Papers

Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize chromatic polynomials of graphs. In 1995, Stanley introduced these symmetric functions and conjectured that they are Schur positive for claw-free graphs. We give examples of a line graphs, which are thus claw-free, whose chromatic symmetric function have a negative coefficient in its Schur expansion. We also give a counterexample to the 2018 conjecture of Monical that Schur positive chromatic symmetric functions have saturated Newton polytope when expanded in any finite number of variables. Both of these examples were found using ChatGPT-5.6 Sol Pro.

This paper studies the statistical properties of the dynamical system generated by a Hecke correspondence on the modular curve over $\mathbb{C}$ and, for every prime number $p$, over $\mathbb{C}_p$. Over $\mathbb{C}$, it proves that the equidistribution to the hyperbolic measure established by Clozel and Otal occurs at an exponential rate. Moreover, it determines the sharp rate, assuming an affirmative solution to the Ramanujan-Petersson conjecture. Over $\mathbb{C}_p$, two distinct types of behavior arise. In the first, every orbit converges towards the Gauss point in the Berkovich affine line, and this paper establishes the sharp exponential convergence rate. In the second, a form of unique ergodicity holds on each orbit closure, and this paper establishes a spectral gap property and, as a consequence, a central limit theorem. This complements the central limit theorem proved by Cantat and Le Borgne over $\mathbb{C}$. Finally, the paper extends and strengthens the results of Goren and Kassaei on the associated random walks.

In this article, we give several families of new examples of isoparametric foliations of $S^n\times S^n$; establish some basic properties of isoparametric foliations of $S^n\times S^n$ of codimension $1$, especially any of them can be realized as level sets of the restriction of a bi-homogeneous polynomial of bi-degree $(1,1),(2,2),(3,3),(4,4)$ or $(6,6)$ to $S^n\times S^n$; and classify bi-homogeneous isoparametric polynomials of bi-degree $(1,1)$ and $(2,2)$ and isoparametric foliations on $S^n\times S^n$ with $n$ even and not equal to $4$.

For a fixed integer $r\ge 2$, we prove that the compressed primitivity problem in the free group $F_r=F(x_1,\dots,x_r)$ is decidable in non-deterministic polynomial time. That is, for a straight-line program $\mathcal A$ over $\{x_1,\dots,x_r\}^{\pm1}$ representing an element $g\in F_r$, the problem of deciding whether $g$ is primitive in $F_r$ belongs to $\mathsf{NP}$, with input measured by the size of $\mathcal A$. For $r=2$, we prove that this problem is decidable in deterministic polynomial time. We also show that, in every fixed rank $r\ge 2$, automorphic minimality of the conjugacy class of a compressed word in $F_r$ is decidable in deterministic polynomial time.

Let $G$ be a graph with adjacency matrix $A$ and degree matrix $D$. Applying $A$ and $D$ repeatedly to the all-ones vector produces a space $M_G$ of functions on the vertices, and these functions are constant on the orbits of the automorphism group. We call $G$ AD-observable if $M_G$ contains the indicator vector of every orbit, and AD-separable if the functions in $M_G$ separate the orbits. Forests are AD-observable, by McKay's leaf-pruning argument. Our main result is that for every cograph, $M_G$ is spanned by the indicator vectors of the cells of the stable partition of color refinement. Hence a cograph is AD-observable, or equivalently AD-separable, exactly when color refinement finds its orbits, and we describe these cographs by their cotrees. We also show that two cographs with the same number of vertices and the same homomorphism counts from all caterpillars have the same counts from all trees. The main result does not extend to distance-hereditary graphs: we give two such graphs on nine vertices that are AD-separable but not AD-observable.

Let $G=F/\langle\langle r\rangle\rangle$ be a one-relator group with the relator $r\in [F,F]$ or $r=[u,v] ~(u,v\in F)$, where $F$ is a finitely generated free group. Baumslag asked whether $G$ is Hopfian, residually finite or automatic. In the case of $r\in[F,F]$, a negative answer to the residual finiteness and automaticity has already been obtained by a result of Olshanskii. In this note, we construct a family of one-relator groups $$G_m=\left\langle a,t\ \middle|\ [t,a[a,t]^{-m}]\right\rangle,$$ whose relators are commutators, each of which has a Baumslag-Solitar subgroup as a retract. These groups provide negative answers to these three questions in both cases.

