Papers

We prove cutoff with an $O(1)$ window for the continuous-time heat-bath Glauber dynamics of the ferromagnetic $q$-state Potts model on the discrete torus $\Lambda_n=(\mathbb Z/n\mathbb Z)^d$ at sufficiently high temperature. More precisely, for every fixed $d\ge2$ and $q\ge3$, there exists $\beta_0=\beta_0(d,q)>0$ such that, for $0<\beta<\beta_0$, the cutoff occurs in an optimal $O(1)$ window centered at \[ t_\star=t_\star^{(n)}:=\frac{1}{2\mathfrak{r}}\log |\Lambda_n|, \] where $\mathfrak{r}\in(0,1)$ is the exact exponential decay rate of the one-site color bias. Thus, the mixing time is determined up to an additive \(O(1)\). It is characterized by the time at which the macroscopic color-density bias from the monochromatic initial condition enters the scale of equilibrium fluctuations. Moreover, our proof shows that the monochromatic initial condition uniquely maximizes the color bias. This is the first implementation of information percolation to prove cutoff for a non-monotone spin system. On a transitive graph, information percolation naturally gives an upper bound on the mixing time of one half of the grand coupling time which is sharp for the Ising model. However, for many non-monotone systems including the Potts model, this misses additional averaging over the update histories giving an upper bound that is off by a constant multiplicate factor. We develop a new framework that treats the information percolation histories as random polymers and show how to average over them to give the effective bias from the initial condition. This accounts for all the additional mixing, recovering a sharp upper bound and cutoff. We believe this approach will be applicable in a much wider range of settings.

In this paper, we investigate the low-depth circuit complexity of Group Isomorphism in the multiplication (Cayley) table model. We prove the first circuit lower bounds for Group Isomorphism: namely, we show that every family of depth-$2$ Boolean circuits deciding Group Isomorphism requires quasipolynomial-size. (We also extend this to a limited class of depth-3 circuits, namely $\textsf{CNF}$-of-$\textsf{XOR}$s.) We complement this with upper bounds of uniform depth-$2\frac{1}{2}$ circuits of quasipolynomial-size: depth 3 circuits with poly-logarithmic bottom fan-in. A sequence of previous results from 1970-2025 progressively reduced the circuit depth from polynomial to $3\frac{1}{2}$; all of these results relied on the generator-enumerator strategy and, in fact, applied more generally to quasigroups. In contrast, our depth-$2\frac{1}{2}$ construction follows a fundamentally different strategy that exploits structure more specific to groups. We guess a composition series for each group, together with generators for its terms and the isomorphism types of its composition factors. We then inductively verify that the corresponding extensions at each level of the two composition series are compatible. A central part in this approach brings to bear the extensive work on the Short Presentation Conjecture, in tandem with the algorithmic theory of group extensions and cohomology.

We prove that $(1-o(1))$-fidelity $N$-qubit Greenberger–Horne–Zeilinger (GHZ) encoding can be performed in time $O(\log N/N)$ using all-to-all 2-local Hamiltonians with bounded 2-qubit interaction terms. This saturates the theoretical lower bound $\Omega(\log N/N)$, and by rescaling, also saturates the lower bound on signaling time for Hamiltonians with power-law interaction strengths $1/r^\gamma$, $\gamma<d$ in dimension $d$. The protocol uses spin-squeezing dynamics combined with quantum signal processing.

Macroscopic hydrogen transport codes model material interfaces with local thermodynamic equilibrium (LTE), imposing continuity of chemical potential as a per-species constraint. Three assumptions hide in that condition: fast interfacial equilibration, a single exchange pathway between the two sides, and a carrier species known in advance on each side. The literature scrutinises the first, but the other two are the more consequential failures, and neither survives at a metal/molten-salt interface. We replace the constraint with reversible reaction channels at the interface that obey mass action, with detailed balance fixing each ratio of rate constants from the thermodynamic data that already parameterise LTE, and we implement the framework in festim. LTE is recovered as the fast-kinetics limit of a single channel, in both its Sieverts/Sieverts and its Sieverts/Henry form, so the framework generalises LTE and does not compete with it. A Damköhler number delimits validity within a channel, a branching ratio between channels. In a representative nickel/FLiBe system, hydrogen partitions kinetically between molecular and fluoride carriers. The apparent interfacial law then drifts between Sieverts and Henry with loading and salt redox state, and an LTE condition underestimates the steady permeating flux. The measured pressure exponent is set by the branching ratio, not by any fixed property of the salt; a redox sweep at fixed temperature should continuously shift it between 0.5 and 1. With two isotopes, two metal-side species feed five salt-side carriers, and a per-species LTE condition is ill-posed.

