Papers

Let $X_N(t)$, $t\in[0,\pi]$, be a stationary Gaussian trigonometric polynomial, and let $\mathcal Z_N$ denote the number of zeros of the rescaled process $X_N(t/N)$ on $[0,N\pi]$. We establish the quantitative central limit theorem \[ d_{\mathrm W}\left( \frac{\mathcal Z_N-\mathbb E[\mathcal Z_N]} {\sqrt{\operatorname{Var}(\mathcal Z_N)}},Z\right) \le C\frac{(\log N)^2}{\sqrt N}, \] where $Z$ is standard Gaussian and $d_{\mathrm W}$ denotes the Wasserstein-1 distance. The proof combines Ornstein–Uhlenbeck regularization of the singular Kac representation with direct estimates of the covariance between the Malliavin–Stein factor and Stein test functions. A lattice lower bound of order $N^{-1/2}$ shows that the convergence rate is optimal up to logarithmic factors.

We study compact static manifolds with boundary from a spectral point of view, with particular emphasis on the relation between Dirichlet, Neumann, and Robin spectra and the nodal structure of static potentials. We establish comparison inequalities linking Dirichlet and Neumann eigenvalues with the scalar curvature and derive sufficient spectral conditions ensuring that a static potential changes sign. We also obtain bounds on the number of connected components of the zero set, sufficient conditions for its connectedness, and sufficient conditions ensuring that its components intersect the boundary. Finally, we study variational properties of Robin static triples and characterize compact Robin static triples whose metrics are critical for the first mixed Robin–Dirichlet eigenvalue under admissible volume-preserving deformations.

The Dotsenko-Fateev equation is a third-order linear ordinary differential equation studied in two-dimensional conformal field theory. We study the confluent Dotsenko-Fateev system, a rank-three system of linear ordinary differential equations obtained by confluence from a first-order system equivalent to the Dotsenko-Fateev equation. It has an irregular singular point at infinity, and its solutions admit double integral representations of confluent Selberg type. Using these representations, we construct sectorial fundamental solution matrices around infinity, establish their asymptotic expansions, and determine the associated Stokes matrices explicitly. We further study three aspects of the confluent Dotsenko-Fateev system: its equivalence to the degree-two polynomial reduction of the Schrödinger equation for the quantum Painlevé equation of type V, its realization as a restriction of a confluent version of the linear Pfaffian system associated with Appell's hypergeometric function $F_4$, and its reduction to the trivial rank-one equation via integral transforms and a quadratic change of the independent variable.

This article develops optimal order (up to logarithmic factors) weighted cubature rules for Monte Carlo numerical integration in Sobolev reproducing kernel Hilbert spaces over common matrix manifolds. Specifically, we consider matrix models of the Stiefel, Grassmann, flag, and Lagrangian Grassmann manifolds each over either real numbers $\mathbb{R}$, complex numbers $\mathbb{C}$, or quaternions $\mathbb{H}$. The proposed cubature rules use evaluation points which are independent and identically distributed uniform on the manifold. The weights are chosen to be optimal with respect to the reproducing kernel Hilbert space worst-case error, under the constraint that the weights sum to one. Crucially, this constraint ensures optimal weights may be found without having to evaluate the kernel mean embedding (the integral of the kernel), which is often difficult to compute. We consider computable Matérn kernels of Frobenius distances between matrices which reproduce the Sobolev reproducing kernel Hilbert spaces with equivalent norms. Open-source numerical experiments validate our theoretical development, including an application to simultaneous approximation of intrinsic volumes in the Cauchy–Kubota formula. We also present preliminary empirical results beyond independent points including transformations of low-discrepancy quasi-Monte Carlo points from the unit cube and jointly optimized points and weights.

We introduce a determinantal linear process on the space of $1$-forms over a weighted graph endowed with a unitary local system, which we call the determinantal linear spanning forest (DLSF). This process is the analog of the uniform spanning tree in the presence of an arbitrary rank $n$ quenched gauge field in ${\rm U}(n)$ (or ${\rm O}(n)$, ${\rm SO}(n)$, ${\rm SU}(n)$, ${\rm Sp}(n)$), and generalizes the determinantal random cycle-rooted spanning forest which corresponds to the case of ${\rm U}(1)$ and ${\rm SU}(2)$. We describe the support of this process by means of a polymatroid which is described in terms of the circular and bicircular matroids. We also explain how the DLSF relates to the computation of currents in an electrical network in the presence of a gauge field. The partition function of the DLSF is shown to be the determinant of the twisted Laplacian, for which we also give a matrix-tree type theorem that computes its characteristic polynomial. We initiate the study of the twisted Laplacian determinant as a functional on the space of connections, and give a first order expansion of it in a neighborhood of the trivial connection, for which we provide a probabilistic interpretation. We extend the construction to infinite graphs, exhibiting the analogs of the classical wired and free distributions on spanning forests. We give a sufficient geometric condition on the local spectral gap for the existence of a Poincaré inequality as well as the uniqueness of the infinite volume measure, and the exponential decay of correlations. We show that on any recurrent graph, the wired and free DLSF coincide. We construct examples where they do not, for example, unlike in the classical case, on $\mathbb Z^d$ for $d\geq 3$.

