Papers

A group has Property (NL) – which stands for “no loxodromics" – if no element of the group acts loxodromically on any hyperbolic space. In this brief note, we provide a complete characterization of which Coxeter groups have Property (NL).

Given a quiver Q with gauge dimension $\bf v$ and framing dimension $\bf w$, one can define the extended quiver variety $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$, which is a smooth family of deformations of the Nakajima quiver variety $\mathcal M(\mathbf v,\mathbf w)$. In this paper we discuss two vertex algebras which chiralize the geometry $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$. We construct a sheaf of $\hbar$-adic vertex superalgebras $\mathscr D^{\mathrm{ch}}_{\widetilde{\mathcal M}(\mathbf v,\mathbf w),\hbar}$ on $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$ which quantizes the jet bundle of $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$, and define a vertex algebra $\mathsf D^{\mathrm{ch}}(\widetilde{\mathcal M}(\mathbf v,\mathbf w))$ to be the $\hbar=1$ specialization of the $\mathbb C^{\times}$-finite part of the vector space of global sections $\Gamma(\widetilde{\mathcal M}(\mathbf v,\mathbf w), \mathscr D^{\mathrm{ch}}_{\widetilde{\mathcal M}(\mathbf v,\mathbf w),\hbar})$. We define another vertex superalgebra $\mathcal V(\mathbf v,\mathbf w)$ by BRST reduction of the tensor product of the $\beta\gamma bc$-system and Heisenberg VOA associated to the quiver Q, and show that there exists a natural vertex superalgebra map from $\mathcal V(\mathbf v,\mathbf w)$ to $\mathsf D^{\mathrm{ch}}(\widetilde{\mathcal M}(\mathbf v,\mathbf w))$. Under certain technical assumptions, we prove that the negative degree BRST cohomologies of the tensor product of $\beta\gamma bc$-systems and Heisenberg VOA associated to the quiver Q are zero, and under stronger assumptions, that the aforementioned vertex superalgebra map is injective. Physically, the vertex superalgebra $\mathcal V(\mathbf v,\mathbf w)$ is closely related to the boundary VOA of the H-twisted 3D $\mathcal N=4$ quiver gauge theory associated to the quiver Q with gauge and framing dimension vectors $\bf v$ and $\bf w$.

Chaotic dynamics play an important role in a number of physical systems. One of the qualitative hallmarks of this behavior is the appearance of a sufficiently "complex" spectrum of energy levels. This also makes it challenging to directly verify the onset of chaos in interacting quantum field theories. We present a class of 4D superconformal field theories (SCFTs) given by orbifolds of 4D $\mathcal{N} = 4$ Super Yang–Mills theory in which operator mixing in a controlled subsector is described by an effective spin chain in one spatial dimension with nearest neighbor interactions tuned by the marginal couplings of the SCFT. Tuning the marginal couplings results in a chaotic spectrum, while generically the spin chain exhibits Anderson localization. We diagnose the onset of chaos by analyzing the statistical distribution of eigenvalues of the dilatation operator, in particular properties such as eigenvalue level repulsion, spectral rigidity, and the spectral form factor. We also show that other diagnostics such as Krylov complexity sometimes do not faithfully capture this information. This structure defines a chaotic billiard in the target space of the stringy realization. We also comment on the large $N$ holographic dual description, where the controlled single spin chain approximation must be supplemented by multi-trace dynamics, i.e., the splitting and joining of multiple spin chains.

In arXiv:2604.19861, we gave general criteria for when one zero-mean Gaussian state can be excited out of another in a (generalized) free field theory. Here we extend this analysis to the case of nonzero mean, i.e., to Gaussian states with vacuum expectation values (VEVs). We prove that excitability is possible exactly when (i) the connected two-point functions satisfy criteria like those in arXiv:2604.19861, and (ii) the difference of the VEVs is bounded relative to the two-point functions. As an application, we give an explicit computation showing that in anti-de Sitter spacetime, a VEV shift can be excited from the Klein-Gordon vacuum if and only if its boundary extrapolation can be excited from the vacuum of the dual conformal field theory.