In [10] it was suggested that the countable chain condition should be defined in the absence of choice as "every predense set contains a countable predense subset". During his tutorial at the "120 Years of Choice" conference in 2024, Philipp Schlicht remarked that this definition does not have an iteration theorem in $\mathsf{ZF}$. We provide a proof of this claim. Specifically, we show that if every finite support iteration of ccc forcings is again ccc, then the Axiom of Choice for countable families of countable sets must hold. However, we prove that under the assumption of the Principle of Dependent Choice, the finite support iteration of ccc forcings is again ccc. This requires correcting some issues with the definition of properness (and therefore Mekler's definition of ccc) from [1]. We then study Knaster principles and their relations to this version of ccc.

Designing machine learning algorithms for wireless resource management is labour-intensive: the architecture, the loss function and the training recipe are all specified by hand. We demonstrate that this design layer can be surrendered to an autonomous agent in its entirety. We adopt the autoresearch protocol, in which an AI coding agent edits a training script, runs a fixed-budget experiment, and retains or discards the change according to a single immutable metric. We grant the agent authority over the architecture family, the input representation, the output parameterization, the loss function and the task-sampling law, and set it a target chosen for its difficulty: sum-least-percentile-rate power control across a multicell network. The formulation targets cell-edge throughput and is non-convex, non-smooth and strongly NP-hard away from its max-min vertex. Safeguards render the results trustworthy: a hash-pinned evaluator, an enforced inference contract and a pre-registered falsifier per experiment. In eighty-one unattended experiments over twenty-six hours, the agent reached $99.5\%$ of a converged minorization-maximization reference in one fixed-cost inference pass, at roughly $600\times$ lower inference cost, closing $94\%$ of the gap from its first working architecture, with one parameter set serving every network size and percentile target. It recovered provable structure rather than tuned constants: the output parameterization it discovered reproduces the exact max-min-optimal allocation at the minimum percentile, for every value of the trained weights.

We prove optimality of the differentiability bound for isotropic positive definite functions on every even-dimensional sphere. If the even continuation of such a function on the $d$-dimensional sphere is $2k$ times differentiable at zero, then the function has $2k+\lfloor(d-1)/2\rfloor$ continuous interior derivatives; previously, optimality was known only in odd dimensions. We construct a function on the two-dimensional sphere whose first derivative does not exist at the equator and transfer it to all even dimensions by turning bands and spherical montée. The resulting examples are strictly positive definite, have $2k$ but not $2k+2$ derivatives at zero, and are not positive definite in the next dimension.

We prove the multivariable Strong Monodromy Conjecture for the local topological zeta function of arbitrary tuples of nonzero nonunit plane curve germs. The entries may be nonreduced and may have common irreducible factors. We also prove that the non-equivariant local motivic and topological zeta functions have the same polar locus and generic pole orders, using a multivariable extension of the Nicaise–Xu definition of motivic pole order. Our arguments combine twisted Poincaré residues with resolution formulas and specialization to one variable. In arbitrary dimension, we prove that a nonzero iterated residue class on a stratum of a log resolution implies the vanishing of the Bernstein–Sato ideal at the corresponding parameter. In particular, every topological polar hyperplane whose order equals the ambient dimension is contained in this zero locus. For algebraic germs, the topological pole order is at most the motivic pole order, and the two attain maximal order simultaneously.

