Papers

In 1955, L. E. Payne conjectured that for a planar clamped plate, each buckling eigenvalue is at least as large as the Dirichlet Laplacian eigenvalue of the same domain with index shifted by one. We prove this conjecture for every odd buckling index.

We study a vertex-labeling analogue of the $1$-$2$-$3$ problem and its list version. For a labeling $\ell:V(G)\to\mathbb N$, let $S_\ell(v)=\sum_{w\in N(v)}\ell(w)$. The additive number $\eta(G)$ is the least $k$ for which there exists $\ell:V(G)\to[k]$ such that $S_\ell(u)\ne S_\ell(v)$ for every $uv\in E(G)$, while the list additive number $\eta_\ell(G)$ is the least $k$ such that the same condition can be satisfied from every assignment of $k$-element lists $L(v)\subset\mathbb N$ with $\ell(v)\in L(v)$. We show that for every $k\ge2$, there is a graph $G$ with $\eta(G)=1$ and $\eta_\ell(G)\ge k$. The separation persists at the minimum possible ordinary value for positive-degree regular graphs: there is a regular graph $H$ with $\eta(H)=2$ and $\eta_\ell(H)\ge k$. We also determine a sharp lower bound for $\eta(G)$ in terms of the order and minimum degree of $G$, and show that the unbounded list gap persists at asymptotically extremal density. Finally, for every fixed $k\ge2$, it is NP-hard to distinguish $\eta(G)=2$ from $\eta(G)>k$, even on asymptotically extremal dense graphs. Consequently, $\eta(G)$ admits no polynomial-time constant-factor approximation unless $\mathrm P=\mathrm{NP}$. Together, these results reveal a robust gap phenomenon: the separation between ordinary and list additive labeling persists at the smallest possible ordinary values and even under asymptotically extremal density, while the ordinary parameter itself remains hard to approximate.

We study some inverse problems in a frequency-averaged model for quantitative photoacoustic tomography, in which the optical parameters are modeled by a finite number of spectral functions and spatial coefficients, and the measurements are given by internal observables averaged over a spectral bandwidth. Using multiple measurements generated by suitable complex geometrical optics solutions, we establish uniqueness and Lipschitz-type stability for the associated linearized inverse problem for diffusion and absorption recovery. This linearized stability further yields local Hölder-type stability for the nonlinear inverse problem via an abstract linear-to-nonlinear argument.

Let $\Omega\subset\mathbb Z^d$ be finite and let $S\subset\mathbb T^d$ be measurable. We study the discrete Fourier concentration operator \[ T_{\Omega,S}=P_\Omega B_SP_\Omega, \] which describes simultaneous localization to a finite set of spatial indices and a prescribed spectral region. For independent exponents $\gamma,\eta\in(0,1]$, we introduce translation seminorms for $\Omega$ and $S$ and use them to bound the trace defect \[ \operatorname{tr}(T_{\Omega,S}-T_{\Omega,S}^2). \] The proof combines an exact trace-defect identity with dyadic Fourier estimates, leading to three regimes: when $\gamma=\eta$, the contributions from the dyadic scales are of the same order, and their sum produces a logarithmic factor, while for $\gamma\ne\eta$ the sum is controlled by one end of the scale range. For discretizations $$ \Omega_R=(RF)\cap\mathbb Z^d $$ of a bounded measurable set $F\subset\mathbb R^d$ satisfying an upper Minkowski-neighborhood estimate with exponent $\gamma$, we obtain trace-defect bounds of order $R^{d-\gamma}\log R$ in the critical case and $R^{d-\min\{\gamma,\eta\}}$ off the critical line. These estimates yield quantitative eigenvalue-counting and plunge-region bounds and recover the discrete Landau asymptotic. For a box model at $\gamma=\eta=1$, the logarithmic factor is sharp.

Let $X$ be a hyperbolic Riemann surface and let $\mu$ and $\nu$ be laminations on $X$ homotopic to measured foliations with finite Dirichlet integral. We prove that $\mu$ and $\nu$ are filling if and only if there exists a homeomorphism to another Riemann surface $f:X\to Y$ and an integrable holomorphic quadratic differential $q$ on $Y$, unique up to the natural equivalence, such that the push-forward laminations are homotopic to the horizontal and vertical foliations of $q$ respectively. This extends a classical theorem of Gardiner-Masur from closed surfaces to arbitrary surfaces. As well, the dual $\mathbb{R}$-tree interpretation yields the solution of an asymptotic Plateau problem for minimal surfaces in a product of two $\mathbb{R}$-trees. We construct examples such that $f:X\to Y$ is not homotopic to a quasiconformal map, and we present sufficient conditions that ensure it is. We deduce applications to main inequalities for locally quasiconformal maps, harmonic maps between surfaces, and big mapping class groups.