Quantum tomography and quantum retrodiction are traditionally viewed as separate inference tasks. We show that the two are manifestations of the same underlying principle. We prove that the Petz recovery map of a measurement channel is a gradient of the log-likelihood used in maximum-likelihood tomography, and we derive its extension to arbitrary quantum channels. Consequently, for measurement channels, repeated applications of the Petz map monotonically increase the likelihood, naturally yielding a method for quantum tomography.

We study the number $\Nu(\Per(G_{\lambda}))$ of distinct monomials with nonzero coefficients in the group permanent of an abelian $p$-group $G_\lambda$ associated with a partition $\lambda$ of a positive integer $N$. First, we derive an explicit formula for $\Nu(\Per(G_{\lambda}))$ in terms of the partial column sums of the Young diagram of $\lambda$. Next, we show that the relative order of the values $\Nu(\Per(G_{\lambda}))$ is determined by a lexicographic comparison of the conjugate Young diagrams. Finally, we investigate congruence properties of $\Nu(\Per(G_{\lambda}))$ for abelian $p$-groups and establish a criterion involving Wolstenholme primes.

It is shown that there exists $c > 0$ so that the Cayley graph over any finite abelian group $Z$ generated by $c \log |Z|$ random elements is properly 3-colorable with high probability (as $|Z| \to \infty$). This is asymptotically tight and improves the best-known bound due to Alon of $\frac{1}{4}\log \log |Z|$ elements. It also settles the abelian case of Alon's suggestion that a bound of $c \log |G|$ may hold for any finite solvable group $G$.

CABAC, the entropy coder of H.264/AVC and the basis for HEVC and VVC, decomposes multi-symbol values into bins via a binarization scheme before a binary arithmetic coder. H.264 uses Truncated Unary plus k-th order Exp-Golomb (UEG); alternatives include canonical Huffman and the entropy-conserving binarization (ECB), which provably preserves entropy mapping m-ary data to m-1 binary strings but has not been evaluated inside a production binary arithmetic coder. We integrate ECB into a from-scratch CABAC implementation alongside UEG, single-context Huffman, and a Huffman variant with per-bin-position contexts (HuffmanPos), all sharing one M-coder backend. We benchmark all four on synthetic sources, DCT residuals from a procedural image, and the full 24-image Kodak suite (2,480 round-trip trials, bit-exact verified). On the procedural image, a sparsity-driven crossover at Q=8 lets ECB overtake single-context Huffman, reaching 27 percentage points below at Q=32. On Kodak the crossover shifts below the tested range and ECB beats single-context Huffman at every Q, the gap growing from 0.031 to 0.113 bits per symbol. HuffmanPos, sharing Huffman's codewords but allocating one context per bin position, beats ECB on 12 of 15 source cells and loses by at most 0.56 percentage points on the other three, despite the same per-symbol bin count as single-context Huffman. This isolates the dominant mechanism: at low source entropy the rate gap is driven primarily by context allocation over the bin stream, not the binarization's per-symbol bin count. ECB's rate efficiency costs 7 to 10x in decoder latency on large alphabets, traced to an O(N*m) decoder; we sketch an interleaved single-pass variant that would close this gap. Code, benchmarks, and raw data are open source.

We study the trinomial equation $x^n +px +q =0$. Here $p$ and $q$ are both real and nonzero. For $n\ge3$, expressions for the roots have been published as hypergeometric series in powers of the parameter $q^{n-1}/p^n$. For the special case of the cubic ($n=3$), we employ Kummer's identities to derive alternative series solutions in powers of the discriminant $D$, and also series in powers of $1/D$. We next derive new series, in powers of $D$ and also in powers of $1/D$, for all $n\ge 3$. The resulting series suggest identities analogous to Kummer's identities, for higher order hypergeometric series.