Esakia duality is the dual equivalence between Heyting algebras and Esakia spaces. However, the traditional proof uses the Prime Ideal Theorem to recover the algebra from its spectrum, a choice principle that is not constructively valid. We build on Townsend's localic Priestley duality to describe a fully constructive, localic Esakia duality. On the algebraic side, we use the recently introduced Heyting frames as the point-free version of Heyting algebras. On the spatial side, Esakia spaces are modeled by two point-free alternatives. First, Esakia locales are defined as a subclass of the ordered Stone locales introduced by Townsend, and it is shown directly that Townsend's equivalence restricts to a duality between Heyting frames and Esakia locales. Second, using the theory of conic frames, in which join-preserving closure operators on frames model localic preorders, we introduce Esakia frames as a frame-theoretic analogue of Esakia spaces. It is shown that Esakia frames are equivalent to Heyting frames and dually equivalent to Esakia locales, and that this factorises the restriction of Townsend's equivalence. This yields a fully constructive, localic Esakia duality.

Let $X$ be a connected smooth complex projective surface. We prove an effective-divisor version of the Arcara–Miles conjecture, together with its strict analogue. For every divisorial Bridgeland stability condition, failure of stability, respectively semistability, of a line bundle or its relevant shift is detected by a natural subobject associated with a non-zero effective Cartier divisor $C$ satisfying $C^2<0$. The proof combines minimal-rank destabilizers, slope Harder–Narasimhan filtrations, the Bogomolov–Gieseker inequality, and an ordered Lorentzian partial-sum estimate that forces the minimal rank to be one. Consequently, strict semistability of a line bundle or its relevant shift is detected by a negative effective divisor, and a numerical semistability condition arising from the twisted deformed Hermitian–Yang–Mills equation is equivalent to stability under all integral scalings.

This is the first of two papers in which we prove the Generalised Cone Conjecture on surfaces.

We present a direct geometric construction which extends the Reshetikhin–Turaev TQFT to circles. We prove that this agrees with the generators-and-relations approach of Bartlett–Douglas–Schommer-Pries–Vicary, thus providing an alternative to the Cerf-theoretic part of their classification of once-extended 3-dimensional TQFTs, and deduce as a result that all once-extended 3-dimensional TQFTs arise via our construction.

We construct a real meromorphic function $F$ on $\mathbb C$ such that $F^{-1}(\{0,1,\infty\})\subset\mathbb R$, while $F$ is not of bounded type in either half-plane. More strongly, for every $a\in\widehat{\mathbb C}\setminus\{0,1,\infty\}$, the $a$-point divisor in either half-plane fails the Blaschke condition. Thus the construction gives a negative answer to the half-plane bounded-type question arising from Nevanlinna's 1925 work. Postcomposition gives the analogous counterexample for any prescribed triple of distinct values in the Riemann sphere. The core construction and proof were generated during an autonomous run of GPT-5.6 Sol Ultra.

Reformulated in terms of the asymptotic rank, a conjecture by Gromov predicts a linear isoperimetric filling inequality in all dimensions greater than or equal to the asymptotic rank of a Hadamard space, in contrast to the Euclidean-type nonlinear behavior below this threshold. We prove the predicted linear inequality for Hadamard spaces with finite asymptotic Nagata dimension and finite asymptotic rank. More precisely, every integral cycle of dimension at or above the asymptotic rank admits a filling whose mass is bounded linearly in the mass of the cycle. Our proof is based on a new self-improvement mechanism for Wenger's sub-Euclidean growth theorem. This approach upgrades the asymptotic rank-one result by Wenger and the recent asymptotic rank-two result by Lang, Stadler, and Urech from exponents arbitrarily close to one to the optimal linear exponent. Moreover, the result extends to arbitrary finite asymptotic ranks.

We consider optimization applications with unknown parameters where the decision maker believes that the optimal value of the nominal problem-the optimization problem they would have solved if the true parameters were known-is unlikely to be large. This belief derives from information that humans have that is not captured in datasets, obtained from domain knowledge and interacting with the physical world. We propose an approach to evaluating policies that provides tighter performance guarantees if the decision maker's belief happens to be correct. Our main result shows that if computing a policy's worst-case performance is a convex program, then the value of human expertise-the maximum improvement in performance guarantees that can be obtained from the belief about the nominal problem-is equal to the minimax gap of a max-min problem. We illustrate our developments in assortment optimization and shortest path problems.