We determine a sharp threshold for the adaptable 2-colorability of a random graph equipped with a uniformly random, not necessarily proper, red/blue coloring of the edges. To accomplish this, we characterize a family of subgraphs along with edge colorings whose inclusion or exclusion determines adaptable $2$-colorability. We further show that above the threshold, a long path with alternating edge colors is formed. We use this path to prove the existence of such a subgraph in the supercritical regime. We then provide and prove symmetric bounds on the critical window for $2$-adaptable colorability. Particularly, we prove bounds matching that of the critical windows for the giant component in the Erd$\H{o}$s-R$\'e$nyi random graph model as well as the satisfiability of a random $2$-SAT instance. Finally, we show that below the critical window, the solution space of adaptable $2$-colorings remains connected, that is one can travel from one adaptable $2$-coloring to another by a sequence of $2$-colorings which differ on $O(\log{n})$ many vertices.

For a DM-type morphism of algebraic stacks $f \colon X \to Y$ with a relative perfect obstruction theory $\mathbb{E} \to \mathbb{L}_{X/Y}$ having a global resolution, we give a formula for the induced virtual pullback in terms of $\mathbb{E}$ and $f$. We apply our formula to prove functoriality properties of the virtual fundamental class in Gromov-Witten theory.

We study regularity, existence, and uniqueness of solutions to vector fields on the two-dimensional torus in Sobolev spaces of dominating mixed smoothness, which measure regularity separately in the two variables. For constant-coefficient vector fields, we obtain families of mixed smoothness estimates describing how the gain or loss of regularity can be distributed between the two variables. In the nonreal case, a gain of one derivative can be distributed between the two directions, whereas for real irrational coefficients the loss is governed by the irrationality measure of the coefficient. We also establish sharpness below the corresponding arithmetic threshold and describe the rational and Liouville obstructions. For real-valued variable coefficients, direct estimates for the periodic Fourier-mode equations yield mixed smoothness regularity results and their isotropic and classical consequences. We then use a periodic conjugation to the averaged constant-coefficient normal form. Although this conjugation introduces an additional loss in the mixed smoothness scale, it preserves isotropic Sobolev orders and therefore transfers the sharp constant-coefficient isotropic theory to the variable-coefficient setting. In the nonresonant regimes, we also obtain existence and uniqueness under the natural zero-mean compatibility condition.

We give a general method for proving Kazhdan's property $(T)$ for quotients $G/K_{[c+1]}$, where $G$ is countable, $K$ is a normal subgroup and $K_{[j]}$ denotes its lower central series. The method combines an affine realization of $G/[K,K]$, a contraction argument, and permanence for nilpotent normal subgroups. We apply this method to several classes of groups by means of homological representations. For the Torelli lower-central series quotients $\operatorname{Mod}(\Sigma_g)/(\mathcal{T}_g)_{[c+1]}$ for $g\ge3$ we show that they have property $(T)$ for every $c\ge1$. We also prove property $(T)$ for $\operatorname{Aut}(F_3)/(\mathrm{IA}_3)_{[c+1]}$ for every $c\ge1$, relating these groups to the tame nilpotent images studied by Lubotzky and Pak. Several other classes of examples are discussed as well. We also prove that the Torelli lower-central series quotients for $c\ge2$ are not residually finite, and thus non-linear.

We study quantitative unique continuation for solutions of the Schrö\-din\-ger equation with complex-valued singular potentials \(V\in L^t\), \(t>d/2\). We obtain a family of scale-invariant \(L^p\to L^q\) Carleman inequalities for the Laplacian and use them to prove the doubling inequalities with explicit dependence on the \(L^t\)-norm of the potential. These estimates are combined with multiscale arguments to obtain propagation of smallness from arbitrary measurable sets of positive measure. As an application of the main results, we prove an upper bound for the BMO norm for the logarithms of Dirichlet-Laplace eigenfunctions in simply connected planar Lipschitz domains.