We give counterexamples to the conjecture of Dais-Henk-Ziegler on projective crepant resolutions of complete-intersection quotient singularities. For finite $G\subset\operatorname{SL}_4(\mathbb{Z})$, we determine whether $\mathbb{C}^4/G$ admits a projective crepant resolution, with the representations considered up to conjugacy in $\operatorname{SL}_4(\mathbb{C})$. Ten explicit criteria decide all cases. The examples also show that the Molien series and the age polynomial do not determine existence. We also discuss questions of Ito-Reid on maximal cyclic subgroups and the generation of cohomology in degree two.

We consider a random subgraph G_p(n,r,<s) of the complete distance graph G(n,r,<s) whose vertices are the r-element subsets of the set {1,...,n} and whose edges join pairs of subsets that intersect in fewer than s elements; each edge survives independently of the others with probability p. The stability of the Erdos-Ko-Rado theorem is the event that the independence number of the random subgraph equals that of the original graph, N = C_{n-s}^{r-s}. For fixed r and s, two-sided bounds for the critical probability, whose ratio depends on r and s, were obtained by P. A. Ogarok and A. M. Raigorodskii. In this paper we establish the order of the critical probability for growing r=r(n) and s=s(n): if min{C_r^s r^2, s r^3} = o(n), then the upper and lower bounds differ only by an absolute constant factor. The proof is elementary; only one of the two branches of the growth condition uses the complete nontrivial-intersection theorem of Ahlswede-Khachatrian.

We determine thirteen entirely CM identity fibers of Shimura parametrizations attached to elliptic curves of rank two and conductor below $2000$. Each yields a proof of the imaginary quadratic class number one theorem. The argument combines the Gross–Zagier formula with the Ribet–Takahashi degree comparison to show that the fiber is completely explained by known CM points.

We study the Solomon–Terao algebra and the Solomon–Terao complex of a central hyperplane arrangement $\mathcal A$ in $\mathbb C^\ell$. Let $\eta$ be a homogeneous polynomial of degree $d\geq2$ whose restriction to every flat of $\mathcal A$ has no critical point away from the origin. We prove that the Solomon–Terao algebra $\mathrm{ST}(\mathcal A,\eta)$ has top degree $|\mathcal A|+\ell(d-2)$ and a one-dimensional top graded component with an explicit generator, solving a conjecture of Abe, Maeno, Murai, and Numata. For $d=2$, we show that the total dimension of $\mathrm{ST}(\mathcal A,\eta)$ is bounded below by that of the Orlik–Solomon algebra. Both results hold without assuming that the arrangement $\mathcal A$ is tame. Finally, we prove that $\mathcal A$ is tame if and only if all positive-degree homology groups of the Solomon–Terao complex associated with $\eta$ vanish.

In this paper, we study various structural properties like girth, chromatic num ber, etc. of the intersection hypergraph on the subgroups of nite non-cyclic abelian group and we characterize for the star and hypertree structures. Fur thermore, we characterize hypergraphs which are embeddable on plane, torus and projective plane.

We study the determinant line bundle for a family of elliptic operators on compact manifolds with boundary with the associated Quillen metric and Bismut-Freed connection, and we prove the global anomaly formula in this context. This extends Witten's holonomy theorem to manifolds with boundary. Witten studied the limiting behavior of the eta invariant of a Dirac operator on the total space of a fibration over $\mathbb{S}^1$ when the metric in the base $\mathbb{S}^1$ is blown up in a certain manner, a process known as taking the adiabatic limit. In this paper, we construct the calculus of adiabatic $b$-pseudodifferential operators, which naturally synthesizes the $b$-calculus and the adiabatic calculus of pseudodifferential operators. Using this calculus, and assuming the fiber operators to be invertible, we compute the adiabatic limit of the natural eta invariant in this setting and relate it to the integral of the Bismut-Freed connection and new correction terms coming from the boundary of the fibers.

