Papers

For graphs $F$, $G$, and $H$, we write $F \to (G, H)$ if every red/blue edge-coloring of $F$ contains either a red copy of $G$ or a blue copy of $H$. The Ramsey number $R(G, H)$ is the smallest integer $N$ such that $K_N \to (G, H)$. Let $H := K_k + nK_h$ be the generalized book graph, and let $G := K_{p+1}(a_1, a_2, \dots, a_{p+1})$ be a complete $(p+1)$-partite graph satisfying $a_1 = 1$, $a_2 \mid (nh-1)$, and $a_i \le a_{i+1}$. In this paper, avoiding the use of Szemerédi's regularity lemma, we prove that for any fixed $h, p \ge 1$, $k \ge 2$, and sufficiently large $n$, $K_{p(nh + a_2k - 1) + 1} \setminus K_{1, nh - a_2(h-2) - 1}\to(G, H).$ This result yields the star-critical Ramsey number $r_*(G, H) = (p-1)(nh + a_2k - 1) + a_2(k + h - 2) + 1.$

We study distribution testing with access to non-adaptive conditional samples. Specifically, we give tight bounds for equivalence testing, determining whether two unknown distributions are equal to or $\varepsilon$-far from each other in total variation distance. Our algorithm and lower bound show that $\tilde \Theta\left(\frac{\log n}{\varepsilon^2}\right)$ queries are necessary and sufficient for this problem. These results demonstrate that the complexity of uniformity, identity, and equivalence testing with non-adaptive conditional samples are all $\tilde \Theta(\log n)$.

Consider a minimal and free topological dynamical system $(X, \Gamma)$, where $\Gamma$ is a discrete amenable group. Assume $(X, \Gamma)$ has the uniform Rokhlin property (URP) and the Cuntz comparison of open sets (COS), then it is shown that $$\mathrm{rc}(\mathrm{C}(X) \rtimes \Gamma) = \frac{1}{2} \mathrm{mdim}(X, \Gamma).$$ The same statement also holds for unital simple AH algebras with diagonal maps. The arguments in both cases are based on the recent preprint of OpenAI dealing with $\mathbb Z$-actions and the topological ideas used in that paper.

Theories of how the brain processes information and returns intelligent outputs are numerous and often difficult to conclusively test. Here, we consider an approach grounded in informatics and algorithmic thermodynamics for how neural systems respond to different patterns of information in different ways, with a focus on the relationship between the information entropy of a signal and a new proposed quantity we term information enthalpy, which represents the internal resource of structured information within a system available to be used for predictions and other forms of work. To evaluate the capacity of the external signals to increase information enthalpy in the system, we introduce the information enthalpy potential (IEP) as a defined metric. We propose and implement a method for quantifying the amount of potential information enthalpy - the IEP - in a given signal across a range of example signals. We offer a conjecture of how information enthalpy may be treated by neural systems within a biological framework before showing how it may integrate and offer falsifiability to existing approaches such as the Free Energy Principle. This framework is also positioned in light of the role of neural criticality. Finally, we postulate a testable and falsifiable framework where the distinct mechanisms within each driver interact through a described n-body-inspired model to govern the "motion" of a neural system through a high dimensional state-space. Through these processes, it is proposed that these features form the most fundamental basis of the neural drivers that give rise to the complexity of adaptive behaviors that are commonly called: intelligence.

Anchored Langevin dynamics (ALD) is useful for non-smooth sampling where the density of the target distribution is possibly non-differentiable and heavy-tailed; reflected anchored Langevin dynamics (RALD) can sample possibly non-differentiable target density on a constrained domain. In this paper, we propose and study non-reversible anchored Langevin dynamics (NALD) for sampling possibly non-differentiable and heavy-tailed target density in the Euclidean space and the non-reversible reflected anchored Langevin dynamics (NRALD) for sampling possibly non-differentiable target density in the constrained space. Our construction adds a circulation drift generated by a possibly state-dependent divergence-free skew-symmetric matrix field and a stream potential. It preserves the target distribution without requiring derivatives of target density, admits a random-time-change representation, and applies both on the whole Euclidean space and on bounded domains with normal reflection. By breaking reversibility, we show that NALD and NRALD can converge to their target distributions faster than their reversible counterparts via finite-time non-asymptotic convergence analysis, a large deviations analysis and asymptotic variance reduction. Numerical experiments demonstrate the efficiency of the proposed algorithms.