We simplify Goodwin's construction of a non-compact closed convex hull in a CAT(0) space and show that three points suffice for the example.

We construct a weakly quasilattice-ordered submonoid of a group whose full semigroup $\mathrm{C}^*$-algebra is nuclear, while the monoid does not embed into any amenable group. Hence, Nica amenability, as well as the nuclearity of its semigroup C*-algebra, do not characterize embeddability into an amenable group for group-embeddable monoids.

We study a class of strongly local, regular Dirichlet forms on the standard unit interval. The aim of our work is to record some Hölder regularity properties that have not been noted in the prior literature. In particular, as our main result, we show that for every $\delta \in (0,1]$ there exists a metric measure space $(X,d,\mu)$ equipped with a strongly local, regular Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^2(\mu)$ with the property that $\delta$ is the supremum of $\alpha \in (0,1]$ for which the domain $\mathcal{F}$ of the Dirichlet form contains a non-constant $\alpha$-Hölder continuous function. To the best of our knowledge, such examples were previously known only for $\delta = 1$. In our construction, the value $\delta$ is characterized by the upper Hausdorff dimension of a certain Radon measure that is used to define $(\mathcal{E},\mathcal{F})$.

We prove the Lipschitz regularity of solutions for a wide class of generalized Bernoulli free boundary problems, in vectorial, higher order, and non-minimizing settings. Our method does not rely on notions of viscosity solutions or comparison methods, which allows us to reach the optimal regularity for Bernoulli-type problems associated to elliptic operators which do not satisfy any maximum principle, namely the elasticity, biharmonic and Stokes equation. With a similar method we obtain the optimal Lipschitz regularity for stationary, non-minimizing solutions of the standard Bernoulli problem in two dimensions.

In this work we consider numerical models with random input parameters, where the model output is a spatially distributed random field. We carry out a statistical sensitivity analysis of spatial sets which arise for instance when the model output exceeds a critical threshold. We consider two approaches: (i) a kernel-based sensitivity analysis working with the Hilbert–Schmidt Independence Criterion, and (ii) a function-valued sensitivity analysis working with generalized Sobol' indices, where the model output is the indicator function of the spatial set of interest. We develop efficient estimators for the sensitivity indices, especially for finite-element-based numerical models, where we approximate volume integrals in the sensitivity measures by finite element quadrature, thereby avoiding Monte Carlo sampling over the spatial domain. This improves the state-of-the-art for the kernel-based indices and makes these methods practically accessible for expensive numerical models. For the generalized Sobol' indices we utilize the same finite element quadrature for efficient computation. We present numerical experiments for a hydrogen combustion process, modeled by a coupled system of convection-diffusion-reaction equations in a two-dimensional spatial domain, and ask whether the temperature remains below a critical value in selected regions of the combustion domain. Our results show that in this example the kernel-based and the generalized Sobol' indices give qualitatively similar input importance rankings. Moreover, the ranking depends on the chosen observation window and can flip between different windows. The algorithms for implementing the proposed methods are provided in the supplementary materials and in an open source code repository.

We develop a local-global theory for visibility of lattice points on families of parameterized curves. We introduce a notion of $p$-adic visibility and ask whether a lattice point is globally visible precisely when it is visible at every prime. We prove that this local-global principle holds for a broad class of families whose parametrizations are homogeneous with respect to positive weights. When the points on these curves fill the entire positive integer lattice, we compute the local and global densities of visible points and show that the global density is the product of the local densities. We then consider polynomial families of the form $y=qP(x)$, with $q\in\mathbb{Q}_{>0}$, and show that visibility can be detected prime by prime exactly when $P$ is a monomial. For non-monomial polynomials the local-global principle can fail, but the set of points where it fails has density zero; for separable polynomials we also obtain a quantitative bound for the number of non-visible points, improving the previously known bound. We further consider families whose lattice points lie on a proper lower-dimensional algebraic subset of the ambient space and show that their visibility densities can behave differently from those of the full lattice. Finally, we extend the theory from visibility from the origin to visibility from one lattice point to another and show that the corresponding local-global principle continues to hold for weighted homogeneous families.