Gromov hyperbolicity measures the geometric complexity of metric spaces and metric graphs through geodesic triangles and four-point conditions. While calculating the exact continuous hyperbolicity constant $\delta^*(G)$ of a graph $G$ typically requires evaluating quadruples of points over an infinite metric continuum, discretizing this domain is a fundamental challenge in metric graph theory. In this paper, we study the continuous Gromov hyperbolicity constant via the Gromov product. Our main result, the Quarter-Point Theorem, establishes that the search space for quadruples $(x,y,z,w)$ attaining the exact hyperbolicity constant $\delta^*(G)$ can be reduced from the entire metric graph to a finite or discrete subset $J'(G)$, comprising solely the vertices, edge midpoints, and quarter-points of $G$. This result holds for every graph, finite or infinite. This theoretical result provides a considerably simplified combinatorial framework for calculating hyperbolicity in complex networks and metric graphs. Furthermore, we exploit this discretization to significantly improve classical results by establishing and structural bounds relating the global hyperbolicity constant $\delta^*(G)$ with its restrictions to vertices ($\delta_V^*(G)$) and midpoints ($\delta_J^*(G)$).

We show that microlocal Arthur packets can be reconstructed from an otherwise underdetermined system of linear equations arising from Kashiwara's local index theorem. The key is a geometric rigidity property of Arthur orbits which, together with Fourier symmetry, uniquely determines the microlocal multiplicities defining these packets. This realizes the associated Arthur distribution as the unique normalized element in the intersection of two filtered spaces of standard stable distributions related by Fourier transform. Under standard enhanced local Langlands hypotheses and Aubert compatibility, this gives an alternative representation-theoretic definition of microlocal Arthur packets for general connected reductive $p$-adic groups that does not use twisted endoscopy and ensures stability through the stability of tempered packet distributions. In the unipotent setting, we use this characterization to prove agreement between endoscopic and microlocal Arthur packets for split $\mathrm{SO}_{2n+1}$ when the residue characteristic is at least $6n+4$. We further use this characterization to compute all unipotent Arthur packets with unramified real infinitesimal character for inner-to-split adjoint exceptional $p$-adic groups. In this exceptional setting, we prove the microlocal weak Arthur packet conjecture and give counterexamples to the microlocal analogues of the least- and greatest-parameter conjectures.

A digital twin is a virtual representation of a physical system, continuously updated with data from the physical system, enabling dynamic information exchange and decision-making. This bidirectional interaction, which distinguishes digital twins from conventional simulation models, introduces new challenges to simulation-based optimization, control, and learning. This paper reviews recent developments in decision-making methodologies motivated by digital twins. We focus on three interconnected areas: simulation optimization, sequential decision making, and risk-aware decision making. For simulation optimization, we discuss data-driven simulation optimization under input uncertainty, streaming input data, decision-dependent observations, and limited computational and data-collection resources. For sequential decision making, we review the connections among digital twins, stochastic optimal control, and reinforcement learning, with emphasis on Bayesian approaches that update models and policies as new information becomes available. For risk-aware decision making, we discuss distributionally robust and Bayesian risk formulations as alternative approaches for addressing model uncertainty. Throughout the paper, we emphasize the common problem of making decisions while the underlying model is being updated. We conclude by identifying research challenges and opportunities.

We study the computational complexity of graph distortion distances, a family of graph metrics measuring how vertex distances differ between two graphs. Formally, these distances are defined in terms of standard matrix norms on the difference of distance matrices under an optimal vertex alignment. The resulting metrics have natural interpretations, corresponding to three ways of measuring the distortion of shortest-path distances: the maximum distortion among all vertex pairs (entrywise maximum norm), global aggregation (entrywise $p$-norms), and local aggregation (operator $p$-norms) of distortions. For maximum distortion, we show that approximation below a factor of two is NP-hard even for highly restricted trees. Due to the close connection to the Gromov-Hausdorff distance, we obtain a similar hardness result for the latter. In both cases, we show essentially optimal additive inapproximability. On the algorithmic side, we give a subcubic-time algorithm for optimal embeddings of connected graphs into paths, based on a structural decomposition along a diametral path. Additionally, we give a simple approximation algorithm when one input graph has a "relatively path-like" metric structure, i.e. has bounded pathlength. For local and global distortion, we derive various hardness results even for restricted inputs. Making use of a close connection between the global distortion distances and Gromov-Wasserstein distances, we give the first formal proof of NP-hardness of the latter, which has been claimed in the literature many times but has, to the best of our knowledge, never been correctly proven.