We prove that topological conjugacy of elements of the Brin–Thompson group $nV$ is undecidable for every $n\geq2$. From a Turing machine $T$, the reduction produces two elements of $nV$, which are conjugate by an involution in $nV$ if $T$ halts on the empty tape. If $T$ does not halt on the empty tape, the topological dynamics of the two group elements are distinguished by their visits to minimal subsystems of the set of (germ-)aperiodic points. In these groups, and any finitely-generated supergroups inside the homeomorphism group of Cantor space, we also obtain undecidability of the Whitehead problem (the problem of being in the same automorphism orbit), and undecidability of group-theoretic conjugacy.

Operator learning aims to learn mappings between function spaces and has been widely used for PDE-related problems. This raises two fundamental questions. First, is there a connection between standard numerical methods and neural operators, given that both are designed to approximate PDE solution operators? Second, when a neural operator is trained on numerically generated data, does it approximate the underlying infinite dimension operator, the training data themselves, or the discretized numerical scheme used to generate them? A key hypothesis in the paper is that the trained neural operator may mimic the discretized numerical schemes that generate the training data. The hypothesis has been partially and preliminary verified by the linear approximation principle proposed in Chen Chen 1995 paper and the following scaling law papers. However, neural operator approximation from the perspective of numerical discretization remains unexplored. In this work, we consider a class of PDEs and derive neural operator architectures from the numerical schemes. Firstly, the numerical basis functions, such as finite element basis functions, can be approximated or directly constructed by learnable neural networks. Then the finite difference schemes for temporal discretization can be unrolled into iterative neural architectures that generate the coefficients of the basis of the resulting neural operator. By leveraging numerical analysis, we establish convergence error estimates for these numerically derived neural operators. Notably, the network size depends polynomial on the approximation error in $L_2$, improving the nested exponential dependence derived in the general neural operator approximation and aligning with the numerical observations. Finally, numerical experiments demonstrate reduced errors and improved parameter efficiency.

Let $M$ be a closed, smooth or topological $n$-manifold, with $n \geq 4$. We construct a homotopy highly cartesian square relating the space ${\mathcal E}(M (\ell))$ of homotopy self-equivalences of $M$ (in a suitable range) over the Postnikov $\ell$-sections of its stable normal microbundle, and an $(\infty + n)$-fold loop space representing an associated (normal) bordism theory. This implies the existence of braids of interlocking exact sequences involving the homotopy groups of ${\mathcal E}(M(\ell))$ and certain Lashof bordism groups, leading to a conceptual explanation and broad generalization of earlier work of Hambleton–Kreck for closed, oriented $4$-manifolds.

Let a finite group act faithfully by symplectic automorphisms on a manifold of $K3^{[n]}$ type for $n\geq2$. We prove Mongardi's conjecture that the strict inequality $\mathrm{rk}(S)+\ell(A_S)<24$ holds for its associated Leech coinvariant lattice $S$. The converse was already proved independently by Huybrechts and Mongardi: any Leech coinvariant lattice $S$ satisfying the strict inequality and of rank at most $20$ is realized by such an action for some $n\geq2$. The key idea is to consider the minimum norms of vectors representing discriminant classes of $S$. The absence of numerical walls gives lower bounds for certain classes, while the structure of Leech lattice implies upper bounds for all discriminant classes. Comparing these bounds reduces the proof to finitely many values of $n$. We exclude the remaining cases using further lattice arguments, together with Höhn–Mason's classification of fixed-point sublattices of Leech lattice, thereby proving Mongardi's conjecture.