We present a family of second-order, linear, unconditionally stable implicit-explicit (IMEX) methods for the Cahn-Hilliard-Navier-Stokes (CHNS) equations modeling matched-density two-phase flows. The proposed semi-discrete scheme combines extrapolation of the nonlinear terms with an auxiliary-variable formulation of the nonlinear free-energy term and a temporal-curvature regularization controlled by a parameter $\epsilon$. We establish a discrete energy estimate showing unconditional long-time stability of the method for $\theta\in(1/2,1]$ and $\epsilon\geq0$. The resulting scheme requires only linear solves at each time step. Numerical experiments demonstrate approximately second-order temporal convergence and examine mass conservation, energy dissipation, numerical robustness, and several representative interfacial-flow problems, including spinodal decomposition, droplet shape relaxation, two-phase lid-driven cavity flow, and Rayleigh-Taylor instability.

Arnold introduced the strong finite type (SFT) property in 1973 while studying the dimension of power series rings. For several classes of rings, polynomial extension is known to preserve the SFT property, but the general question remained open. We answer it negatively by constructing an SFT ring $R$ such that $R[X]$ is not SFT.

This paper proves the short-time existence of $G_2$-Laplacian flow for closed, asymptotically conical $G_2$-structures on a complete manifold, shows that this solution is unique among closed solutions, and gives a lower bound for the existence time depending on the initial geometry.

We identify the critical boundary operator-norm profile of finite-prime composition operators on the Hardy–Hilbert space \(\mathcal H^2\) of Dirichlet series. For \[ \varphi_{\delta,\boldsymbol\rho}(s) = \frac12+\delta + \delta\sum_{j=1}^d\rho_jp_j^{-s}, \qquad \boldsymbol\rho\in B_d, \] the renormalized positive coefficient operators converge uniformly in operator norm, with \(O(\delta)\) error, to an explicit multivariate weighted Hankel operator \(\mathcal H_{\boldsymbol\rho}\); consequently, \[ 2\delta\|C_{\varphi_{\delta,\boldsymbol\rho}}\|^2 = \|\mathcal H_{\boldsymbol\rho}\| + O(\delta) \] uniformly over \(B_d\). We show that the limiting operator admits the total-degree reduction \[ \mathcal H_{\boldsymbol\rho} \simeq D_{\boldsymbol\rho} H_{R_{\boldsymbol\rho}/2} D_{\boldsymbol\rho}\oplus\mathbf{0}, \] where the diagonal factors are convolution-collision norms of the normalized prime weights. This structure, together with the affine comparison principle of Brevig and Perfekt, yields an explicit concentration inequality for \(\|\mathcal H_{\boldsymbol\rho}\|\), identifies the one-prime configurations as the exact equality cases in the limiting norm estimate, and gives a quantitative deficit away from them. For fixed \(\sigma>\frac12\), we also obtain a second-order expansion of the squared norm and fully finite-dimensional approximations with explicit total-degree and Dirichlet-sum truncation errors. Together, these results show that a single coefficient-operator structure governs the singular boundary profile, the fixed-\(\sigma\) perturbative regime, and certified finite-dimensional approximation.

Decentralized wireless collectives including vehicular swarms, IoT clusters, and edge AI networks require communication protocols that maintain robustness under dynamic topologies and heterogeneous link quality. While Random Linear Network Coding (RLNC) provides algebraic resilience against packet erasures, its performance degrades significantly when peers exhibit diverse channel conditions. This paper presents Adaptive Peer Clustering with Hierarchical RLNC (APC-RLNC), a system that dynamically groups peers by exponentially weighted moving average (EWMA) reliability metrics and applies multi-tier network coding within and across clusters. We formalize the clustering optimization problem, derive closed-form decoding probability bounds for Markov erasure channels, and prove O(sqrt(T)) regret for online reconfiguration under the Follow-the-Regularized-Leader (FTRL) framework. Our implementation includes both a high-fidelity network simulator and a proof-of-concept testbed deployment on Jetson Nano edge devices. Evaluation across diverse scenarios including high-mobility vehicular networks, burst-error channels, and adversarial interference demonstrates 5.2-9.8 percentage-point packet delivery ratio (PDR) improvements, 10-23% latency reductions, and up to 30% higher node retention compared to state-of-the-art baselines. The system exhibits linear scalability to 500+ nodes and maintains real-time reconfiguration overhead below 3%. APC-RLNC establishes adaptive clustering as a foundational primitive for AI-native 6G wireless systems.