A (3, 6)-fullerene is a cubic planar graph whose faces all have 3 or 6 sides. We give an exact enumeration of (3, 6)-fullerenes with V vertices and each of five possible symmetry types: *332, 332, 2*2, *222, and 222, in orbifold notation. We use this enumeration, together with some constructions, to resolve three conjectures about the existence of (3, 6)-fullerenes with 2*2$, *222, and 222 symmetry for given numbers of vertices.

For $G=SU(N)$ at finite and large $N$, with a totally symmetric representation, we realize the homological block $F_K$ for a knot complement $S^3 \backslash K$, given in the form of the inverted Habiro series, as a half-index of a 3d $\mathcal{N}=2$ theory $T[M_3]$ by studying some examples, which we expect to extend to general knots. From the half-index expression, it is also possible to realize the colored HOMFLY-PT polynomial by taking a certain set of poles. Through the half-index realization, we describe a method for obtaining the $G=SU(N)$ homological block and its $a$-deformed version for $S^3 \backslash K$ from a Habiro series expression for the colored HOMFLY-PT polynomial. We also discuss some properties of partition functions for arbitrary $N$.

Persistent entropy provides a compact summary of persistence diagrams, but discards geometric information inherent to the data. This limitation creates a gap between scalar summaries, which are computationally efficient but geometrically coarse, and directional topological transforms, which are expressive but high-dimensional and computationally demanding. In this work, we introduce the Persistent Entropy Transform (PET), a novel directional topological descriptor which can be interpreted as an entropy-based compression of directional topological transforms. We establish basic theoretical properties of PET. In particular, we prove translation invariance, scale invariance under positive uniform scalings, and orthogonal equivariance. Empirically, we use synthetic shapes to assess directional sensitivity, consistency with rotational equivariance, robustness under controlled perturbations, and dependence on directional sampling density. In addition, we test the novel tool on two real time-series benchmark datasets, the TwoLead- ECG and the MIT-BIH, to provide a proof of concept showing that PET embeddings can be used succesfully as compact feature vectors on real time-series benchmarks. These experiments support and validate PET as a compact and computationally tractable descriptor.

Fat minors are the metric analog of graph minors that are tailored to the analysis of metric (edge-weighted) graphs and, more generally, metric spaces having a suitable notion of shortest paths. Despite a large interest in this notion, not much is known about the structure of metric graphs excluding a fixed fat minor. We prove that if a metric graph $G$ excludes a fixed graph $H$ as a $\delta$-fat minor, for some $\delta>0$, then $G$ enjoys the metric analog of flatness (aka uniform quasi-wideness) - a structural property from the field of Sparsity. In essence, our flatness result says that for any $\alpha\geq \beta$ large enough compared to $\delta$, in every large enough set $A$ in $G$ one can find a sizable subset $B$ that becomes $\alpha$-scattered after removing a bounded number of balls of radius $\beta$. We call this property drill-flatness. Notably, the proof only relies on excluding shallow fat minors: every branch set has radius at most $2\alpha$. As a corollary, we prove that metric graphs that exclude a fixed $\delta$-fat minor have bounded $\varepsilon$-scatter dimension if we consider only $\varepsilon$-scatters at distances large enough compared to $\delta$. By combining this with the results of Abbasi et al. [FOCS 2023], we infer that the $k$-Center problem on instances excluding $H$ as a $\delta$-fat minor admits an approximation algorithm that finds a solution of cost at most $(1+\varepsilon)\cdot\mathsf{OPT}+{\cal O}(\delta/\varepsilon^2)$ in time ${\cal O}_{H,\varepsilon}(n^{{\cal O}(1)})$. This is one of the first algorithmic results for general fat-minor-free metrics. We also study drill-flatness in hereditary classes of (unweighted) graphs, where we obtain a characterization equating drill-flatness with excluding shallow induced minors. This is an induced analog of the equivalence between flatness and nowhere denseness - one of central results of Sparsity.