Let $1 < p < \infty$. We identify the optimal decay exponent of the $p$-harmonic measure $\omega_p ( B( \xi, t) \cap \partial \mathbb{B}, 0, \mathbb{B}) $ for small $t$, where $\mathbb{B}$ is the unit ball in $\mathbb{R}^n$, $\xi \in \partial \mathbb{B}$ and $B(\xi, t)$ denotes the euclidean ball centered at $\xi$ of radius $t$. As a consequence and, under suitable geometrical restrictions on a domain $\Omega \subset \mathbb{R}^n $, we identify the optimal decay exponent of $\omega_p ( B( \xi, t) \cap \partial \Omega, a, \Omega)$ for small $t$, where $a \in \Omega$ and $\xi \in \partial \Omega$. When restricted to the range $1 < p < n$ we provide new estimates complementing previous results of different authors in the unit ball, the unit disc or the upper half-space.

We study a class of weakly convex optimization problems in which the objective is the sum of a smooth convex term and a weakly convex term that may be nonsmooth. To exploit this structure, we develop a splitting technique based on the alternating direction method of multipliers (ADMM), which decouples the minimization of the two components into tractable subproblems. Because the update associated with the smooth term may require an inner iterative solver, we further linearize this term, yielding a linearized ADMM (LADMM) scheme with an inexpensive one-step update. Under mild conditions, we establish the subsequence convergence of both ADMM and LADMM methods to directional stationary solutions, which are equivalent to critical points and Clarke stationary solutions for our weakly convex problem. Numerical experiments on two low-dimensional test functions and a high-dimensional logarithmic regularized logistic regression model demonstrate that the proposed approaches are computationally efficient and produce solutions of comparable quality to baseline methods.

In this paper, we develop and analyze a fully discrete local discontinuous Galerkin (LDG) method with IMEX-Euler time discretization for a class of multi-dimensional quasilinear stochastic convection-diffusion-type equations driven by multiplicative $\mathcal Q$-Wiener noise. The leading diffusion matrix may depend on the solution as well as the spatial and temporal variables, while the lower-order drift and noise coefficients may depend on both the solution and its gradient. Under a suitable stochastic parabolicity condition, we establish unconditional high-moment stability estimates for the fully discrete scheme in the quasilinear setting. In the semilinear setting, where the leading diffusion matrix is independent of the solution but may vary in space and time, we further prove optimal high-moment strong error estimates of order $\mathcal O(h^{r+1})$ in space and $\mathcal O(k^{1/2})$ in time. A pathwise error estimate is then derived by combining the high-moment error bound with a discrete Kolmogorov argument. Numerical experiments are presented to illustrate the stability and convergence properties of the proposed method.

This is an updated version of a paper by the author which originally appeared in 1997. The original paper was a survey of the closely related fields of taut and Dupin submanifolds of Euclidean space, and this updated version includes many results in the field that have appeared since the publication of the original version. The emphasis is on stating results in their proper context and noting areas for future research, and relatively few proofs are given. The important class of isoparametric hypersurfaces is surveyed in detail, as is the relationship between the two concepts of taut and Dupin. Also included is a brief introduction to submanifold theory in Lie sphere geometry which is needed to state many known results on Dupin submanifolds accurately. The paper concludes with detailed descriptions of the main known classification results for both Dupin and taut submanifolds.

The Lehmer transform of a positive signal is the ratio of its power sums at consecutive orders, read as a function of the order, and it is a strictly increasing analytic curve from the smallest observation to the largest. Composing this curve with a map whose inverse is the Lambert function turns it into a probability law on the real line, the Lehmer-Lambert distribution, which we introduce and study. Its distribution, density and quantile functions are closed-form, its random variates are exact, and the law is invariant under changes of gain and of power-law calibration of the signal. Its modes lie at orders at which the signal changes scale, and for a sample its tails are exponential with rates equal to the two extreme logarithmic gaps of the sample. We derive moments, generating functions and limit theorems, and we investigate applications of the Lehmer-Lambert distribution in the analysis of electroencephalograms of patients with major depressive disorder.

We compute the Fourier expansions of the Kudla-Millson theta lifts of vector-valued Hilbert cusp forms of parallel weight. These expansions are with respect to 0-dimensional cusps of non-compact orthogonal Shimura varieties defined over any totally real number field, and computed using a mixed model of the adelic Weil representation under a partial Fourier transform. As a consequence, we obtain the injectivity of the theta lifts under some additional hypothesis on the variety. Furthermore, we show that the theta lifts are square-integrable harmonic differential forms, and deduce lower bounds for dimensions of cohomology groups of such varieties.