Let $S_h^\varphi(x)$ denote the number of integers $n\le x$ for which $\varphi(n)=\varphi(n+h)$. For the unit shift, we prove $S_1^\varphi(x)\ll x\exp{-(1/2-o(1))\sqrt{\log x,\log_2 x}}$. More generally, put $A=\log_3 x+\log_4 x-\log 2$, $G=\sqrt{\log x,A}$, and $V=\log x/G$. For every fixed integer $J\ge 1$, uniformly for $1\le h\le \exp{G/\sqrt{J}}$, we obtain $S_h^\varphi(x)=D_{h,>Y_J}^\varphi(x)+O_J(x\exp{-\sqrt{J},G+o_J(V)})$, where $Y_J=\exp{\sqrt{J},G}$. Here $D_{h,>Y_J}^\varphi(x)$ is the above-cutoff part of the classical Graham–Holt–Pomerance same-support family; it is empty for odd $h$. A moving choice $J\asymp \log_2 x/\log_3 x$ gives the unit-shift estimate and an analogous decomposition for a uniform range of shifts. The proof combines the smooth-totient theorem of Banks–Friedlander–Pomerance–Shparlinski with labelled supplier systems, a shifted divisor convolution, and an injective encoding of large supplier products into weighted friable tuples. The results of this paper have been formally verified in Lean.

AdamW is the de facto optimizer for training large language models (LLMs), yet the theory behind it still lives mostly in finite-variance regimes. This is increasingly unsatisfying, as empirical evidence indicates that stochastic gradient noise in LLM pretraining is typically heavy-tailed. Recent work shows that sign-based optimizers such as Lion and Muon achieve sharp heavy-tailed rates, and that AdaGrad can also converge under heavy-tailed noise. However, no rigorous convergence theory for AdamW has yet been established in this regime. Can AdamW converge under the same heavy-tailed assumptions, or does its second-moment accumulator create a genuine obstruction? We formulate this as an open problem, prove a positive weighted-metric benchmark, and give a corridor lower-bound mechanism showing how denominator memory can hide large gradients.

Following the idea of Jungsoo Kang and Jun Zhang, we prove the strong Arnol'd chord conjecture for the boundary of a uniformly convex domain in $\mathbb{R}^{4}$, using an ellipsoid embedding construction due to Oliver Edtmair. We prove a general structural result for Legendrians $L$ which are eventually equivariantly essential (E3), in the sense that the $k$th Gutt-Hutchings capacity $c_{k}(D^{*}TL)$ is infinite for $k$ large enough. We show that any E3 Legendrian in the boundary of a Liouville domain $\Omega$ bounds a chord of length at most $\liminf c_{k}(\Omega)/k$.

For fixed $k\ge 2$, let $g_k(n)$ be the greatest excess $a_1+\cdots+a_k-n$ among positive integers $a_i$ satisfying $a_1!\cdots a_k!\mid n!$. We prove that, for every $\varepsilon>0$, all but $o(x)$ integers $n\le x$ satisfy \[ g_k(n)\ge \left(\frac{3(k-1)}{\log 12}-\varepsilon\right)\log n. \] We also prove, as $n\to\infty$, the pointwise upper bound \[ g_k(n)\le (k-1)\log_2 n+\log_2\log n+O_k(1). \] The central analytic input is uniform phase separation for one or two frequencies on fixed-prime $S$-unit progressions, deduced directly from the finite exceptional-subspace alternative of Drmota and Spiegelhofer, and the resulting uniform digit-sum normal-order theorem. A mixed $2$–$3$ representation, quantitative two-block estimates, and a large-prime Kummer sieve produce the stated coefficient.

We resolve Erdős Problem 550, originally asked as question (2) of Erdős, Faudree, Rousseau, and Schelp. Precisely, for fixed integers $k\geq 2$ and $1\leq m_1\leq \cdots \leq m_k$, we prove that, for every sufficiently large $n$ and every $n$-vertex tree $T$, $R(T,K_{m_1,\ldots,m_k}) \leq (k-1)(R(T,K_{m_1,m_2})-1)+m_1$. The proof combines an off-Turán tree-embedding theorem, proved by regularity and whole-edge allocation, with a compactness theorem for bounded-rank hypergraph obstructions. The full and unconditional proof has been formally verified in Lean.