We propose the first algorithm to morph geometric graphs on hyperbolic surfaces. It is based on a generalization of Tutte's spring embedding theorem on essentially 3-vertex-connected graphs. We describe the algorithms in detail and show experiments with triangulations and graphs on a hyperbolic surface of genus two, the Bolza surface, and a hyperbolic surface of genus three, the Klein quartic.

We construct a new reduction process which takes a $(132,213)$-avoiding permutation to a shorter one that is cyclic if and only if the original was. Iterating it determines whether a given $(132,213)$-avoiding permutation is cyclic. Reversing it gives four moves that build every cyclic $(132,213)$-avoiding permutation, uniquely, from $1$ if $n$ is odd, and $21$ if $n$ is even. Our main application is the first non-trivial lower bound for the growth rate of $\mathcal{C}_n(132,213)$, the cyclic permutations of length $n$ avoiding $132$ and $213$. We also give several other consequences of the reduction, including a bijection between the odd and even size classes and an exact enumeration for those permutations with a restricted number of layers.

This article studies the restricted Hitchin map $h_V$ of a stable rank 2 vector bundle $V$ on a smooth projective curve $C$. This map associates to a trace-free twisted endomorphism $\varphi : V \to V \otimes K_C$ its determinant, which is a quadratic differential. We show that if $V$ is a general wobbly vector bundle, then it has a single nilpotent twisted endomorphism, up to scalars. As a consequence we show that $h_V$ is generically finite and we compute its degree. Furthermore, we show that if $C$ is not hyperelliptic, then the image of $h_V$ contains a quadratic differential with simple zeros. This is equivalent to saying that there is a smooth spectral curve associated to a twisted endomorphism of $V$.

Let $M=\mathbb{H}^n\times\mathbb{S}^m$, where $n\geq 2$, $m\geq 1$, and $N=n+m$. Let $P_k$ be the order-$2k$ GJMS operator, with $1\leq k<N/2$, and assume that $\Lambda_0=\inf\sigma_{L^2(M)}(P_k)>0$. We study$$P_kU-\lambda U=|U|^{q-2}U,\qquad q=\frac{2N}{N-2k},\qquad 0<\lambda\leq\Lambda_0,$$and attainment of the associated critical quotient $S_{\lambda,k}(M)$. Let $S_{N,k}$ be the Euclidean best Sobolev constant. For $0<\lambda<\Lambda_0$, the inequality $S_{\lambda,k}(M)<S_{N,k}$ implies attainment and a nontrivial weak solution. Localized Euclidean extremals establish this inequality when $N\geq4k$, or when $2k+2\leq N<4k$ and $\lambda>\Lambda_{\mathrm{loc}}$, where $\Lambda_{\mathrm{loc}}$ is explicit. If $N\geq2k+2$ and $S_{\Lambda_0,k}(M)<S_{N,k}$, attainment also holds at $\lambda=\Lambda_0$ in the threshold form completion. At the threshold, $L^2$-coercivity fails precisely on the constant spherical eigenspace. We combine cocompactness for its hyperbolic coefficient with a profile decomposition relative to the critical transformations preserving $\mathcal{A}=\mathbb{R}^{n-1}\times{0}$. Under the threshold hypotheses above, the strict Euclidean inequality excludes concentration escaping $\mathcal{A}$ from normalized minimizing sequences. If $r_j^{(J)}$ denotes the remainder after the first $J$ extracted profiles, then$$\lim_{J\to\infty}\limsup_{j\to\infty}|r_j^{(J)}|_{L^q(\mathbb{R}^N)}=0,$$which yields compactness modulo the axis-preserving transformations.

In this paper we prove that a square root of a cyclic normal operator is unitarily equivalent to a block multiplication operator on a vector-valued Lebesgue space with a $2$–normal symbol. In addition, we show that a cyclic operator which admits a cyclic $2$–normal extension is unitarily equivalent to a block multiplication operator on a corresponding vector-valued Hardy space with $2$–normal symbol. We consider the existence of bounded point evaluations in the vector-valued Hardy space setting; as an application, we prove that an operator admitting a cyclic $2$–normal extension has nontrivial invariant subspaces. We also study uniqueness for minimal $n$–normal extensions of operators that have cyclic $2$–normal extensions.