In a wireless network, the initial/random access mechanism (RACH) allows idle/new users to join the network and (possibly) request allocated transmission resources for subsequent traffic. Building on our own previous work, for cell-free user-centric networks, we consider location-dependent random access codebooks such that users in a certain geographic area (location) make use of the corresponding set of random access preambles (codewords). We expand our previous work in two ways: (1) we consider multipath channels with line-of-sight (LoS) propagation within a given radius; (2) we consider two different approaches. The first makes use of Zadoff-Chu (ZC) sequences and GLRT detection to cope with the unknown delay, and it is conceptually similar to the 3GPP 2-step RACH specification (here extended to the cell-free case). The second builds on our previous work on multisource approximate message passing (AMP). For both schemes, we also consider a novel near Maximum-Likelihood approach for localization of the random access users directly from the detected RACH preambles, implicitly using angle of arrival and time difference of arrival information embedded into the LoS components. Simulation results show that the AMP approach achieves generally better performance for random access user detection, while both approaches have similar localization capability with a slight superiority for the frequency-domain scheme.

We study random access (RACH) schemes for cell-free (CF) user-centric networks to handle many geographically distributed users with sporadic traffic and intermittent activity. The RACH must allow the system to: 1) detect preambles sent by the (yet unknown) random access users in the RACH slot; 2) localize them for fast allocation of user-centric radio-unit (RU) clusters. Most prior work uses simplified models, neglecting frame-synchronous but chip-asynchronous transmission, possible line-of-sight (LoS) propagation for certain user-RU pairs, and multipath non-line-of-sight (NLoS) propagation yielding frequency-selective channels. Building on our previous work, we consider location-dependent partitioned random access codebooks where users in a geographic area (location) use the corresponding subset of random access preambles. We present a unified framework for joint detection and localization over a spatially consistent network-wide channel model, incorporating these neglected aspects. We evaluate two schemes: 1) a “legacy” scheme using Zadoff-Chu (ZC) sequences, extending the 3GPP 2-step RACH to the CF case; 2) our multisource approximate message passing (AMP) approach extended to multipath frequency-selective fading. For both schemes, we develop novel approximated GLRT preamble detection and Maximum-Likelihood position estimators with super-resolution refinement, implicitly exploiting received signal strength, angle of arrival, and time-difference of arrival information embedded into LoS components. Numerical results show that the AMP-based scheme achieves superior preamble detection, while both schemes have similar and excellent localization capability.

We define numerical adelic characteristic classes for vector bundles on projective varieties over number fields. The metric data are carried by the tautological quotient line bundle on the associated projective bundle. The construction uses the established intersection theory of adelic line bundles. Higher characteristic classes are multilinear intersection functionals. They are continuous for simultaneously controlled model sequences that are Cauchy for the supremum norm. We construct the resulting tautological numerical intersection algebra. We also prove controlled numerical Bogomolov–Gieseker inequalities. We first treat the case of a curve over $K$, where the numerical inequality is combined with the Deligne pairing and the adelic Hodge index theorem. We then pass to higher dimension, using Moriwaki's model-level dimension induction before taking the controlled Zhang limit. The curve theorem includes equality and uniform-gap criteria.

Let $G$ be a graph. The circumference of $G$, denoted by $cir(G)$, is the length of a longest cycle in $G$, or zero if $G$ is acyclic. In 1993, Broersma, van den Heuvel, Jung, and Veldman conjectured that, for every $t>0$, there is a constant $A=A(t)>0$ such that every 2-connected $t$-tough graph of order $n$ has circumference at least $A\log n$; the conjecture is recorded as Conjecture~2 in the 2006 survey on toughness by Bauer, Broersma, and Schmeichel. In this note, we confirm the conjecture. More precisely, every 2-connected $t$-tough graph $G$ of order $n$ satisfies $cir(G)\ge \lceil \log_k((k-1)n+1)\rceil$, where $k=\lceil 1/t\rceil+2$. The proof combines Win's bounded-degree spanning tree theorem with the theorem of Briański, Joret, Majewski, Micek, Seweryn, and Sharma that the treedepth of a 2-connected graph is at most its circumference.