We present an efficient protocol for learning an unknown $k$-local Lindblad generator on $n$ qubits using only product-state preparations, short-time evolution, and single-qubit Pauli measurements, without prior knowledge of the interaction structure. For fixed $k$ and a supplied weighted interaction-strength bound $\bar\alpha$, the protocol estimates all Hamiltonian and dissipative Pauli–GKSL coefficients with total coefficient error at each site at most $\varepsilon$ with probability at least $1-\delta$ using $\widetilde{\mathcal O}_k(\bar\alpha^2n^{2k-2}\varepsilon^{-2}\log(n/\delta))$ samples and polylogarithmically many distinct evolution times. This guarantee requires no degree, sparsity, or tail promise. A convex optimization converts these estimates into a valid $k$-local Lindblad generator with diamond-norm error at most $\varepsilon$ using $\widetilde{\mathcal O}_k(\bar\alpha^2n^{2k}\varepsilon^{-2}\log(n/\delta))$ samples and polynomial-time classical postprocessing. With a supplied local effective-sparsity bound $r$ and a sufficiently small sitewise subthreshold tail $\mathcal O_k(\varepsilon)$, coefficient recovery requires only $\widetilde{\mathcal O}_k(\bar\alpha^2r^2\varepsilon^{-2}\log(n/\delta))$ samples, without a supplied threshold or coefficient locations and without a coefficient-gap assumption. In particular, exactly sparse bounded-degree models have logarithmic sample dependence on $n$. We also provide guard-band support recovery and complementary supplied-candidate guarantees, extend the guarantees to model misspecification, and prove complementary sample-complexity lower bounds.

In this paper, we study half-space observables in the massive Euclidean $\phi^4_4$ theory. We prove that the renormalized correlators can be multiplied by half-space truncations without requiring any additional renormalization. More precisely, products with smooth approximations of the half-space indicator converge to well-defined distributional limits, uniformly in the ultraviolet cutoff. The proof relies on a position-space renormalization framework for the Wilson–Polchinski flow equation based on a hierarchy of power-counting spaces adapted to the singularity structure of the correlators. This yields uniform power-counting bounds and Besov–Hölder regularity estimates for the renormalized correlators. As a consequence, the correlators converge as the ultraviolet cutoff tends to infinity, and the convergence takes place in explicitly identified Besov spaces.

We solve fairly explicitly an optimal stopping problem for a Wiener process with unobserved Bernoulli drift, in the presence of a cost on terminal position which is symmetric and increases with distance from the origin, and of a fixed positive cost per unit time \(c > 0\). After filtering, the problem reduces to Markovian optimal stopping with complete observations for the state process “centered” by its starting position $x \in \mathbb R$. However, the solution becomes possible only after foliating by an additional state-parameter \(y \in \mathbb{R}\), representing the displacement from the initial position; this foliation “lifts” the problem from the real line to the plane, solves the augmented problem for each fixed initial position \(x\), characterizes fairly explicitly the optimal stopping region in \((x,y)\)-space, and finally obtains the solution of the original problem by “slicing” along \(y=0\). Following this procedure, we show that, under suitable structural assumptions on the terminal cost, each fixed-\(x\) continuation section is either empty or a single bounded interval, whose endpoints are determined uniquely by a balancing condition; the corresponding value function is then given in semi-explicit form. The two-dimensional continuation region is obtained by gluing these fixed-\(x\) intervals over \(x\); its two free boundaries satisfy natural monotonicity properties and, at regular points, can be described by a coupled system of ordinary differential equations. The resulting description yields a threshold-type solution of the original one-dimensional problem whenever the horizontal slice \(y=0\) enters the two-dimensional continuation region.

Quantum instruments constitute the general description of quantum dynamics, encompassing both quantum measurements and quantum channels as special cases. Consequently, the incompatibility of quantum instruments represents a fundamental manifestation of nonclassicality in quantum theory. Here, we establish the operational significance of this notion by demonstrating communication tasks with classical inputs and outputs that enable the semi-device-independent certification of incompatible quantum instruments. We introduce a class of three-party communication tasks involving a sender, a relayer, and a receiver, and derive the tight upper bounds of the figure of merits of these tasks achievable by all compatible instruments implemented by the relayer. Furthermore, we show that these bounds coincide with the optimal performance attainable in a classical communication subject to the same dimensional constraints. Violation of this bound certifies the incompatibility of the pair of quantum instruments implemented by the relayer. This identifies certification of incompatible instruments as a manifestation of quantum advantage in communication. This certification protocol is genuine as it is able to certify the incompatibility of a pair of instruments where the set of induced measurements and the set of induced channels are compatible and, therefore it does not depend on the incompability of measurements and channels induced by the instruments. Finally, we identify the simplest instance of our communication scenario that enable the certification of incompatible quantum instruments.