For $H^1(\mathbb{R}^2)$-orthonormal families $\{u_n\}_{n=1}^N$ with one-body density $\rho=\sum_{n=1}^N |u_n|^2$, we prove the whole-space endpoint Moser–Trudinger inequality $$ \int_{\mathbb{R}^2}\frac{e^{4\pi\rho(x)}-1-4\pi\rho(x)}{\rho(x)}\,dx \le C\left(N-1+\int_{\mathbb{R}^2}\rho(x)\,dx\right), $$ where the integrand on the left-hand side is defined to be zero on $\{x: \rho(x)=0\}$. We develop a heat-semigroup refinement of Rumin's density method to treat both the endpoint coefficient $4\pi$ and the whole-space setting. Our inequality is sharp in two respects: the optimality of the coefficient $4\pi$ and the linear growth in the rank $N$. Moreover, through a semiclassical analysis of the density generated by normalized Dirichlet eigenfunctions, we establish the collective asymptotic behaviors of the quantities appearing in the inequality in various regimes. These semiclassical behaviors show that the rank-dependent term $N-1$ on the right-hand side is indispensable.

Many dense matrices in scientific computing have hierarchically low-rank off-diagonal structure but are accessible only through routines that apply the matrix and, in general, its transpose or adjoint to vectors, that is, through matrix–vector products (matvecs). We present a fully matrix-free algorithm that compresses such a matrix into Hierarchically Semi-Separable (HSS) form and factorizes it simultaneously with the off-diagonal ranks chosen adaptively from a prescribed tolerance. The method uses $\mathcal{O}(k)$ matvecs with each operator, where $k$ is the largest off-diagonal rank found. Building on fixed-rank block nullification, it samples the matrix in small random batches and enlarges each local basis until a randomized error estimate falls below the tolerance. Unitary nullspace updates give $\mathcal{O}(k^3)$ local dense work, the same asymptotic cost as when the ranks are known in advance. Because compression and elimination proceed together, no separate factorization step is required. Excluding the matvecs, the factorization uses $\mathcal{O}(N)$ arithmetic and storage when the off-diagonal ranks are bounded, and its inverse is applied in $\mathcal{O}(N)$ time. Experiments on Poisson and Helmholtz Schur complements, Matérn covariance matrices, and a complex Lippmann–Schwinger equation demonstrate the factorization as both a direct solver and a preconditioner on problems with up to millions of unknowns.

We consider several families of sequences of Baire-one self-maps of $[0,1]$ investigated by A. Alikhani-Koopaei. They form a chain of inclusions where each space is closed and nowhere dense in the next one. We prove that each space is lower $2/3$-porous in the next one. If one modifies these families where sequences of Baire-one functions are real-valued uniformly bounded, then they are Banach spaces and each space of the respective chain is c-porous in the next one.

An Orlicz ball is the set of points where the sum of a fixed even convex function of the coordinates does not exceed a given radius, here proportional to the dimension. For a broad class of such functions, including non-homogeneous ones, we find the exponential rate of the intrinsic volumes whose order is proportional to the dimension and express it by a two-parameter variational formula involving only one-dimensional integrals. As a function of the order-to-dimension ratio, this rate is concave and continuous on the closed unit interval, and the convergence is uniform in the order. This proves the exponential-profile conjecture of Kabluchko, Marynych, and Prochno for these bodies and recovers their result for $\ell_p$ balls with $p>1$. The proof relies on the theory of large deviations. An exact formula for the Laplace transform of the intrinsic volumes in the radius, extending theirs for $\ell_p$ balls, plays the role of a moment generating function, and a Laplace method gives its exponential rate. An exponential change of measure, analogous to that used in Cramér's theorem, then recovers the intrinsic volumes. The key step is a lower bound on the tilted density at its mean, obtained from log-concavity rather than a law of large numbers.

We study the possible dimensions of solution sets in linearly perturbed convex optimization problems and their behavior under convex duality. The dimensions of solution sets and inverse solution sets capture two complementary forms of nonuniqueness: that of optimal solutions for a given perturbation, and that of perturbations for which a given point is optimal. Although these two families are linked by Fenchel conjugacy, we show that their collections of dimensions can be prescribed independently for finite supercoercive convex functions. As a geometric consequence, we prove that the facial dimension signatures of a convex body and its polar can be prescribed independently. The connection between these two realization results is obtained by homogenizing the epigraphs of a function and its conjugate. In contrast, under simultaneous facial exposedness, we establish a dimensional obstruction relating the smallest positive proper face dimensions of the body and its polar. These results resolve two open questions posed by Roshchina and Tunçel.