We establish a quantum super Schur–Weyl duality for the mirabolic Hecke algebra \(H_n^{\mathrm{mir}}(q)\) with \(q\) indeterminate and determine the corresponding bimodule decomposition. The centralizer of the mirabolic action is shown as a direct sum of homogeneous quantum Schur superalgebras. We describe the tensor-space annihilator and the resulting faithful quotient. When nonzero, the annihilator is generated by an explicit spectral idempotent attached to the smallest rectangle excluded by the hook condition. From the quantum super Schur–Weyl duality, we derive a super Frobenius formula for mirabolic Hecke algebras. This formula then yields a Murnaghan–Nakayama rule and Regev-type formulas for the irreducible characters. We also construct a super mirabolic RSK bijection from words in a super alphabet with an additional even letter to hook semistandard insertion tableaux together with recording pairs that distinguish the positions of the additional letter. This correspondence yields a Roichman formula for irreducible characters of \(H_n^{\mathrm{mir}}(q)\). Finally, we establish a parameter-inversion isomorphism between \(H_n^{\mathrm{mir}}(q^{-1})\) and the \(q\)-rook monoid algebra \(R_n(q)\) and use it to transport the duality, annihilators, and character formulas in both directions. The transported tensor decomposition provides a representation-theoretic interpretation of hook and two-row character sums for \(R_n(q)\).

We establish a uniform small cancellation theorem for non-elementary torsion-free hyperbolic groups: given any prescribed small cancellation constant, finitely many fixed words suffice to produce a small cancellation element from every pair of noncommuting elements. As an application, we prove that the first-order theory of every non-elementary torsion-free hyperbolic group does not admit a model companion.

We prove that almost all finite groups are $2$-groups of nilpotency class two, when groups of order at most $x$ are counted up to isomorphism and $x\to\infty$. More precisely, uniformly for $2^m\leq x<2^{m+1}$, all but an $O(2^{-cm^2})$ proportion have order $2^m$ and class two, for some absolute constant $c>0$. For each fixed prime $p$, we give a formula for the number of groups of order $p^m$, with relative error $O(p^{-m/3})$ as $m\to\infty$. This refines the classical logarithmic estimates for the number of $p$-groups. For every fixed prime $p$ and sufficiently large $m$, we also prove that all but a $p^{-m^2/300}$ proportion of groups of order $p^m$ have central elementary abelian Frattini subgroup. For fixed $p$, the distribution of the minimum number of generators among groups of order $p^m$ is asymptotically supported on one or two adjacent values, with explicit probabilities depending on $m$ modulo three.

Consider a $d$-sphere $\mathbb{S}^d$ and denote by $\Gamma(T\mathbb{S}^d)$ the set of all vector fields on $\mathbb{S}^d$. We study the Navier-Stokes equations (NSE): $$ \partial_t u + \nabla_u u + grad \pi = \mathbf{\Delta} u + \operatorname{div} f(\cdot, t);\, \operatorname{div} u=0,$$ for the vector field $u(\cdot, t)\in \Gamma(T\mathbb{S}^d)$, where $\mathbf{\Delta}$ denotes the Ebin-Marsden Laplace operator defined by $\mathbf{\Delta} u= \operatorname{div} (\nabla u + \nabla u^t)^{\sharp}$, and $\operatorname{div} f(\cdot, t)$ is the periodic external force. We investigate the Navier-Stokes equations in the framework of $L^p$-spaces over vector fields on $\mathbb{S}^d$ and prove the existence and uniqueness of a periodic solution to such equations. Moreover, exploiting the distribution of eingenvalues of Ebin-Marsden Laplace operator we show the existence of an inertial manifold for solutions around that solution.

Denote by $\mathrm{Kl}(a,n)$ the normalized Kloosterman sum modulo $n$. In this paper, we study the moments of $|\mathrm{Kl}(1,n)|$ on average over $n\leqslant X$, and show that \begin{align*} \sum_{n\leqslant X}|\mathrm{Kl}(1,n)|\asymp X(\log X)^{\frac{8}{3\pi}-1}. \end{align*} The lower bound is proven by considering a subfamily of moduli which factorize in a nice way such that vertical Sato–Tate distributions of Kloosterman sums (as variants of Katz) apply. The upper bound is based on a new density version of upper bound sieves, combining with averages of divisor functions in arithmetic progressions and a similar vertical Sato–Tate distribution.