We introduce an analytically invertible framework for wavefront construction based on the scattering properties of periodic waveguide networks governed by a gauge-shifted Helmholtz operator. By determining the exact transmission coefficients of the network, we express the lattice reactance as a Fourier expansion whose coefficients are analytically mapped onto the underlying graph architecture, allowing the required bond connections, refractive indices, lengths, and gauge phases to be determined directly from a prescribed target transmission coefficient. In contrast to conventional inverse-design approaches, the present formulation provides a closed-form route from desired wave transmission profiles to physically realisable structures. The framework extends naturally from one-dimensional angular filtering to two-dimensional image synthesis, where arbitrary transmitted intensity patterns are reconstructed through exact spectral control of the network scattering response.

Matrix-aware optimizers such as Muon and Muown have recently shown strong empirical performance for pre-training Transformers. In particular, Muown separates each weight matrix into row magnitudes and an un-normalized direction variable, updating the former with Adam and the latter with Muon. We show that the directional update of Muown is equivalent to a Riemannian step on the normalized directions, while the magnitude of the un-normalized parameterization only modulates the angular step size. This explains the step-size stability of Muown and suggests making the angular step size explicit. The resulting method, AngularMuown, optimizes directly over the normalized directions and uses a schedulable angular multiplier decoupled from the radial magnitude update. AngularMuown improves over Muown and, at the time of writing, a preliminary version is leading the per-optimizer category of the modded nanoGPT speedrunning competition. Further experiments on Qwen2-0.5B, and 1.1B parameter mixture-of-experts models confirm the algorithm scales beyond small models. An implementation of the algorithm is available at https://github.com/fhueb/angular-muown

We solve Kadison's problem on the existence of maximal abelian subalgebras in irreducible subfactors $N \subset M$ when $N$ is AFD or when $N$ is with expectation in $M$. In the latter case, we first show that the relative bicentralizer flow of a type $\mathrm{III}_1$ irreducible subfactor with expectation is always ergodic. This also implies that for every irreducible subfactor with expectation $N \subset M$, there exists an AFD subfactor with expectation $P \subset N$ that is irreducible in $M$.

We study sharp inequalities for ratios of products of principal minors of real positive definite matrices. Our main result gives a closed-form solution to a family of nonconvex optimization problems over the positive definite cone. As a special case, we prove that the infimum of the Ingleton ratio over $4\times 4$ positive definite matrices is $16/27$, confirming a conjecture of Hall and Johnson. We also show that the cone of absolutely bounded ratios of products of principal minors is not polyhedral for $n\ge 4$, and that it is not semialgebraic over $\mathbb{Q}$.

Introduced by Buchweitz, the singularity category of an algebra $A$ measures its homological singularity, vanishing if and only if $A$ has finite global dimension. This notion extends naturally to the context of $k$-categories. In this paper we study the singularity category of a triangular matrix category $\Lambda:=\left[ \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]$. By utilizing the framework of recollements, we provide a characterization of this category, proving that when certain homological conditions are satisfied, there exists an equivalence of singularity categories $D_{sg}(\mathrm{Mod(\Lambda)})\simeq D_{sg}(\mathrm{Mod}(\mathcal{U}))$. This result generalizes the one obtained by Pin Liu and Ming Lu in [16].

Diffusion models are known to exploit unknown low-dimensional structure to accelerate sampling. However, existing convergence theory under low-dimensional data structure has largely focused on update rules with narrowly prescribed coefficient choices. This raises a fundamental question: is adaptation to low-dimensional structure sensitive to the precise choice of update coefficients? In this paper, we show that such adaptation is a robust property of diffusion models. For a broad class of update coefficients, we prove that $\widetilde{O}(k/\varepsilon)$ iterations suffice to generate an $\varepsilon$-accurate sample in total variation (TV) distance, independently of the ambient dimension. Our framework substantially broadens the class of diffusion samplers known to enjoy low dimensional adaptation and applies to several commonly used methods in practice. These results provide a theoretical justification for the empirical effectiveness of diffusion samplers across different coefficient choices when applied to structured, high-dimensional data.