We study vector-valued local minimizers of the borderline double phase functional whose integrand is $|D\mathbf{w}|^p+a(x)|D\mathbf{w}|^p\log^\gamma(e+|D\mathbf{w}|)$, where $p>1$, $\gamma>0$ and the coefficient $a(\cdot)$ is nonnegative and bounded. If the modulus of continuity $\omega(\cdot)$ of $a(\cdot)$ satisfies $\omega(r)\log^\gamma(1/r)\to0$ as $r\to0$, then minimizers are locally Hölder continuous for every exponent less than one. If $a(\cdot)$ is Hölder continuous, then the gradients of minimizers are locally Hölder continuous. In the vector-valued setting these results were previously available only for $\gamma=1$. The proof combines a comparison with frozen functionals, whose regularity theory applies uniformly in the frozen coefficient, with the higher integrability of minimizers.

We study the number of improving steps that local search may require for $(2,2,r)$-MCA (Maximum Constraint Assignment), a bounded-occurrence version of MAX-CSP with valued constraints in which every constraint involves at most two variables, every variable occurs in at most two constraints, and every variable takes one of $r$ values. We consider the flip neighborhood, where two assignments are neighbors if they differ in the value of exactly one variable; moving to such a neighbor is called a flip. Local search repeatedly performs improving flips, that is, flips that strictly increase the objective value, until a local optimum is reached. Our main result is that $(2,2,5)$-MCA has the all-exp property: there is an infinite family of instances with initial assignments for which every sequence of improving flips ending in a local optimum has length exponential in the number of variables. Thus, on these instances, local search requires exponentially many steps regardless of which improving flip is chosen at each step. This is the first all-exp result for $(2,2,r)$-MCA for any value of $r$, and our result extends to every $r\ge 5$. In contrast, for $(2,2,2)$-MCA every sequence of improving flips has length at most quadratic in the number of variables~\cite{Kaz2020,KCJ2020}. We give a short, direct proof of this result.

We establish quantitative weak convergence for the joint terminal law of the solution to a stochastic wave equation and the normalized error of its exponential Euler approximation. For multiplicative trace-class noise and spatial regularity $\alpha\in(1,2]$, the normalized error converges to its limiting distribution at rate $O(m^{-(\alpha-1)})$, where $m$ is the number of time steps. This estimate identifies the leading weak bias and yields a first-order weak-error expansion with remainder $O(m^{-\alpha})$. Moreover, by canceling the leading weak-error term, we construct a Richardson extrapolation scheme with weak convergence order $\alpha$ for $\alpha\in(1,2]$ and show that this rate is sharp. To overcome the lack of parabolic smoothing and control the remainders amplified by normalization, we combine dimension-uniform regularity estimates for the joint Kolmogorov equation with a conditional weak expansion and backward transport of coefficient directions.

Variational inequalities (VIs) capture a broad class of optimization problems central to modern machine learning, including min-max optimization and equilibrium computation in multiplayer games. Traditionally, algorithms for solving VIs, such as Extragradient (EG) and Past Extragradient (PEG), have been analyzed under the standard $L$-Lipschitz assumption. However, this condition is often overly restrictive, and recent works have instead turned to the more relaxed $(L_0,L_1)$-Lipschitz setting. Existing guarantees for $(L_0,L_1)$-Lipschitz VIs require a quasi-sharpness condition [Vankov et al., 2024], while monotone guarantees for EG cover only unconstrained root finding [Choudhury and Loizou, 2025]. In this work, we provide the first convergence guarantees for EG and PEG for monotone $(L_0,L_1)$-Lipschitz VIs. We also improve upon prior convergence guarantees for strongly monotone VIs by supporting larger stepsizes in the constrained setting and eliminating exponential dependence on the initial distance from the solution for convergence rates in the unconstrained setting. Lastly, we propose line-search variants for when Lipschitz parameters are unknown. Numerical experiments complement our theoretical results and demonstrate the practical efficacy of our methods.