The kernel bandit problem consists of sequentially optimizing an unknown function with noisy feedback, where the function has bounded norm in a given Reproducing Kernel Hilbert Space (RKHS). A central quantity in the regret analysis of kernel bandits is the maximum information gain $\gamma_T$. In particular, the best existing upper bounds scale as $\sqrt{T\gamma_T}$ up to log factors, and nearly-matching lower bounds have been derived for specific kernels such as squared exponential and Matérn. However, lower bounds for general kernels are lacking, thus making it unclear in what generality the upper bounds are near-optimal. In this paper, we establish a general $\Omega(\sqrt{T\gamma_T/\log T})$ minimax regret lower bound for non-constant continuous kernels on compact domains, establishing near-optimality (within log factors) in a very general sense. We show that the log factor appearing in this bound is unavoidable in general, but that it can be removed under certain conditions. Among other things, our findings imply that the minimax-optimal scaling is exactly $\Theta(\sqrt{T\gamma_T})$ (i.e., within constant factors) for the Matérn-$\nu$ kernel with $\nu \in (0,2)$, $\gamma$-exponential kernel with $\gamma \in (0,2)$, and certain piecewise-polynomial kernels.

The Whitehead problem, which asks whether every abelian group $A$ satisfying $\mathrm{Ext}^1(A,\mathbb{Z}) = 0$ is free, is independent of ZFC. However, this problem has an analogue in condensed mathematics that can be answered affirmatively. In this note, we give a new proof that an abelian group $ A $ of size $\kappa$ is free if and only if $\underline{\mathrm{Ext}}^1(\underline{A}, \underline{\mathbb{Z}})(S_\kappa) = 0$, where $ S_\kappa $ is the Stone space of the Boolean completion of $\operatorname{Add}(\omega,\kappa)$.

We give a convergent explicit representation of the Ricci-flat Kähler metrics supplied by the Calabi–Yau theorem in a one-parameter family of classes on a fixed Kummer K3 surface. An explicit radial background has normalized Monge–Ampère residual \(O(a^4)\). Its scalar Green operator \(G_a\) is represented by periodic Ewald kernels, separated radial kernels, finite-dimensional Schur complements, and convergent Neumann series. The coefficients are defined by \(U_{a,1}=-G_af_a\) and \(U_{a,n}=G_a\sum_{j=1}^{n-1}Q_a(U_{a,j},U_{a,n-j})\), where \(Q_a\) is the polarized quadratic Monge–Ampère term. A Green estimate of order \(a^{-1}\) gives a first correction of order \(a^3\) and a convergence parameter of order \(a^2\). A Catalan majorant proves absolute convergence on the entire smooth surface for every sufficiently small fixed \(a\). The sum defines a positive form solving the volume equation; comparison identifies the sum with the smooth normalized Calabi–Yau potential. We also record sufficient analytic conditions for the same recursion on collapsing elliptic K3 surfaces.

We provide an elementary approach to computing the quadratic Gauss sum, emphasizing the method employed to compute the sum corresponding to multiples of 4. The proof method employed does not rely on the law of quadratic reciprocity, but only on exploiting symmetries among the sum's components, together with a recursive approach.

We construct a quasi-Banach function lattice $X$ on $\mathbb R$ whose quasi-norm is order continuous and invariant under every translation. Nevertheless, $X$ contains a bounded sequence with common compact support which is equicontinuous in $X$ and has no norm-convergent subsequence. This provides a counterexample to Conjectures 2.5 and 2.11 proposed by Brudnyi in \cite{Brudnyi2015}.

For $n\ge3$ and $\gamma>-1$, we study normal solutions of \begin{equation*} (-\Delta)^{n/2}u=(n-1)!\,|x|^{n\gamma}e^{nu} \quad\hbox{in }\R^n. \end{equation*} We reformulate this problem on the cylinder via the Emden–Fowler transform. We first establish Kelvin symmetry for normal solutions. We then prove existence and uniqueness, up to dilation, of the radial normal solution and establish its radial nondegeneracy. Next, we carry out a detailed spectral analysis of the linearized operator and construct branches of nonradial normal solutions by the Crandall–Rabinowitz theorem. Finally, we prove that every normal solution is radial when $\gamma>0$ is sufficiently small.