Reliable electric vehicle (EV) charging infrastructure is a cornerstone of sustainable, low-carbon cities, yet urban climate stress such as extreme heat, heavy precipitation, and humidity increasingly raises equipment fault risk and undermines the resilience of urban energy and mobility services. Shifting operation from reactive repair to preventive maintenance depends on accurate, forward-looking fault-risk prediction, a task complicated by the heterogeneous time scales of physical, behavioral, contextual, and historical signals and by forecasting over a multi-week horizon. We develop FGDSE, a feature-governed dynamic stacking ensemble that forms an interpretable decision-support system for climate-resilient charging-asset management. It partitions heterogeneous signals into four feature families, assigns each to a domain expert whose inductive bias matches the data, and adds two deep temporal experts for short-term pulses and long-term degradation; a horizon-wise gating mechanism then learns adaptive weights to forecast daily fault risk over 1 to 30 days. SHAP attribution and an X-learner extend the probabilistic output into causal decision support with post-level treatment effects. On 25 months of data from 13 stations, FGDSE surpasses twelve baselines beyond the ten-day horizon, sustains about 85% macro-recall at 30 days with an AUC decay of only 3.2 points, and reveals a shift of dominance from fault history toward climate stress. It identifies extreme heat as the sole exposure whose causal effect amplifies over time, flagging roughly 30% of posts as heat-sensitive and yielding quantitative thresholds for climate-adaptive maintenance that strengthens urban mobility resilience and sustains low-carbon travel.

KPZ-type extremal fluctuations have recently been proved for several models of random walks in space-time random environments (RWRE) in $1+1$ dimensions. A general moment criterion predicts the spatial scale at which this behavior should occur, but does not by itself guarantee non-trivial fluctuations at that scale. In this paper, we show that the averaging process provides an instance in which this criterion is not sharp: because of a degeneracy in the update mechanism, the actual KPZ limit appears only beyond the predicted scale. The relevant critical contribution arises from the interplay of two distinct fluctuation mechanisms, a phenomenon that appears to be rather special within the RWRE setting. Our proof builds on Dobrushin-type local-time limit theorems for zero-sum additive functionals, refined estimates for tilted $k$-point motions, and the recent moment-based axiomatic characterization of Cole–Hopf solutions to the one-dimensional KPZ equation. We also identify the behavior on the two sides of the critical scale, thereby sharply separating the subcritical, critical, and supercritical regimes of the model.

Demand response (DR) operation of electrolysis devices is gaining traction to capitalize on volatile electricity markets, but their dynamic operation poses challenges to the durability and lifespan of these systems. Our recently developed multi-scale optimization framework for grid-integrated electrolysis systems studied the impacts of DR and effects on device durability. A major hurdle in that work is tractably scaling the model to include participation in high-frequency electricity markets and/or longer time horizons. To address this, in this work, we developed an aggregate-informed Benders decomposition method to tractably solve large instances of this problem structure. To illustrate the efficacy, we solve a case study with market participation in the day-ahead market (DAM) and real-time markets (RTM) for up to 40 years to effectively capture the effects of device lifespan change. We compare our decomposition algorithm to both commercial solvers and traditional Benders decomposition. We find that incorporating aggregate subproblem information accelerates convergence, reducing the final optimality gap by up to ~81% relative to traditional Benders on the 40-year horizon instances that otherwise stall near 85%. We additionally apply the algorithm to find that RTM participation offers economic advantages under the higher price volatility.

We study the evolution of a viscous incompressible fluid whose initial vorticity is supported on a smooth open curve. We show that, for $\nu t\ll 1$ and sufficiently small Reynolds number $\Gamma/\nu$, the vorticity is described at leading order by a Lamb–Oseen type vortex concentrated around a curve evolving according to the binormal flow predicted by the localized induction approximation. The solution is written as an explicit leading-order profile plus a lower-order perturbation, which is controlled in a Morrey $\mathcal{M}^{\infty}$ norm. Then, we apply this construction to the Hasimoto soliton. In this case, the estimates are uniform with respect to the torsion parameter, allowing us to consider a large-torsion regime in which the soliton undergoes a macroscopic displacement. We show that the corresponding Navier–Stokes solution contains a localized portion of the kinetic-energy distribution, associated with the binormal velocity, which remains concentrated inside a moving physical region and undergoes an order-one displacement during an admissible time interval.