Borzellino and Brunsden introduced a general notion of suborbifold and proved that a suborbifold is the image of a complete orbifold embedding if and only if it is both saturated and split. Their definitions and proofs are formulated in the classical language of orbifold charts. In this paper we translate these notions into the language of proper étale Lie groupoids.We introduce the groupoid analogues of saturation and splitting, which we call orbit saturation and stabilizer splitting, and prove that they coincide with the chart-based conditions. Our main theorem states that a subgroupoid is the image of a complete orbifold embedding if and only if it is both orbit saturated and stabilizer split. We then prove that these notions are Morita invariant: if two orbifold groupoids are Morita equivalent, then their orbit saturated and stabilizer split subgroupoids correspond naturally to each other. This solves an open problem raised by Mestre and Weilandt and explains why the embedding definition of Cho–Hong–Shin fails to be Morita invariant. Along the way we provide detailed examples, including the diagonal embedding, the non-split example of Borzellino and Brunsden, and explicit Morita equivalences.

We study a stochastic optimization problem in which the random vector follows a semiparametric elliptical distribution with unknown location and shape parameters and an unspecified radial distribution. Assuming that the decision maker only has access to samples of the random vector, we develop a framework to jointly address parameter uncertainty and distributional ambiguity arising from this composite structure. To update the posterior distribution of the parameters, a key challenge is that the density of an elliptical distribution depends on the unknown radial distribution, and hence conventional likelihood-based Bayesian inference is infeasible. To overcome this challenge, we propose a novel estimating-equation-based Gibbs inference framework that updates posterior distributions of model parameters solely through estimating equations. For elliptical distributions, we construct a pair of estimating equations for the location and shape parameters that are invariant to the radial component, which enables posterior learning without specifying the density. For the distributional uncertainty, we construct an ambiguity set for the radial distribution by transforming observed samples into pseudo-samples of the radial variable. Combining the above components, we formulate a composite Bayesian risk–distributionally robust optimization model, in which the inner problem captures worst-case radial distributions within the ambiguity set, and the outer problem evaluates the decision risk induced by the parameter uncertainty through a user-specified risk measure. We further develop a cutting-plane algorithm to solve the resulting optimization problem. Numerical experiments demonstrate that the proposed approach provides substantial performance improvements over benchmark methods, including sample average approximation and distributionally robust optimization.

Let $G$ be a finite abelian group of order $n$ and $X$ a finite $G$-set. We give a necessary and sufficient condition on $G$ for any $X$ to be determined up to isomorphism by certain combinatorial data. Specifically, if $M_H^X$ is the number of $H$-orbits in $X$, and $a_d^X$ is the number of $G$-orbits in $X$ of cardinality $d$, we characterize when equalities $M_H^{X_1} = M_H^{X_2}$ and $a_d^{X_1}=a_d^{X_2}$ for all $H \leq G$ and $d|n$, respectively, force an isomorphism $X_1 \simeq X_2$. Our main result is that this occurs if and only if $G$ is cyclic or $G$ is the direct sum of a finite cyclic group and a cyclic group of prime order. We conclude with an arithmetic application to $L$-function factorization.

Let $F$ be a nonarchimedean local field of characteristic zero and residue characteristic $p$. Let $E$ be a quadratic field extension of $F$. Let $D$ be the unique quaternion division algebra defined over $F$. Let $G=\GL_{2}(E)$ and $H=D^\times$. This paper studies the modulo distinction problem for the pair $(G,H)$ with $\ell\neq p$, which gives a complete classification of $H$-distinguished representations of $G$, including the cases $\ell=2$ and $p=2$. In terms of type theory, we give a complete classification of $H$-distinguished supercuspidal representations when $p$ is odd.

Can the Alexander-Orbach conjecture hold on hierarchical lattices? We construct Gromov-Hausdorff-Prokhorov scaling limits of critical percolation clusters on hierarchical lattices in the chemical metric and determine the typical spectral dimensions of the associated diffusions. These dimensions are dense in $(1,2)$; however, outside a narrow region in dimension space, we prove that the Alexander-Orbach conjecture fails on every hierarchical lattice under consideration.

We establish the existence and uniqueness, up to translation, of monotone viscous-dispersive shock profiles for the one-dimensional Navier-Stokes-Fourier-Korteweg (NSFK) system, provided the two constant end states satisfy the Rankine-Hugoniot and Lax entropy conditions and the shock strength is small. Treating the small shock strength as a singular perturbation parameter, we construct the profiles via Fenichel's geometric singular perturbation theory. We further provide pointwise estimates for the profiles, which are used in the nonlinear stability analysis of generic Riemann solutions to the NSFK system in a companion work.