We study moduli spaces of modular vector bundles on projective irreducible holomorphic symplectic manifolds of $K3^{[n]}$-type. Under suitable numerical assumptions, we exhibit connected components of these moduli spaces which are again irreducible holomorphic symplectic manifolds of $K3^{[n]}$-type. Moreover, the corresponding universal families induce derived equivalences with the original manifolds. This produces smooth components of moduli spaces of modular vector bundles on irreducible holomorphic symplectic manifolds of any even dimension.

We prove several novel connections between properties of nonarchimedean groups and fragments of the axiom of choice which hold in their associated permutation model.

Quantum tunneling is expected to provide a computational speedup in quantum computing, a phenomenon that Adiabatic Quantum Optimization (AQO) aims to leverage. While some academic proofs of concept have been studied, such as the "Hamming weight with a spike" (HWS) problem, the algorithmic gains of this effect remain underexplored. In this work we extend the analysis underlying HWS to more general potentials. In the first half of the work, we establish (discrete) log-concavity of the ground state as a key structural property in this context. We devise a framework for establishing log-concavity of the ground state for a large family of discrete, 1-dimensional Schrödinger operators. The family includes convex potentials, but also certain potentials with local minima. In the convex case, this provides a discrete version of a continuous result by Brascamp and Lieb ('76). We demonstrate the utility of our result by establishing new spectral gap bounds, going beyond related results by Jarret and Jordan ('14) for convex potentials. In the second half of the work, we use our results on log-concavity to extend the perturbative analysis of HWS by Reichardt ('04) to the larger family of potentials with log-concave ground state. As a concrete instantiation, we use our result to extend the HWS analysis from a linear potential (which is exactly solvable) to a quadratic potential (which is no longer solvable). Our result strongly suggests the broader applicability of tunneling to convex potentials with spikes

We present a posteriori error bounds for a general class of stabilized finite element approximations of time-dependent mean field games. We first show the equivalence between the norm of the error and the dual norm of the residual in the coupled Hamilton-Jacobi-Bellman and Kolmogorov-Fokker-Planck equations. We then derive a reliable and efficient a posteriori error estimator that is based on residual estimators, along with the temporal jump estimator, and an estimator for the stabilization terms in the numerical discretization. Finally, for stabilizations based on mass-lumping in time and affine-preserving spatial stabilizations, we show that the stabilization estimators can be bounded in terms of the residual and temporal jump estimators, thus yielding an improved reliable, locally computable, and locally efficient estimator.

We consider finite-time singularity formation for classical solutions to the following parameterized nonlinear wave equation: \[ u_{tt}=c(u)^2u_{xx}+\lambda c(u)c'(u)(u_x)^2, \] where \(\lambda\in[0,2]\) is a parameter. In previous works, it is known that finite-time blow-up solutions exist for \(0<\lambda\le1\) and for \(\lambda=2\). The cases \(\lambda=1\) and \(\lambda=2\) are known as the variational wave equation and the \(p\)-system, respectively, and they possess symmetric structures or conservation laws. For \(0<\lambda<1\), the blow-up construction is based on the fact that, after decomposing the wave into two Riemann variables, one component can be kept sufficiently small and hence does not prevent the other component from blowing up. In the present paper, we treat the remaining intermediate case \(1<\lambda<2\). This case is essentially different from the previous one, since both Riemann variables may grow. The key idea is to introduce suitable supremum functions for the Riemann variables and to derive a comparison principle through right Dini derivatives. This allows us to control the relative size of the two components and to obtain a Riccati-type differential inequality for the dominant supremum function.

We prove the Virtual Surjection Conjecture for profinite groups. Namely, given a product of $n$ profinite $\mathrm{FP}_{k}$ groups, a subgroup that virtually surjects onto $k$-tuples must be $\mathrm{FP}_{k}$ as well. We also prove the $n$-$(n+1)$-$(n+2)$ Conjecture for profinite groups, as well as a few other $\mathrm{FP}_{n}$ permanence results for fibre products. Our main tool is a numerical criterion for property $\mathrm{FP}_{n}$ of modules of profinite groups. Our work suggests a new finiteness property to investigate.