Let $n\ge3$ and $1<p<n$. We first prove the local trace analogue of the sharp one-bubble critical-point stability theorem of Liu and Zhang~\cite{LiuZhang2025}: near a positive trace-bubble, the Euler–Lagrange residual controls the gradient distance to the normalized trace-bubble manifold with the sharp power $\max\{1,p-1\}$. Then, we establish a Struwe-type compactness theorem for the critical trace functional, which gives the trace counterpart of the Mercuri–Willem decomposition~\cite{MercuriWillem2010}. Combining Struwe-type compactness with the local stability estimate yields a sharp quantitative one-bubble critical-point stability theorem.

We study a network-archaeology problem for a dynamic graph whose latent substrate is a random recursive tree and whose observed topology is enriched by an independent homogeneous Erdős-Rényi shortcut layer. From a single unlabeled snapshot, the goal is to construct a confidence set of deterministic size for the first vertex. Since shortcut edges create cycles, the usual tree-based arguments using Jordan centrality do not apply directly. Our method uses auxiliary subcritical bond percolation to expose a tree-like renormalized structure: retained recursive-tree clusters form heavy-tailed blobs, retained shortcuts connect these blobs through a subcritical rank-one random graph, and large components are leading backbone blobs decorated by subcritical shortcut pieces. Applying Jordan centrality inside the largest auxiliary percolation components then gives a deterministic-size root confidence set for the cyclic observed network.

Cooperative multi-task semantic communication (CMT-SemCom) improves task execution performance by leveraging shared representations. However, as we demonstrated in [1], cooperative multi-tasking can be either constructive or destructive, depending on the semantic relationships among tasks. To ensure constructive cooperation, we propose a semantic-aware task clustering method for CMT-SemCom. We have formulated a sequential multi-stage optimization problem in which semantically aligned tasks are clustered once after a short initial training phase, and then end-to-end (E2E) joint training is conducted exclusively within the discovered groups. Specifically, the problem decomposes into two stages: (i) a semantic clustering problem leveraging hierarchical density-based spatial clustering, and (ii) an intra-cluster E2E CMT-SemCom learning problem. Simulation results demonstrate that the proposed framework effectively mitigates destructive cooperation and negative transfer, yielding accuracy gains compared to unclustered multi-tasking and individual training baselines.

We define doubling conditions for measured Lorentzian length spaces in terms of chronological diamonds, and prove that such conditions are implied by suitable timelike curvature bounds. In particular, for the first time, we relate doubling to curvature bounds in Lorentzian geometry. As a consequence, we obtain compactness results for Lorentzian generalized cones with respect to the Lorentzian Gromov–Hausdorff convergence introduced by Mondino–Sämann.

We work on two conjectures on tensor eigenvalue multiplicities. By using the language of algebraic geometry we give stronger, more refined versions of the conjectures and we prove them in many new cases, notably for all $2\times 2\times\dots\times 2$ tensors. We also establish a connection between the rank of a tensor and the multiplicities of the zero eigenvalue.

We prove Reiner's conjecture for higher Bruhat orders in corank 3: the facial intervals of B(n,n-3) are precisely the spherical intervals, and all other intervals are contractible.

We prove Borsuk-Ulam-type results governed by selection structures. Selection structures extend the matroidal framework for colorful theorems in discrete geometry and include non-matroidal examples such as chessboard complexes. Motivated by Frick and Wellner's Radon-type strengthening of Fan's theorem and its colorful variants, we prove selection-structure analogues whose conclusions are determined by Radon partitions. We also prove a prime-power selection-structure covering version of Volovikov's theorem, governed by Tverberg partitions. We include applications to fair partitions, including selection-structure versions of the ham sandwich and necklace splitting theorems.