We develop an explicit operator and coefficient calculus for admissible (g,K)-modules of SU(2,2), where g = sl(4,C) and K = S(U(2) x U(2)). The K-types are indexed by triples (n,k,m). We introduce operators A_delta and B_delta associated with the noncompact roots, together with auxiliary operators Q and R. We establish their commutator relations, the central coefficient relations, and, in the unitary setting, the corresponding adjoint and norm-factor identities. This reduces part of the analysis of (g,K)-modules to the study of scalar coefficients arising from compositions of these operators. The coefficient formula applies to weights of multiplicity one, including all boundary weights. We derive necessary sign restrictions and explicit relations constraining the coefficient space. These coefficients also determine irreducibility. Several families of unitary (g,K)-modules are analyzed according to the minimal value of n+m. For N=0, the construction yields two-parameter families of unitary modules and describes reducibility when certain coefficients vanish. For N>0, the method produces both larger families of K-types and multiplicity-free ladder-type families. The resulting patterns are compatible with the known description of the unitary dual of SU(2,2) due to Knapp and Speh. The paper provides an explicit framework for recovering unitary (g,K)-modules from their K-type structure and suggests a possible approach to other real reductive groups.

We study stationary states for actions of countable discrete groups on unital separable $C^*$-algebras. We prove that the stationary state space associated with an inner action of a subgroup of the unitary group that generates the algebra is a Choquet simplex. We also give a locality criterion covering Bernoulli shifts. The simplex structure yields a canonical decomposition of stationary states into tracial and purely nontracial parts. For inner actions, we characterize extreme stationary states by factoriality of their GNS von Neumann algebras. We further show that a stationary state is tracial if and only if its GNS von Neumann algebra is finite, and that it is purely nontracial if and only if this algebra is of type~$\mathrm{III}$. As applications, for $2\leq d\leq\infty$ the stationary state simplex of $C^*(\mathbb{F}_d)$ has a Poulsen face, while for a nontrivial property~$(T)$ group the stationary state simplex of $C^*(\Gamma)$ is not Poulsen. Finally, we give a sufficient spectral-gap condition for $S_{\mu}(A)$ to be a Bauer simplex and construct a family $(A_d,\Gamma_d,\mu_d)$ satisfying this condition.

Sterboul and Deming gave classical matching-based characterizations of non-Kőnig–Egerváry graphs through flower–posy and blossom-pair configurations. We consider two classical configuration families, denoted \(T\) and \(S\), and introduce a new walk-based family \(J\), based on \(J\)-flowers and \(J\)-posies. Our main result proves that, for every graph \(G\), \[ \SD_T(G)=\SD_S(G)=\SD_J(G). \] Thus the additional flexibility of the \(J\)-framework preserves the set of vertices detected by the classical configurations. The proof is vertex-preserving and passes through strict-Hall structure in traces of \(J\)-posies. As a consequence, every prescribed vertex of a connected matchable strict-Hall graph lies in a rigid \(T\)-posy for a suitable perfect matching, linking the theory naturally with matching-covered graphs.

We introduce the curvature-distortion number, a scale-invariant parameter measuring the least multiplicative spread of positive edge weights required to make a weighted discrete curvature nonnegative everywhere. We develop the theory for Lin–Lu–Yau curvature when the transport distance is the fixed combinatorial graph distance. For trees, the invariant admits an explicit nonlinear fixed-point description, which produces a canonical optimal weight that is unique up to scaling. More importantly, the curvature-distortion number controls the branching topology of the tree: for every finite tree $T$, \[ |B(T)|\le \left\lceil \DN_{\LLY}(T)\right\rceil, \] where $B(T)$ is the set of branch vertices. Thus the amount of weight distortion required to achieve nonnegative curvature imposes a direct quantitative restriction on the topological complexity of the tree. For locally finite infinite trees, finite distortion is classified completely: it occurs precisely for the double ray and for one-ended trees obtained from a finite tree by attaching a single ray. This connection between curvature distortion and tree topology extends naturally to general connected graphs through the subgraph formed by edges lying in no cycle of length $3$, $4$, or $5$. Whenever the curvature-distortion number is finite, this tree-like part is a forest unless the whole graph is a cycle of length at least $6$, and each of its tree components inherits the corresponding distortion and topological bounds. In particular, the tree theory yields complete finite-distortion classifications for graphs of girth at least $6$.