Let $a(n)$ be the number of standard Young tableaux of shape $(n,n,n)$ whose entry in row $2$, column $1$ is odd; equivalently, the number of those whose first descent is even. This is entry A011553 of the On-Line Encyclopedia of Integer Sequences, contributed in 1996, and after thirty years of curation it carries no formula. We supply one, $a(n) = 8\,\bigl(n!\,(n+2)!\bigr)^{-1}\sum_{m=1}^{\lfloor n/2\rfloor} m(m+1)(3n-2m-1)!/(n-2m)!$, and use it to settle both of the conjectures the entry records. A creative telescoping certificate shows that $a$ satisfies a linear recurrence of order two with polynomial coefficients; the order-three recurrence conjectured by R. J. Mathar in 2023 is a left multiple of it, with explicit cofactor $(4S^{-1}-3)/(7n-9)$. We also prove $a(n)\sim 3^{3n+7/2}/(64\pi n^{4})$, the asymptotic conjectured by V. Kotesovec in 2014. The second proof gives slightly more than the conjecture asks: the position of the first descent has a limiting distribution, the probability that the $(2,1)$ entry equals $r+1$ tending to $r(r+2)/3^{\,r+1}$. Summing the even terms, a uniformly random tableau of shape $(n,n,n)$ has an odd $(2,1)$ entry with probability tending to $27/64$, whereas for a uniformly random tableau of $n$ cells of unrestricted shape the corresponding limit is $1/e$.

Let $M$ be a closed manifold equipped with a collection of smooth real vector fields $X_1,\ldots, X_k$ and a smooth measure $\mu$. Let $P:=\sum_j X_j^\dagger X_j $ be the associated sublaplacian. Under the assumption that each pair of points of $M$ can be connected by a path obtained by joining integral curves of the vector fields, the equation \[ Pu=f \] admits unique zero-average solutions $u$ in the natural energy space, for all zero average data $f\in L^2$. We say that exact regularity holds for the above equation if $u\in H^k(M)$ whenever $f\in H^k(M)$, where $H^k(M)$ is the $L^2$ based Sobolev space of any order $k\in \mathbb{N}$. We investigate how the presence of a characteristic submanifold, namely a submanifold tangent to all vector fields $X_j$, may cause a failure of exact regularity. We prove that this indeed happens for a class of "worm sublaplacians", which are real analogues of Kohn Laplacians on Diederich–Fornaess worm domains, of interest in several complex variables. Our main tool is a scaling lemma inspired by work of Barrett and Christ on the $\bar\partial$-Neumann problem.

We develop a representation-theoretic approach to determine the Morse index and nullity of an embedded minimal surface based only on its topology and symmetry group. The method combines an equivariant comparison between the second variations of area and energy with the representation theory of the dihedral group. In the Euclidean unit ball we compute for every $k\geq3$ the Morse index and nullity of any free boundary minimal $k$-noid with prismatic symmetry of order $4k$, and of any free boundary minimal surface of genus one with connected boundary and antiprismatic symmetry of order eight. In the round sphere we obtain an alternative computation for the Morse index and nullity for the first family of the Lawson surfaces, using only half of their symmetry group. The results are independent of any particular construction of the surfaces.