For matrices drawn from the standard Gaussian orthogonal ensemble (GOE) and Gaussian unitary ensemble (GUE), we prove that the normalized zero counting measure of the permanental characteristic polynomial $Per(zI_N-H_N)$ converges almost surely to the standard Wigner semicircle law on $[-2,2]$, rotated by $\pi/2$ onto the imaginary axis. This proves the conjecture proposed by Fyodorov in \cite{Fyodorov2006}.

The purpose of the paper is to place Whiting's classical growing mode stability argument, extended to real frequencies by Shlapentokh-Rothman for the scalar wave equation and by Andersson, Ma, Paganini and Whiting in general, in the framework of classical PDE theory. The key steps are: a description of the dual or adjoint modes, a singular phase space pairing argument which is technically executed via the Fourier transform, followed by a standard unique continuation result. One part of our description of the dual modes connects them directly to the standard mode solutions which are smooth over the event horizon. In the zero spin case, there is a geometric interpretation of this: quasinormal modes (for non-zero real frequencies) which are smooth across the future event horizon can be extended as distributional QNM solutions by 0 through the past event horizon. We also describe the behavior of mode solutions at the bifurcate sphere.

Sparse domination is a central tool in modern harmonic analysis, offering a unified approach to weighted inequalities for Calderón–Zygmund operators and related operators such as commutator operators, rough singular integrals, square functions etc. In this expository note, we briefly survey the main ideas behind sparse bounds on $L^p$ and weighted $L^p$ spaces, and discuss boundedness results for sparse operators on Morrey and generalized Morrey spaces. We establish a model result on the boundedness of sparse operators on Komori–Shirai type weighted Morrey spaces $L^{p,\kappa}(w)$, $w\in A_p$. These bounds offer a simpler framework for deriving Morrey-space estimates for Calderón–Zygmund operators and related operators via sparse domination. We conclude with remarks about extensions and connections to other Morrey space settings.

We show that every continuous and isometry invariant valuation on spherical polytopes is a linear combination of the spherical intrinsic volumes. The proof relies on a weak differentiability property satisfied by valuations on polytopes in $\mathbb{R}^n$ with a natural smoothness property with respect to the action of the affine group. This enables us to transfer several results established by Alesker for quasi-smooth valuations to the polytopal setting, and to reduce the problem to the translation invariant case of measurable valuations on polytopes.

Classical Nesterov acceleration employs different choices of damping and inertial parameters in the convex and strongly convex settings, both for continuous-time dynamics and for discrete algorithms. When the strong convexity parameter is small, directly using the strongly convex damping or inertial coefficient may lead to slower early-stage convergence than the corresponding convex choice, despite its favorable asymptotic exponential or linear rate. We develop a unified continuous-discrete framework that encompasses both classical regimes and provides systematic transitions between them. The resulting coefficient families retain the accelerated convex behavior at early stages while attaining the strongly convex asymptotic rate. The continuous-time dynamics arise from a two-state coupling and are analyzed within a unified Lyapunov framework that yields simultaneous $\mathcal{O}(1/t^2)$ and exponential convergence estimates, thereby recovering the classical convex and strongly convex rates. We further derive two classes of accelerated forward-backward algorithms by discretizing the proposed dynamics and establish convergence estimates covering the convex, strongly convex, and intermediate regimes. The framework recovers the classical Nesterov inertial coefficients and generates hyperbolic, exponential, algebraic, and polynomial transition families. Numerical experiments demonstrate the effectiveness of the proposed methods when the strong convexity parameter is small.