Let $M$ be a nonzero type III von Neumann algebra with separable predual, and let $N$ be a semifinite von Neumann algebra. We prove that $M_*$ is not isomorphic to a subspace of $N_*$. For $1<p<\infty$, $p\ne2$, we prove that $L_p(M)$ is not isomorphic to a complemented subspace of $L_p(N)$. In particular, the Banach isomorphism class of $L_p(M)$ distinguishes type III algebras from semifinite algebras for $1\le p<\infty$, $p\ne2$.

We obtain the necessary and sufficient conditions on input functions for a family of positive exponent power weight $L^1$ Stein-Weiss inequalities. Earlier work by De Nápoli and Picon showed that a certain cocanceling condition was necessary and sufficient in $L^1$, but it was unknown if this condition remained necessary in the positive exponent power weight setting. This present article identifies weaker necessary conditions for positive exponent power weights, some of which were shown to be sufficient for positive, non-integer exponent power weights in the author's earlier work.

Haemers' conjecture for the generalized spectrum asserts that almost all graphs are determined by their generalized spectrum, that is, for almost all graphs $G$, every graph with the same spectrum as $G$ whose complement has the same spectrum as the complement of $G$ is isomorphic to $G$. We prove this conjecture by showing that the random graph $G(n,1/2)$ is determined by its generalized spectrum with probability tending to one. The proof combines a local theory of the denominators of the rational orthogonal matrices that relate generalized cospectral graphs, Fourier analysis of random symmetric matrices over finite rings, Seidel switching, inverse Littlewood–Offord theory over finite fields, rank estimates for random symmetric matrices modulo primes, and a study of the rational factors of the characteristic polynomial of a random symmetric $\pm1$ matrix with zero diagonal.

Given a matrix $A\in M_{k\times N}(\Bbb{F}),$ we examine finite frames $\{f_j\}_{j=1}^N$ with vanishing $A$-moments and investigate their connections with the frame dilation property. We establish several equivalent characterizations for the existence of vanishing $A-$moment dilations, including conditions involving dual frames, operator ranges, and linear independence properties associated with the matrix $A$. In particular, we show that every frame with redundancy $k$ admits a dual frame with vanishing $A$-moments for some totally nonsingular matrix $A$. A key takeaway of this is that by invoking such a dual frame as the encoding frame, any frame with redundancy $k$ (erasure robustness is not needed) allows perfect signal reconstructions for any number of $\ell$-erasures ($\ell \leq k$) which only involves inverting a matrix of small size ($\ell$) instead of inverting the subframe operator which is a matrix of large size ($N-\ell$). A special case of this type of frames are $k$-th vanishing moment frames which have natural connections with graph frames. We characterize the existence of dual frames with $k-th$ vanishing moments through polynomial cancellation identities and operator theoretical conditions. We further discuss some applications to problems in erasure recovery and graph frames. These results provide a unified framework linking vanishing moment conditions, dual frame theory, dilation theory, and graph theoretic constructions of frames.

We study general maximal hard-core models on periodic lattice-type graphs. While for standard models, phase coexistence may arise for high activity, for maximal models this behavior may occur also for sufficiently low activity values. Relying on the concept of volume allocation, we develop a unified set of assumptions that imply the Peierls condition, a convergent cluster expansion for the partition function, and hence the conclusions of Pirogov-Sinai theory for this class of models. We further check these assumptions for a number of examples including standard lattice-type periodic graphs, proving crystallization for both high and low activity. We also derive estimates for the values of activity that bound the phase coexistence regions in the phase diagram, rewriting for this purpose the proof of a technical result in the derivation of Pirogov-Sinai theory.