We exhibit a topologically mixing continuous-alphabet subshift of the free group $F_2$ whose sofic mean dimension depends on the chosen homomorphic sofic approximation, which gives an answer of Li \cite[Remark 2.7]{Li13}. The system is the hard-core subshift \[ X_{\rm hc}=\{x\in [0,1]^{F_2}:x_gx_{gs}=0\text{ for every }g\in F_2\text{ and }s\in\{a,b\}\}. \] We construct one sofic approximation from finite quotients compatible with the parity homomorphism $F_2\to\Z/2\Z$; all of its action graphs are bipartite and give sofic mean dimension exactly $1/2$. A second approximation is selected from two independent uniform random permutations and gives a value in $[1/5,9/20]$. Consequently a mixing action can have two distinct positive sofic mean dimensions.

The target-free clique conjecture asserts that the supports of stable fixed points of a nondegenerate combinatorial threshold-linear network (CTLN) are exactly its target-free cliques: bidirected cliques for which no outside vertex receives an edge from every clique vertex. We give an explicit six-neuron counterexample. For every sufficiently small $\varepsilon>0$, the CTLN defined by one fixed graph at $\delta=29\varepsilon/25$ is nondegenerate and has a stable fixed point with nonclique full support. Its values of $q=\delta(1-\varepsilon)/\varepsilon$ tend to $29/25$. In the complementary direction, for any CTLN on $n\geq3$ vertices, we prove that in the parameter range \[ q\geq n-2-\frac{n-3}{2}\varepsilon, \] no nonclique support can satisfy both the fixed-point positivity and linear stability conditions. Consequently, throughout this range, every nondegenerate CTLN has exactly its target-free cliques as supports of stable fixed points. In particular, this holds when $\varepsilon\leq\delta/(\delta+n-2)$.

In this paper, we establish a moderate deviation bound for Gaussian maxima and the variance asymptotics of the Sherrington-Kirkpatrick free energy at criticality based on Gaussian convexity. First, let $(X_1,\ldots,X_N)$ be centered Gaussian vector with $\operatorname{Var}(X_i)\leq 1$. Suppose that, for fixed $\alpha\in(0,\sqrt 2)$ and $\kappa>0$, $\mathbb{E}\max_iX_i\geq\alpha\sqrt{\log N}$ and $\mathbb{E}\max_iX_i+\kappa\sqrt{\log N}\leq\sqrt{2\log N}$. We prove that $$ \mathbb{P}\left(\max_iX_i\geq \mathbb{E}\max_iX_i+\kappa\sqrt{\log N}\right) \leq N^{-\kappa^2/(2-\alpha^2)+o(1)}. $$ This answers a question of Ding, Eldan and Zhai. The exponent is sharp, as witnessed by an equicorrelated Gaussian field. Second, for the Sherrington–Kirkpatrick model at the critical inverse temperature $\beta_c=1/\sqrt2$, we prove $$ \operatorname{Var}\bigl(F_N(\beta_c)\bigr)=\frac16\log N+O(1). $$ Our argument provides the variance asymptotics at the critical temperature from an entropy perspective, via a route distinct from that of Du and Huang. For the upper bound, we express the variance as an entropy under exponential tilting and identify this entropy with the Kullback–Leibler divergence of a Gaussian synchronization model. Its derivative is then bounded using the I-MMSE formula, information percolation, and estimates for the susceptibility of the critical Erdős–R'enyi random graph. For the lower bound, we combine Gaussian convexity applied at the replica parameter with an estimate for inverse moments on the sphere and an identity relating GOE eigenvalue densities in consecutive dimensions.

This paper investigates inverse two-phase Stefan-type problems for parabolic heat equations with unknown time-dependent source and reaction coefficients. Suitable transformations reduce the original free-boundary problems to equivalent equations on fixed spatial domains, and Fourier spectral expansions are used to derive reconstruction formulas for the unknown coefficients. In the first formulation, the source coefficients are identified from integral, pointwise, and nonlocal additional data, leading to Volterra integral equations. In the second formulation, an exponential transformation is applied to recover the reaction coefficients. A principal advantage of the model with two moving boundaries is that each boundary provides its own Stefan condition which sufficient to determine the two unknown time-dependent coefficients without additional overdetermination conditions. Under appropriate assumptions the existence, uniqueness, boundedness and regularity of weak and strong solutions are established. Illustrative examples confirm the applicability of the reconstruction procedure and show that the coefficients remain stable under the noisy data. The proposed approach provides a rigorous base for identifying time-dependent thermal parameters in two-phase phase-change processes.