In this paper, we resolve two conjectures posed by Yu and Tan in 2020 (in two separate papers published in the IEEE Trans. Inf. Theory). Specifically, we establish that: 1) the exact common information for a pair of $\rho$-correlated Gaussian sources is given by the conjectured expression $\frac{1}{2}\log\frac{1+\rho}{1-\rho}+\frac{\rho}{1+\rho}$; and 2) the admissible region for the shared randomness rate and the communication rate in exact channel synthesis is exactly the conjectured one. These results yield two important consequences. First, for any $\rho>0$, the exact common information of a correlated Gaussian pair strictly exceeds Wyner's common information. Second, for $\rho>0$, the exact channel synthesis of such a pair requires strictly higher rates than the total-variation version. The proof combines an exact optimal-transport representation of the worst-case Gaussian cross-entropy, Fathi's Gaussian transport inequality, and a determinant inequality arising from the covariance structure of the conditional means.

We compute the local integrals appearing in the refined Gan–Gross–Prasad conjecture for Fourier–Jacobi periods of $\mathrm{Sp}_4$ in new ramified cases and use this to formulate an explicit conjectural identity relating Petersson norms of degree 2 Siegel cusp forms and associated half-integral weight forms. We note consequences of our identity for the growth of Petersson norms, the size of Fourier coefficients, and non-vanishing of central $L$-values.

We study the connectedness and enumeration of reconfiguration graphs of valid triangulations of convex polygons whose vertices are cyclically colored with $j \ge 3$ colors, where every triangle has vertices of three pairwise distinct colors. For $j = 3$, we settle a conjectural expectation of Acharya, Mütze, and Verciani: we prove that the twist graph $\mathcal{H}_{3k+2}$ is connected for every $k \ge 4$, whereas $\mathcal{H}_8$ and $\mathcal{H}_{11}$ are disconnected. Using a colored root-edge decomposition that induces Cartesian products in the state space, we obtain coupled recurrences for $T(3k)$ and $T(3k+2)$. The corresponding generating functions reduce to the equation $U(x) = 1 + xU(x)^4$, and the difference between the two consecutive families is given by the Raney number $T(3k+3) - T(3k+2) = R_{4,5}(k-1)$. For $j \ge 4$, reconfiguration is performed by validity-preserving diagonal flips. We extend the root-edge decomposition to all admissible classes $N \not\equiv 1 \pmod{j}$, obtaining, for each fixed $j$, a finite algebraic system of functional equations. We further prove that the flip graph $\mathcal{G}_N^{(j)}$ is connected whenever valid triangulations exist. Thus, the root-edge decomposition provides a unified structural framework for the enumeration and reconfiguration of cyclically colored triangulations.

Let $(R,\mathfrak{m})$ be a one-dimensional Cohen-Macaulay local ring. In this paper, we find the reduction number $r_{z}(\mathfrak{m}^d)$ of $\mathfrak{m}^d$ with respect to a witness $z\in \mathfrak{m}^d \setminus \mathfrak{m}^{d+1}$ to the generalized Loewy length $\text{g}\ell\ell(R)$ for several infinite families of hypersurfaces $\left\{(R,\mathfrak{m}) \right\}$. For every principal reduction $w$ of $\mathfrak{m}^d$, we have $\text{g}\ell\ell(R) \leq d(r_{w}(\mathfrak{m}^d)+1)$. We give examples of families $\left\{ (R,\mathfrak{m}) \right\}$ such that $d(r_{z}(\mathfrak{m}^d)+1)-\text{g}\ell\ell(R)=0$ and $d(r_{z}(\mathfrak{m}^d)+1)-\text{g}\ell\ell(R)=1$. In particular, we prove that for every prime $p$, there are local hypersurfaces $R$ over $\mathbb{F}_p$ such that $d(r_{z}(\mathfrak{m}^d)+1)-\text{g}\ell\ell(R)=0$ and $d(r_{z}(\mathfrak{m}^d)+1)-\text{g}\ell\ell(R)=1.$ We also prove that the difference $d(r_{z}(\mathfrak{m}^d)+1)-\text{g}\ell\ell(R)$ can vary independently of $\text{g}\ell\ell(R)-e(R)$.

For surfaces of genus $g \geq 3$ with at most one boundary component, we give explicit generating sets for the kernel of the Birman-Craggs-Johnson homomorphism and the commutator subgroup of the Torelli group. As an application, we give a new proof of Johnson's calculation of the abelianization of the Torelli group.