In this paper, we first classify complete four-dimensional shrinkers under a pinching condition on the $2$nd-Ricci curvature. We then classify closed four-dimensional shrinkers with (weighted) integral parameter-pinching of the half Weyl curvature under several pinching assumptions on the biorthogonal curvature. Finally, we establish a Hitchin-Thorpe type inequality for closed non-trivial four-dimensional shrinkers under some assumptions on the half Weyl curvature and the biorthogonal curvature.

We investigate the consequences of requiring a posterior approximation to preserve a specified downstream decision problem. A target posterior $P$ and loss determine a regret geometry on actions, a baseline approximation $Q_0$ determines the forward-Kullback-Leibler information required to induce action changes, and a restricted approximation family $\mathcal{Q}$ determines which such changes are available. Contracting KL divergence over Bayes-action fibers gives exact distances to decision adequacy and decision failure together with the least-informative posterior deformations that reach either side of the decision boundary. In regular finite-dimensional problems, the target and baseline constructions have quadratic local limits: a target regret Hessian $G$ and a baseline information metric $J_I$ . Their generalized eigenproblem $Gv = \gamma J_Iv$ orders local decision directions by regret consequence per unit information cost and induces a tolerance-dependent effective dimension. For restricted approximation families, the tangent image separates decision coverage from information efficiency: a family may miss consequential decision directions, or it may realize reachable directions only at excess Fisher cost. The resulting framework provides decision-relative criteria for comparing and designing posterior approximation families.

We study the convergence of minimising movement approximations to weak solutions of a fourth-order viscoelastic phase separation system with degenerate concave mobility. The underlying semi-implicit variational scheme combines a dynamic transport distance for the phase-field variable with a weighted $L^2$-distance for a transformed stress variable. We establish subsequential convergence of the scheme and show that the limiting solutions satisfy the energy-dissipation inequality. To this end, we derive a coercivity estimate for the metric slope, which yields the required space-time regularity for the De Giorgi interpolant, and construct constitutive fluxes whose dissipation is controlled by the squared metric slope. These estimates allow us to pass to the limit in the discrete energy-dissipation inequality. The weak formulation of the evolution equation for the phase-field variable is derived using the flow interchange technique. Under additional assumptions on the data, we further show that the limiting solutions satisfy an entropy-dissipation inequality

Let $K$ be a function field over an $F$-finite field $k$ of characteristic $p > 0$ and let $\nu$ be a valuation of $K/k$. We show $\nu$ is an Abhyankar valuation of $K/k$ if and only if the corresponding valuation ring is Frobenius split. The proof proceeds by analyzing when ideals of a valuation ring are uniformly $F$-compatible and establishing a general ramification-theoretic characterization of Frobenius split valuation rings of $F$-finite fields. In addition, when the ground field $k$ is not $F$-finite, we show that the equivalence between Frobenius splitting and the Abhyankar property can fail by constructing an example of an excellent Frobenius split DVR of a function field whose corresponding valuation is not divisorial. Our methods do not rely on local uniformization results.

A long-standing hypothesis in artificial intelligence and neuroscience posits that intelligence is closely related to compression: the ability to compress information efficiently intuitively reflects capacities associated with intelligence and learning. Indeed, recent experimental works verify this intuition by showing connections between the capabilities of large language models (LLMs) and their ability as compressors: for instance, Deletang et al. (ICLR'24) demonstrate that LLMs can be used as powerful compressors, and Huang et al. (COLM'24) show that the compression ability of LLMs is highly correlated with their performance on benchmarks for knowledge and reasoning. In this work, we initiate a theoretical study of this connection. Our main result is that LLMs (formally modeled as next-token predictors) are equivalent to monotone (a.k.a. order-preserving) compression algorithms—namely, compression algorithms where the encoding process preserves the ordering of the inputs—in the sense that the one can be constructed from the other while preserving the same error up to an additive gap of 2. We next show that the monotonicity is required for this equivalence to hold if and only if cryptographic (infinitely-often) one-way functions exist. As a direct corollary, we get a cryptographic result of independent interest: the notion of next-bit pseudoentropy (a computational analogue of entropy) of a distribution is equivalent to monotone incompressibility of the distribution. (Previously, it was only known (Haitner et al., ITCS'23) that incompressibility implies next-bit pseudoentropy.)